Creating Galois Representations#
- ZeroRepresentation(K): FldPad -> GalRep#
Galois representation \(0\) over a \(p\)-adic field \(K\). It is 0-dimensional, and \(0+A=A\), \(0\otimes A=0\) for every Galois representation \(A\).
- Example: Galrep Zerorepresentation (ex-153a0b)#
> K:=pAdicField(3,20); > zero:=ZeroRepresentation(K); > zero; Galois representation 0 with G=C1, I=C1 over Q3[20] > zero + CyclotomicCharacter(K) eq CyclotomicCharacter(K); true > zero*CyclotomicCharacter(K) eq zero; true
- PrincipalCharacter(K): FldPad -> GalRep#
Principal character \(1\) of the absolute Galois group of \(K\), as a Galois representation. It is a 1-dimensional unramified representation, same as
UnramifiedCharacter(K,1). Thus \(1\otimes A=1\) for every Galois representation \(A\).
- Example: Galrep Principalcharacter (ex-48649a)#
Take \(K={\mathbb{Q}}_3\), and \(F/K\) the unramified extension of degree 4, so that \(G=\mathop{\rm Gal}\nolimits(F/K)\cong C_4\). The 4 irreducible representations of \(G\) can be viewed as Galois representations, and the first one of these is the principal character (for any group).
> K:=pAdicField(3,20); > one:=PrincipalCharacter(K); > one; 1-dim trivial Galois representation 1 over Q3[20] > F:=ext<K|4>; > A1,A2,A3,A4:=Explode(GaloisRepresentations(F,K)); > A1 eq one; true
- CyclotomicCharacter(K): FldPad -> GalRep#
Cyclotomic character over \(K\). It is an unramified character (trivial on inertia) and takes the value \(q\), the size of the residue field of \(K\), on any Frobenius element.
- Example: Galrep Cyclotomiccharacter (ex-4a5d9f)#
> K:=pAdicField(3,20); > chi:=CyclotomicCharacter(K); > chi,EulerFactor(chi); 1-dim unramified Galois representation Unr(1/3) over Q3[20] -1/3*x + 1 > chi^3,EulerFactor(chi^3); 1-dim unramified Galois representation Unr(1/27) over Q3[20] -1/27*x + 1
- UnramifiedCharacter(K, c): FldPad, Any -> GalRep#
Galois representation over \(K\) given by an unramified character that sends the arithmetic Frobenius element \(\mathop{\rm Frob}\nolimits_K\mapsto c^{-1}\) (and, so, the geometric Frobenius element \(\mathop{\rm Frob}\nolimits_K^{-1}\mapsto c\).) The parameter \(c\) must be a non-zero complex number.
- Example: Galrep Unramifiedcharacter (ex-6bab13)#
> K:=pAdicField(3,20); > assert UnramifiedCharacter(K,1) eq PrincipalCharacter(K); > assert UnramifiedCharacter(K,1/3) eq CyclotomicCharacter(K); > C<i>:=ComplexField(); > UnramifiedCharacter(K,2+i); 1-dim unramified Galois representation Unr(2+i) over Q3[20]
- UnramifiedRepresentation(K, CharPoly): FldPad, RngUPolElt -> GalRep#
Unique unramified Galois representation \(\rho\) over \(K\) with Euler factor \(\det(1-\mathop{\rm Frob}\nolimits_K^{-1}|\rho)={\rm CharPoly}\).
- Example: Galrep Unramifiedrepresentation (ex-b1231d)#
> K:=pAdicField(37,20); > R<x>:=PolynomialRing(Rationals()); > rho:=UnramifiedRepresentation(K,(1-37*x)*(1-3*x)); > rho; 2-dim unramified Galois representation Unr(1-40*x+111*x^2) over Q37[20] > rho eq CyclotomicCharacter(K)^(-1)+UnramifiedCharacter(K,3); true
- UnramifiedRepresentation({K,dim,}{dimcomputed,CharPoly}): FldPad, RngIntElt, RngIntElt, RngUPolElt -> GalRep#
Unramified Galois representation over \(K\) of dimension dim, with Euler factor CharPoly computed up to and inclusive degree dimcomputed.
- Example: Galrep Notfullycomputed (ex-f653e8)#
Consider the hyperelliptic curve \(C: y^2=x^5+x+1\) over the \(p\)-adic field \({\mathbb{Q}}_{10007}\).
> _<x>:=PolynomialRing(Rationals()); > p:=10007; > K:=pAdicField(p,20); > _<X>:=PolynomialRing(K); > C:=HyperellipticCurve(X^5+X+1); C; Hyperelliptic Curve defined by y^2 = x^5 + O(10007^20)*x^4 + O(10007^20)*x^3 + O(10007^20)*x^2 + x + 1 + O(10007^20) over pAdicField(10007, 20)
The Galois representation \(A\) associated to \(H^1(C)\) is unramified, of dimension 4, and could be defined by
UnramifiedRepresentation(K,1-ap*x+bp*x^2-p*ap+p^2);if we find \(a_p\) and \(b_p\) by counting points of \(C\) over \({\mathbb{F}}_p\) and \({\mathbb{F}}_{p^2}\). The coefficient \(a_p\) can be computed very quickly:
> k:=ResidueClassField(Integers(K)); > _<X>:=PolynomialRing(k); > Ck:=HyperellipticCurve(X^5+X+1); > ap:=p+1-#Ck; ap; -21
However, \(b_p\) would take a long time. If we are only interested in working with \(A\) up to degree 1 (e.g. to compute \(L\)-series of \(C/{\mathbb{Q}}\) with \(<10^8\) terms), there is no reason to compute it. Instead, we can define an unramified Galois representation of degree 4, which is known to be computed only up to degree 1:
> A:=UnramifiedRepresentation(K,4,1,1-ap*x); > A; 4-dim unramified Galois representation Unr(1+21*x+O(x^2)) over Q10007[20]
One can still take direct sums, and tensor products of such representations with (possibly ramified) Galois representations, and the Euler factors will still be correct up to degree 1:
> A*A; 16-dim unramified Galois representation Unr(1-441*x+O(x^2)) over Q10007[20] > EulerFactor(A*A+1/CyclotomicCharacter(K)); -10448*x + 1
- SP(K, n): FldPad, RngIntElt -> GalRep#
The \(n\)-dimensional indecomposable Galois representation
SP(n)over a \(p\)-adic field \(K\); see §Notation and Printing for its description.
- Example: Galrep Sp (ex-8d3d78)#
> K:=pAdicField(3,20); > SP(K,1) eq PrincipalCharacter(K); true > rho:=SP(K,2); rho; 2-dim Galois representation SP(2) over Q3[20] > Degree(rho); 2 > Semisimplification(rho); 2-dim unramified Galois representation Unr(1-4/3*x+1/3*x^2) over Q3[20] > $1 eq PrincipalCharacter(K)+CyclotomicCharacter(K); true > InertiaInvariants(rho); 1-dim unramified Galois representation Unr(1/3) over Q3[20] > EulerFactor(rho); -1/3*x + 1
- SP(K, f, n): FldPad, RngUPolElt, RngIntElt -> GalRep#
Unramified twist \(\psi\otimes{\hbox{\tt SP}(n)}\) over a \(p\)-adic field \(K\), with \(\psi\) specified by its Euler factor \(f\).
- Example: Galrep Sp 1 (ex-3bd7e7)#
> K:=pAdicField(2,20); > R<x>:=PolynomialRing(Rationals()); > SP(K,1-x^2,2); 4-dim Galois representation Unr(1-x^2)*SP(2) over Q2[20] > $1*$1; // Tensor product with itself 16-dim Galois representation Unr(1-1/2*x^2+1/16*x^4) + Unr(1-2*x^2+x^4)*SP(3) over Q2[20]
Representations from Finite Extensions#
- GaloisRepresentations(F, K): FldPad, FldPad -> SeqEnum[GalRep]#
For a \(p\)-adic extension \(F/K\), compute all irreducible Galois representations that factor through the (Galois closure of) \(F/K\).
- Example: Galrep Galoisrepresentations 1 (ex-314f0e)#
We take \(F\) to be a degree 16 dihedral extension of \(K={\mathbb{Q}}_2\), and compute the irreducible characters of \(\mathop{\rm Gal}\nolimits(F/K)\), viewed as Galois representations over \(K\).
> K:=pAdicField(2,20); > R<x>:=PolynomialRing(K); > F:=ext<K|x^8+2>; > list:=GaloisRepresentations(F,K); > list; [ 1-dim trivial Galois representation 1 over Q2[20], 1-dim Galois representation (1,1,-1,-1,1,1,1) with G=D8, I=D8, conductor 2^2 over Q2[20], 1-dim Galois representation (1,1,-1,1,1,-1,-1) with G=D8, I=D8, conductor 2^3 over Q2[20], 1-dim Galois representation (1,1,1,-1,1,-1,-1) with G=D8, I=D8, conductor 2^3 over Q2[20], 2-dim Galois representation (2,2,0,0,-2,0,0) with G=D8, I=D8, conductor 2^8 over Q2[20], 2-dim Galois representation (2,-2,0,0,0,-zeta(8)_8^3+zeta(8)_8, zeta(8)_8^3-zeta(8)_8) with G=D8, I=D8, conductor 2^10 over Q2[20], 2-dim Galois representation (2,-2,0,0,0,zeta(8)_8^3-zeta(8)_8, -zeta(8)_8^3+zeta(8)_8) with G=D8, I=D8, conductor 2^10 over Q2[20] ]
The first 5 characters are not faithful, and we can descend them to smaller quotients of \(\mathop{\rm Gal}\nolimits(F/K)\).
> min:=[Minimize(rho): rho in list | not IsFaithful(Character(rho))]; > min; [ 1-dim trivial Galois representation 1 over Q2[20], 1-dim Galois representation (1,-1) with G=C2, I=C2, conductor 2^2 over Q2[20], 1-dim Galois representation (1,-1) with G=C2, I=C2, conductor 2^3 over Q2[20], 1-dim Galois representation (1,-1) with G=C2, I=C2, conductor 2^3 over Q2[20], 2-dim Galois representation (2,-2,0,0,0) with G=D4, I=D4, conductor 2^8 over Q2[20] ]
- GaloisRepresentations(f): RngUPolEltFldPad -> SeqEnum[GalRep]#
For a polynomial \(f\) over a \(p\)-adic field \(K\) and splitting field \(F\), returns irreducible representations of \(\mathop{\rm Gal}\nolimits(F/K)\).
- Example: Galrep Galoisrepresentations (ex-da3085)#
We construct 4 one-dimensional characters of \({\mathbb{Q}}_2(\zeta_8)/{\mathbb{Q}}_2\).
> K:=pAdicField(2,20); > R:=PolynomialRing(K); > GaloisRepresentations(R!CyclotomicPolynomial(8)); [ 1-dim trivial Galois representation 1 over Q2[20], 1-dim Galois representation (1,-1,1,-1) with G=C2^2, I=C2^2, conductor 2^3 over Q2[20], 1-dim Galois representation (1,1,-1,-1) with G=C2^2, I=C2^2, conductor 2^2 over Q2[20], 1-dim Galois representation (1,-1,-1,1) with G=C2^2, I=C2^2, conductor 2^3 over Q2[20] ]
- PermutationCharacter(F, K): FldPad, FldPad -> GalRep#
For a \(p\)-adic extension \(F/K\), compute \({\mathbb{C}}[\mathop{\rm Gal}\nolimits(\bar{K}/K)/\mathop{\rm Gal}\nolimits(\bar{K}/F)]\) as a Galois representation over \(K\) of degree \([F:K]\).
- Example: Galrep Permutationcharacter (ex-ba722b)#
Take \(K={\mathbb{Q}}_2\) and \(F={\mathbb{Q}}_2(\root 3 \of 2)\). Then
PermutationCharacter(F,K)is a 3-dimensional representation which is the trivial representation plus a 2-dimensional irreducible one.> K:=pAdicField(2,20); > R<x>:=PolynomialRing(K); > F:=ext<K|x^3-2>; > PermutationCharacter(F,K); 3-dim Galois representation (3,1,0) with G=S3, I=C3, conductor 2^2 over Q2[20] > $1 - PrincipalCharacter(K); 2-dim Galois representation (2,0,-1) with G=S3, I=C3, conductor 2^2 over Q2[20]
- A !! chi: GalRep, AlgChtrElt -> GalRep#
- A !! chi: GalRep, SeqEnum -> GalRep#
Change a Galois representation by a finite representation with character \(\chi\), which must be a character of
Group(A), or a list of values that determine such a character.
- Example: Galrep Smash (ex-b9793f)#
Take \(K={\mathbb{Q}}_2\) and \(F={\mathbb{Q}}_2(\root 3 \of 2)\), so that \(G=\mathop{\rm Gal}\nolimits(F/K)\cong S_3\). Using
!!we can start with any Galois representation whose finite part comes from this Galois group, and replace it by any other character of \(G\).> K:=pAdicField(2,20); > R<x>:=PolynomialRing(K); > F:=ext<K|x^3-2>; > rho:=PermutationCharacter(F,K); > rho!![1,1,1]; 1-dim trivial Galois representation 1 over Q2[20] > rho!![6,0,0]; 6-dim Galois representation (6,0,0) with G=S3, I=C3, conductor 2^4 over Q2[20] > rho!![0,0,0]; // but not [-1,0,0] - may not be virtual Galois representation 0 with G=S3, I=C3 over Q2[20]
Local Representations of Global Objects#
- GaloisRepresentation(chi, p): GrpDrchElt, RngIntElt -> GalRep#
Precision: RngIntElt Default: 40
Local Galois representation at \(p\) of a Dirichlet character \(\chi\).
- Example: Galrep Dirichlet (ex-a9f044)#
Local components of a Dirichet character \(\chi\) of order 6 at \(p=2,3,7\).
> G<chi>:=FullDirichletGroup(7); > GaloisRepresentation(chi,2); 1-dim unramified Galois representation (1,-zeta(3)_3-1,zeta(3)_3) with G=C3, I=C1 over Q2[40] > GaloisRepresentation(chi,3); 1-dim unramified Galois representation (1,-1,-zeta(3)_3-1,zeta(3)_3,-zeta(3)_3, zeta(3)_3+1) with G=C6, I=C1 over Q3[40] > GaloisRepresentation(chi,7); 1-dim Galois representation (1,-1,-zeta(3)_3-1,zeta(3)_3,-zeta(3)_3,zeta(3)_3+1) with G=C6, I=C6, conductor 7^1 over Q7[40]
By our convention, the character \(\chi\), the associated Artin representation, and the Galois representations associated to them all have the same Euler factors.
> loc1:=EulerFactor(chi,2); > loc2:=EulerFactor(ArtinRepresentation(chi),2); > loc3:=EulerFactor(GaloisRepresentation(chi,2)); > loc4:=EulerFactor(GaloisRepresentation(ArtinRepresentation(chi),2)); > [PolynomialRing(ComplexField(5))| loc1,loc2,loc3,loc4]; [ (0.50000 - 0.86603*$.1)*$.1 + 1.0000, (0.50000 - 0.86603*$.1)*$.1 + 1.0000, (0.50000 - 0.86603*$.1)*$.1 + 1.0000, (0.50000 - 0.86603*$.1)*$.1 + 1.0000 ]
- GaloisRepresentation(A, p): ArtRep, RngIntElt -> GalRep#
Precision: RngIntElt Default: 40 Minimize : BoolElt Default: false
Local Galois representation at \(p\) of an Artin representation \(A\). This is the representation of a decomposition group at \(p\) of \(\mathop{\rm Gal}\nolimits(\bar {\mathbb{Q}}/{\mathbb{Q}})\) on the dual vector space of \(A\). (The reason for the dual is that, by our convention, the global and the local Euler factors agree; see §Conventions and Example Example: Galrep Dirichlet.) If
Minimizeistrue, choose the field through which it factors to be as small as possible (automatic for faithful representations).
- Example: Galrep Artin (ex-e01700)#
Local components of an Artin representation. We take the Trinks’ polynomial \(x^7-7x-3\) with Galois group PSL\((2,7)\) over \({\mathbb{Q}}\), one of its 7-dimensional representations \(A\) of conductor \(3^87^8\), and compute its local components over \({\mathbb{Q}}_2\), \({\mathbb{Q}}_3\), \({\mathbb{Q}}_5\) and \({\mathbb{Q}}_7\).
> R<x>:=PolynomialRing(Rationals()); > K:=NumberField(x^7-7*x-3); > GroupName(GaloisGroup(K)); PSL(2,7) > A:=ArtinRepresentations(K)[5]; > GaloisRepresentation(A,2); 7-dim unramified Galois representation (7,0,0,0,0,0,0) with G=C7, I=C1 over Q2[40] > GaloisRepresentation(A,3); 7-dim Galois representation (7,-1,1) with G=S3, I=S3, conductor 3^8 over Q3[40] > GaloisRepresentation(A,5); 7-dim unramified Galois representation (7,0,0,0,0,0,0) with G=C7, I=C1 over Q5[40] > GaloisRepresentation(A,7); 7-dim Galois representation (7,1,1,0,0) with G=C7:C3, I=C7:C3, conductor 7^8 over Q7[40] > Conductor(A) eq 3^8*7^8; true
- GaloisRepresentation(E): CrvEll -> GalRep#
Minimize: BoolElt Default: true
Local Galois representation of (the first \(l\)-adic étale cohomology group of) an elliptic curve over a \(p\)-adic field. If
Minimizeistrue(default), choose the field through which it factors to be as small as possible.
- Example: Galrep E (ex-29c470)#
Take an elliptic curve \(E/{\mathbb{Q}}_5\), with additive (potentially good) reduction of type II.
> K:=pAdicField(5,20); > E:=EllipticCurve([K|0,5]); > E; Elliptic Curve defined by y^2 = x^3 + O(5^20)*x + (5 + O(5^21)) over pAdicField(5, 20) > loc:=LocalInformation(E); loc; <5 + O(5^21), 2, 2, 1, II, true>
Its Galois representation is an unramified twist of a representation with finite image that factors through the dihedral extension \({\mathbb{Q}}_5(\zeta_6,{\root 6\of 5})\) of \({\mathbb{Q}}_5\).
> A:=GaloisRepresentation(E); A; 2-dim Galois representation Unr(sqrt(5)*i)*(2,-2,0,0,-1,1) with G=D6, I=C6, conductor 5^2 over Q5[20] > Field(A); Totally ramified extension defined by the polynomial x^6 - 5 over Unramified extension defined by the polynomial x^2 + 4*x + 2 over 5-adic field mod 5^20
- GaloisRepresentation(E, p): CrvEll, RngIntElt -> GalRep#
Precision: RngIntElt Default: 40 Minimize : BoolElt Default: true
Local Galois representation of (the first \(l\)-adic étale cohomology group of) an elliptic curve over \({\mathbb{Q}}\) at \(p\). If
Minimizeistrue(default), choose the field through which it factors to be as small as possible.
- Example: Galrep E P (ex-94d575)#
We take the elliptic curve 20a1 over \({\mathbb{Q}}\) and compute its local Galois representation at a prime \(p=3\) of good reduction, \(p=5\) of (non-split) multiplicative reduction and \(p=2\) of additive reduction.
> E:=EllipticCurve("20a1"); > GaloisRepresentation(E,3); 2-dim unramified Galois representation Unr(1+2*x+3*x^2) over Q3[40] > GaloisRepresentation(E,5); 2-dim Galois representation Unr(-5)*SP(2) over Q5[40] > GaloisRepresentation(E,2); 2-dim Galois representation Unr(sqrt(2)*i)*(2,0,-1) with G=S3, I=C3, conductor 2^2 over Q2[40] > EulerFactor($3),EulerFactor($2),EulerFactor($1); 3*x^2 + 2*x + 1 x + 1 1
- GaloisRepresentation(E, P): CrvEll, RngOrdIdl -> GalRep#
Precision: RngIntElt Default: 40 Minimize : BoolElt Default: true
Local Galois representation of (the first \(l\)-adic étale cohomology group of) an elliptic curve \(E\) over a number field \(F\) at a given prime ideal \(P\). If
Minimizeistrue(default), choose the field through which it factors to be as small as possible.
- Example: Galrep E P (ex-f7aec3)#
> K:=CyclotomicField(5); > E:=BaseChange(EllipticCurve("75a1"),K); > P:=Ideal(Decomposition(K,5)[1,1]); > GaloisRepresentation(E,P); 2-dim Galois representation Unr(sqrt(5)*i)*(2,0,-1) with G=S3, I=C3, conductor pi^2 over ext<Q5[10]|x^4-15*x^3-40*x^2-90*x-45>
- GaloisRepresentation(C): CrvHyp[FldPad] -> GalRep#
Degree : RngIntElt Default: Infinity() Minimize: BoolElt Default: false
Galois representation associated to (\(H^1\) of) a hyperelliptic curve \(C\) over a \(p\)-adic field.
Degreespecifies that Euler factors of unramified pieces should only be computed up to that degree. (See Example Example: Galrep Notfullycomputed.) SettingMinimize:=trueforces the representation to be minimized. (SeeMinimize.)
- Example: Galrep Crvhyp Qp (ex-1fdb67)#
We take a curve \(C\) over \(K={\mathbb{Q}}_{23}\) of conductor \(23^2\) and compute its Galois representation.
> K:=pAdicField(23,20); > R<x>:=PolynomialRing(K); > C:=HyperellipticCurve(-x,x^3+x^2+1); // genus 2, conductor 23^2 > A:=GaloisRepresentation(C); A; 4-dim Galois representation Unr(1-46*x+529*x^2)*SP(2) over Q23[20]
If \(F/K\) is a finite extension, then the base change of \(A\) to \(F\) is the same as the Galois representation of \(C/F\):
> F:=ext<K|2>; > BaseChange(A,F); 4-dim Galois representation Unr(1-1058*x+279841*x^2)*SP(2) over ext<Q23[20]|2> > GaloisRepresentation(BaseChange(C,F)); 4-dim Galois representation Unr(1-1058*x+279841*x^2)*SP(2) over ext<Q23[20]|2>
- GaloisRepresentation(C, p): CrvHyp[FldRat], RngIntElt -> GalRep#
Degree : RngIntElt Default: Infinity() Minimize: BoolElt Default: false
Galois representation associated to (\(H^1\) of) a hyperelliptic curve \(C/{\mathbb{Q}}\) at \(p\).
Degreespecifies that Euler factors of unramified pieces should only be computed up to that degree. (See Example Example: Galrep Notfullycomputed.) SettingMinimize:=trueforces the representation to be minimized. (SeeMinimize.)
- Example: Galrep Crvhyp Q (ex-1f2ef0)#
> R<x>:=PolynomialRing(Rationals()); > C:=HyperellipticCurve((x^2+5)*(x+1)*(x+2)*(x+3)); > GaloisRepresentation(C,5); // bad reduction 4-dim Galois representation Unr(1+2*x+5*x^2) + Unr(5)*SP(2) over Q5[20] > GaloisRepresentation(C,11); // good reduction 4-dim unramified Galois representation Unr(1-2*x+6*x^2-22*x^3+121*x^4) over Q11[5] > GaloisRepresentation(C,997: Degree:=1); // don't count pts over GF(997^2) 4-dim unramified Galois representation Unr(1+26*x+O(x^2)) over Q997[5]
- GaloisRepresentation(C, P): CrvHyp[FldNum], RngOrdIdl -> GalRep#
Degree : RngIntElt Default: Infinity() Minimize: BoolElt Default: false
Galois representation associated to (\(H^1\) of) a hyperelliptic curve \(C\) over a number field at a prime ideal \(P\).
Degreespecifies that Euler factors of unramified pieces should only be computed up to that degree. (See Example Example: Galrep Notfullycomputed.) SettingMinimize:=trueforces the representation to be minimized. (SeeMinimize.)
- Example: Galrep Crvhyp Nf (ex-e4e4fd)#
We take a curve of genus 4 over \({\mathbb{Q}}(\zeta_{11})\) and compute its Galois representation at a unique prime \(P\) above 11.
> K<zeta>:=CyclotomicField(11); > R<x>:=PolynomialRing(K); > C:=HyperellipticCurve(x^9+x^2+(zeta-1)); > P:=Ideal(Decomposition(K,11)[1,1]); > GaloisRepresentation(C,P); 8-dim Galois representation Unr(1-44*x^3+1331*x^6) + Unr(11)*SP(2) over ext<Q11[2]|x^10+22*x^9+55*x^8+44*x^7-33*x^6-22*x^5-22*x^4-33*x^3+44*x^2+ 55*x+11>
- GaloisRepresentation(f, p): ModFrmElt, RngIntElt -> GalRep#
Precision: RngIntElt Default: 40
Local Galois representation at \(p\) of a modular form \(f\). Currently only implemented when \(p^2\) does not divide the level.
- Example: Galrep F P (ex-e7791f)#
We take a rational modular form of weight 4 and level 5, and compute its Galois representations at \(p=3\) (unramified principal series) and \(p=5\) (Steinberg).
> f:=Newforms("5k4")[1,1]; > GaloisRepresentation(f,3); 2-dim unramified Galois representation Unr(1-2*x+27*x^2) over Q3[40] > GaloisRepresentation(f,5); 2-dim Galois representation Unr(-25)*SP(2) over Q5[40]