# Creation Functions ## Ambient Spaces An ambient space of modular symbols is created by specifying a base ring, character, weight, and optional sign. Note that spaces of modular symbols may be defined directly over any base ring (unlike `ModularForms`, which can only be base extensions of spaces over the integers). The most general signature is `ModularSymbols(eps, k, sign)`, where `eps` is a Dirichlet character (the level is taken to be the modulus of `eps`, and the base ring is taken to be the base ring of `eps`). For information on Dirichlet characters, see Section [Dirichlet Characters](../../BasicRings/ResidueClassRings/dirichlet-characters.md#rngint-dirichlet). **Warning:** Certain functions, such as `DualVectorSpace`, may fail when given as input a space of modular symbols over a field of positive characteristic, because the Hecke operators $T_p$, with $p$ prime to the level, need not be semisimple. ### `ModularSymbols(N): RngIntElt -> ModSym` The space of modular symbols of level $N$, weight $2$, and trivial character over the rational numbers. ### `ModularSymbols(N, k): RngIntElt, RngIntElt -> ModSym` The space of modular symbols of level $N$, weight $k$, and trivial character over the rational numbers. ### `ModularSymbols(N, k, F): RngIntElt, RngIntElt, Fld -> ModSym` The space of modular symbols of level $N$, weight $k$, and trivial character over the field $F$. ### `ModularSymbols(N, k, sign): RngIntElt, RngIntElt, RngIntElt -> ModSym` The space of modular symbols of level $N$, weight $k$, trivial character, and given $sign$ (as an integer) over the rational numbers. ### `ModularSymbols(N, k, F, sign): RngIntElt, RngIntElt, Fld, RngIntElt -> ModSym` The space of modular symbols of level $N$, weight $k$, trivial character, and given $sign$ (as an integer), over the field $F$. ### `ModularSymbols(eps, k): GrpDrchElt, RngIntElt -> ModSym` The space of modular symbols of weight $k$ and character $\varepsilon$ (as an element of a Dirichlet group). Note that $\varepsilon$ determines the level and the base field, so they do not need to be specified. ### `ModularSymbols(eps, k, sign): GrpDrchElt, RngIntElt, RngIntElt -> ModSym` The space of modular symbols of weight $k$ and character $\varepsilon$ (as an element of a Dirichlet group). The level and base field are specified as part of $\varepsilon$. The third argument “sign” allows for working in certain quotients. The possible values are $-1$, $0$, and $+1$, which correspond to the $-1$ quotient, full space, and $+1$ quotient, respectively. The $+1$ quotient of $M$ is $M/(*-1)M$, where $*$ is `StarInvolution(M)`. ### `Example: Creation Ambient (ex-3cf3aa)` We create spaces of modular symbols in several different ways. ```magma > M37 := ModularSymbols(37); M37; Full modular symbols space for Gamma_0(37) of weight 2 and dimension 5 over Rational Field > Basis(M37); [ {-1/29, 0}, {-1/22, 0}, {-1/12, 0}, {-1/18, 0}, {oo, 0} ] ``` As `M37` is a space of modular symbols, it is not incorrect that its dimension is different than that of the three-dimensional space of modular forms $M_{2}(\Gamma_0(37))$. We have $$ \dim \bf M_{2}(\Gamma_0(37)) = 2\times {\rm (dim\,\, cusp\,\, forms)} + 1\times {\rm (dim\,\, Eisenstein\,\, series)} = 5. $$ ```magma > MF := ModularForms(Gamma0(37),2); > 2*Dimension(CuspidalSubspace(MF)) + Dimension(EisensteinSubspace(MF)); 5 ``` Next we decompose `M37` with respect to the Hecke operators $T_2$, $T_3$, and $T_5$. ```magma > D := Decomposition(M37,5); D; [ Modular symbols space for Gamma_0(37) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(37) of weight 2 and dimension 2 over Rational Field, Modular symbols space for Gamma_0(37) of weight 2 and dimension 2 over Rational Field ] ``` The first factor corresponds to the standard Eisenstein series, and the second corresponds to an elliptic curve: ```magma > E := EllipticCurve(D[2]); E; Elliptic Curve defined by y^2 + y = x^3 + x^2 - 23*x - 50 over Rational Field > Rank(E); 0 ``` We now create the space $\bf M_{12}(1)$ of weight $12$ modular symbols of level $1$. ```magma > M12 := ModularSymbols(1,12); M12; Full modular symbols space for Gamma_0(1) of weight 12 and dimension 3 over Rational Field > Basis(M12); [ X^10*{0, oo}, X^8*Y^2*{0, oo}, X^9*Y*{0, oo} ] ``` ```magma > DimensionCuspFormsGamma0(1,12); 1 > R:=PowerSeriesRing(Rationals()); > Delta(z)+ O(z^7); z - 24*z^2 + 252*z^3 - 1472*z^4 + 4830*z^5 - 6048*z^6 + O(z^7) ``` As a module, cuspidal modular symbols equal cuspforms with multiplicity two. ```magma > M12 := ModularSymbols(1,12); > HeckeOperator(CuspidalSubspace(M12),2); [-24 0] [ 0 -24] > qExpansionBasis(CuspidalSubspace(M12),7); [ q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 + O(q^7) ] ``` For efficiency purposes, since one is often interested only in $q$-expansion, it is possible to work in the quotient of the space of modular symbols by all relations $*x = x$ (or $*x=-x$), where $*$ is `StarInvolution(M)`. In either of these quotients (except possibly in characteristic $p>0$) the cusp forms appear with multiplicity one instead of two. ```magma > M12plus := ModularSymbols(1,12,+1); > Basis(M12plus); [ X^10*{0, oo}, X^8*Y^2*{0, oo} ] > CuspidalSubspace(M12plus); Modular symbols space for Gamma_0(1) of weight 12 and dimension 1 over Rational Field > qExpansionBasis(CuspidalSubspace(M12),7); [ q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 + O(q^7) ] ``` The following is an example of how to create Dirichlet characters in Magma, and how to create a space of modular symbols with nontrivial character. For more details, see Section [Dirichlet Characters](../../BasicRings/ResidueClassRings/dirichlet-characters.md#rngint-dirichlet). ```magma > G := DirichletGroup(16*7,CyclotomicField(EulerPhi(16*7))); > Order(a); 2 > Conductor(a); 4 > Order(b); 4 > Conductor(b); 16 > Order(c); 6 > Conductor(c); 7 > eps := a*b*c; > M := ModularSymbols(eps,2); M; Full modular symbols space of level 112, weight 2, character a*b*c, and dimension 32 over Cyclotomic Field of order 48 and degree 16 > BaseField(M); Cyclotomic Field of order 48 and degree 16 ``` ## Labels It is also possible to create many spaces of modular symbols for $\Gamma_0(N)$ by passing a “descriptive label” as an argument to `ModularSymbols`. The most specific label is a string of the form `[Level]k[Weight][IsogenyClass]`, where `[Level]` is the level, `[Weight]` is the weight, `[IsogenyClass]` is a letter code: `A`, `B`, …, `Z`, `AA`, `BB`, …, `ZZ`, `AAA`, …, and `k` is a place holder to separate the level and the weight. If the label is `[Level][IsogenyClass]`, then the weight $k$ is assumed equal to $2$. If the label is `[Level]k[Weight]` then the cuspidal subspace of the full ambient space of modular symbols of weight $k$ and level $N$ is returned. The following are valid labels: `11A`, `37B`, `3k12A`, `11k4`. The ordering used on isogenies classes is `lt`; see the documentation for `SortDecomposition`. **Note:** There is currently no intrinsic that, given a space of modular symbols, returns its label. **Warning:** For $146$ of the levels between $56$ and $450$, our ordering of weight $2$ rational newforms disagrees with the ordering used in [[Cremona, 1997](../../references.md#cite-cremona-algs)]. Fortunately, it is easy to create a space of modular symbols from one of Cremona’s labels using the associated elliptic curve. ``` > E := EllipticCurve(CremonaDatabase(),"56A"); > M1 := ModularSymbols(E); ``` Observe that Cremona’s `"56A"` is different from ours. ``` > M2 := ModularSymbols("56A"); > M2 eq M1; false ``` ### `ModularSymbols(s, sign): MonStgElt, RngIntElt -> ModSym` ### `ModularSymbols(s): MonStgElt -> ModSym` The space of modular symbols described by a label, the string $s$, and given $sign$. ### `Example: Creation Spaces (ex-999030)` The cusp form $\Delta(q)$ is related to the space of modular symbols whose label is `"1k12A"`. ```magma > Del := ModularSymbols("1k12A"); Del; Modular symbols space for Gamma_0(1) of weight 12 and dimension 2 over Rational Field > qEigenform(Del,5); q - 24*q^2 + 252*q^3 - 1472*q^4 + O(q^5) ``` Next, we create the space corresponding to the first newform on $\Gamma_0(11)$ of weight $4$. ```magma > M := ModularSymbols("11k4A"); M; Modular symbols space for Gamma_0(11) of weight 4 and dimension 4 over Rational Field > AmbientSpace(M); Full modular symbols space for Gamma_0(11) of weight 4 and dimension 6 over Rational Field > qEigenform(M,5); q + a*q^2 + (-4*a + 3)*q^3 + (2*a - 6)*q^4 + O(q^5) > Parent($1); Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^2 - 2*a - 2 ``` We next create the $+1$ quotient of the cuspidal subspace of weight-$4$ modular symbols of level $37$. ```magma > M := ModularSymbols("37k4",+1); M; Modular symbols space for Gamma_0(37) of weight 4 and dimension 9 over Rational Field > AmbientSpace(M); Full modular symbols space for Gamma_0(37) of weight 4 and dimension 11 over Rational Field > Factorization(CharacteristicPolynomial(HeckeOperator(M,2))); [ , ] ``` ## Creation of Elements Suppose $M$ is a space of weight $k$ modular symbols over a field $F$. A modular symbol $P(X,Y)\{\alpha,\beta\}$ is input as `M!`, where $P(X,Y)\in{}F[X,Y]$ is homogeneous of degree $k-2$, and $\alpha, \beta\in \bf P^1({\mathbb{Q}})$. Here is an example: ### `Example: Creation Elements 1 (ex-c14194)` First create the space $M = \bf M_4(\Gamma_0(3);{\mathbb{F}}_7)$. ```magma > F7 := GF(7); > M := ModularSymbols(3,4,F7); > R := PolynomialRing(F7,2); ``` Now we input $(X^2-2XY)\{0,1\}$. ```magma > M!; 6*Y^2*{oo, 0} ``` Note that $(X^2-2XY)\{0,1\} = 6Y^2\{\infty,0\}$ in $M$. When $k=2$, simply enter `M!<1,[alpha,beta]>`. ```magma > M := ModularSymbols(11,2); > M!<1,[Cusps()|0,Infinity()]>; -1*{oo, 0} > M![<1,[Cusps()|0,Infinity()]>, <1,[Cusps()|0,1/11]>]; -2*{oo, 0} ``` Any space $M$ of modular symbols is finitely generated. One proof of this uses that every modular symbol is a linear combination of *Manin symbols*. Let $\bf P^1({\mathbb{Z}}/N{\mathbb{Z}})$ be the set of pairs $(\overline{u},\overline{v}) \in {\mathbb{Z}}/N{\mathbb{Z}}\times {\mathbb{Z}}N{\mathbb{Z}}$ such that ${\rm GCD}(u,v,N)=1$, where $u$ and $v$ are lifts of $\overline{u}$ and $\overline{v}$ to ${\mathbb{Z}}$. A Manin symbols $\langle P(X,Y),(\overline{u},\overline{v})\rangle$ is a pair consisting of a homogeneous polynomial $P(X,Y)\in F[X,Y]$ of degree $k-2$ and an element $(\overline{u},\overline{v})\in \bf P^1({\mathbb{Z}}/N{\mathbb{Z}})$. The modular symbol associated to $\langle P(X,Y),(\overline{u},\overline{v})\rangle$ is constructed as follows. Choose lifts $u,v$ of $\overline{u},\overline{v}$ such that ${\rm GCD}(u,v)=1$. Then there is a matrix $g=\begin{pmatrix}w&z\\ u&v\end{pmatrix}$ in ${\operatorname{SL}}_2({\mathbb{Z}})$ whose lower two entries are $u$ and $v$. The modular symbol is then $g(P(X,Y)\{0,\infty\})$. The intrinsic `ConvertFromManinSymbol` computes the modular symbol attached to a Manin symbol. Ever modular symbol can be written as a linear combination of Manin symbols using the intrinsic `ManinSymbol`. ### `Example: Representation Conversion (ex-b624a4)` In this example, we convert between Manin and modular symbols representations of a few elements of a space of modular symbols of weight $4$ over ${\mathbb{F}}_5$. ```magma > F5 := GF(5); > M := ModularSymbols(6,4,F5); > R := PolynomialRing(F5,2); > ConvertFromManinSymbol(M,); (3*X^2 + 3*X*Y + 2*Y^2)*{-1/2, 0} + (X^2 + 4*X*Y + 4*Y^2)*{-1/3, 0} + (X^2 + X*Y + 4*Y^2)*{1/3, 1/2} > ManinSymbol(M.1-3*M.2); [ , <2*X^2, (1 2)> ] ``` Thus the element `M.1-3*M.2` of $M$ corresponds to the sum of Manin symbols $\langle X^2, (0,1)\rangle + 2\langle X^2,(1,2)\rangle$. ### `M ! x: ModSym, . -> ModSymElt` The coercion of $x$ into the space of modular symbols $M$. Here $x$ can be either a modular symbol that lies in a subspace of $M$, a $2$-tuple that describes a modular symbol, a sequence of such $2$-tuples, or anything that can be coerced into `VectorSpace(M)`. If $x$ is a valid sequence of such $2$-tuples, then `M!x` is the sum of the coercions into $M$ of the elements of the sequence $x$. ### `ConvertFromManinSymbol(M, x): ModSym, Tup -> ModSymElt` The modular symbol associated to the $2$-tuple $x = \langle P(X,Y),[u,v] \rangle$, where $P(X,Y) \in F[X,Y]$ is homogeneous of degree $k-1$, $F$ is the base field of the space of modular symbols $M$, and $[u,v]$ is a sequence of $2$ integers that defines an element of $\bf P^1({\mathbb{Z}}/N{\mathbb{Z}})$, where $N$ is the level of $M$. ### `ManinSymbol(x): ModSymElt -> SeqEnum` An expression for the modular symbol $x$ in terms of Manin symbols, which are represented as $2$-tuples $\langle P(X,Y),[u,v] \rangle$. ### `Example: Creation Elements 2 (ex-4a3707)` ```magma > M := ModularSymbols(14,2); M; Full modular symbols space for Gamma_0(14) of weight 2 and dimension 5 over Rational Field > Basis(M); [ {oo, 0}, {-1/8, 0}, {-1/10, 0}, {-1/12, 0}, {-1/2, -3/7} ] > M!<1,[1,0]>; 0 > M!<1,[0,1/11]>; {-1/10, 0} + -1*{-1/12, 0} > M![<1,[0,1/2]>, <-1,[0,1/7]>]; // sequences are added {-1/8, 0} + -1*{-1/12, 0} + -1*{-1/2, -3/7} > M!<1,[0,1/2]> - M!<1,[0,1/7]>; {-1/8, 0} + -1*{-1/12, 0} + -1*{-1/2, -3/7} > M!<1,[Cusps()|Infinity(),0]>; // Infinity() is in Cusps(). {oo, 0} ``` We can also coerce sequences into the underlying vector space of M. ```magma > VectorSpace(M); Full Vector space of degree 5 over Rational Field Mapping from: Full Vector space of degree 5 over Rational Field to ModSym: M given by a rule [no inverse] Mapping from: ModSym: M to Full Vector space of degree 5 over Rational Field given by a rule [no inverse] > Eltseq(M.3); [ 0, 0, 1, 0, 0 ] > M![ 0, 0, 1, 0, 0 ]; {-1/10, 0} > M.3; {-1/10, 0} ``` The “polynomial coefficients” of the modular symbols are homogeneous polynomials in $2$ variables of degree $k-2$. ```magma > M := ModularSymbols(1,12); > Basis(M); [ X^10*{0, oo}, X^8*Y^2*{0, oo}, X^9*Y*{0, oo} ] > R := PolynomialRing(Rationals(),2); > M!; X^9*Y*{0, oo} > M!; -25/48*X^9*Y*{0, oo} > Eltseq(M!); [ -19171, -58050, -30970 ] > M![1,2,3]; X^10*{0, oo} + 2*X^8*Y^2*{0, oo} + 3*X^9*Y*{0, oo} > ManinSymbol(M![1,2,3]); [ , <2*X^8*Y^2, (0 1)>, <3*X^9*Y, (0 1)> ] ```