# Newforms

In this section we describe how to compute both cuspidal and Eisenstein newforms.

The intrinsics below require that the base ring of $M$ has characteristic $0$. To compute mod $p$ eigenforms, use the `Reduction` intrinsic (see Section [Reductions and Embeddings](reductions-embeddings.md#sec-reductions-and-embeddings)).

## `NumberOfNewformClasses(M : parameters): ModFrm -> RngIntElt`

```magma
Proof: BoolElt                    Default: true
```

The number of Galois conjugacy-classes of newforms associate to the modular forms space $M$, which must have base ring ${\mathbb{Z}}$ or ${\mathbb{Q}}$. By “associated to” we mean that the newform lies in $M{\otimes}{\mathbb{C}}$.

## `Newform(M, i, j : parameters): ModFrm, RngIntElt, RngIntElt -> ModFrmElt`

```magma
Proof: BoolElt                    Default: true
```

The $j$th Galois-conjugate newform in the $i$th Galois-orbit of newforms in the space of modular forms $M$, which must have base ring ${\mathbb{Z}}$ or ${\mathbb{Q}}$.

## `Newform(M, i : parameters): ModFrm, RngIntElt -> ModFrmElt`

```magma
Proof: BoolElt                    Default: true
```

The first Galois-conjugate newform in the $i$th orbit in the space of modular forms $M$, which must have base ring ${\mathbb{Z}}$ or ${\mathbb{Q}}$.

## `Newforms(M : parameters): ModFrm -> List`

```magma
Proof      : BoolElt                    Default: true
OnePerOrbit: BoolElt                    Default: false
```

Sort list of the newforms associated to the space of modular forms $M$ divided up into Galois orbits. If `OnePerOrbit` is `true`, return only one representative from each Galois orbit.

## `Newforms(I, M): [Tup], ModFrm -> ModFrm`

Use this intrinsic to find the newforms associated to the space of modular forms $M$ with prespecified eigenvalues. Here $I$ is a sequence $[\langle p_1, f_1(x)\rangle,...,\langle p_n,f_n(x)\rangle]$ of pairs. Each pair consists of a prime number that does not divide the level of $M$ and a polynomial. This intrinsic returns the set of newforms $\sum a_n q^n$ in $M$ such that $f_n(a_{p_n})=0$. (This intrinsic only works when $M$ is cuspidal and defined over ${\mathbb{Q}}$ or ${\mathbb{Z}}$.)

## `NewformDecomposition(M): ModFrm -> List`

Given a newspace $M$, returns representatives of each of the Galois orbits of newforms in $M$.

## `Example: Newforms (ex-ecf1ff)`

We compute the newforms in $M_5(\Gamma_1(8))$.

```magma
> M := ModularForms(Gamma1(8),5); M;
Space of modular forms on Gamma_1(8) of weight 5 and dimension 11 over
Integer Ring.
> NumberOfNewformClasses(M);
4
> Newforms(M);
[* [*
q + 4*q^2 - 14*q^3 + 16*q^4 - 56*q^6 + O(q^8)
*], [*
q + 1/24*(a - 30)*q^2 + 6*q^3 + 1/12*(-a - 162)*q^4 + 1/3*(-a + 6)*q^5
+ 1/4*(a - 30)*q^6 + 1/3*(2*a - 12)*q^7 + O(q^8),
q + 1/24*(b - 30)*q^2 + 6*q^3 + 1/12*(-b - 162)*q^4 + 1/3*(-b + 6)*q^5
+ 1/4*(b - 30)*q^6 + 1/3*(2*b - 12)*q^7 + O(q^8)
*], [*
57/2 + q + q^2 + 82*q^3 + q^4 - 624*q^5 + 82*q^6 - 2400*q^7 + O(q^8)
*], [*
q + 16*q^2 + 82*q^3 + 256*q^4 + 624*q^5 + 1312*q^6 + 2400*q^7 + O(q^8)
*] *]
> Newform(M,1);
q + 4*q^2 - 14*q^3 + 16*q^4 - 56*q^6 + O(q^8)
> Newform(M,2);
q + 1/24*(a - 30)*q^2 + 6*q^3 + 1/12*(-a - 162)*q^4 + 1/3*(-a + 6)*q^5
+ 1/4*(a - 30)*q^6 + 1/3*(2*a - 12)*q^7 + O(q^8)
> Parent(Newform(M,2));
Space of modular forms on Gamma_1(8) of weight 5 and dimension 2 over
Number Field with defining polynomial x^2 - 12*x + 8676 over the
Rational Field.
> Newform(M,2,2);
q + 1/24*(b - 30)*q^2 + 6*q^3 + 1/12*(-b - 162)*q^4 + 1/3*(-b + 6)*q^5
+ 1/4*(b - 30)*q^6 + 1/3*(2*b - 12)*q^7 + O(q^8)
> IsEisensteinSeries(Newform(M,1));
false
> IsEisensteinSeries(Newform(M,2));
false
> IsEisensteinSeries(Newform(M,3));
true
> IsEisensteinSeries(Newform(M,4));
true

```

The following example demonstrates picking out a newform in $S_2(\Gamma_0(65))$ with prespecified eigenvalues.

```magma
> S := CuspForms(65,2);
> R<x> := PolynomialRing(IntegerRing());
> I := [<3,x+2>];
> Newforms(I,S);
[* [*
q - q^2 - 2*q^3 - q^4 - q^5 + 2*q^6 - 4*q^7 + O(q^8)
*] *]
> Factorization(HeckePolynomial(S, 2));
[
    <x + 1, 1>,
    <x^2 - 3, 1>,
    <x^2 + 2*x - 1, 1>
]
> I := [<2,x^2-3>];
> Newforms(I,S);
[* [*
q + a*q^2 + (-a + 1)*q^3 + q^4 - q^5 + (a - 3)*q^6 + 2*q^7 + O(q^8),
q + b*q^2 + (-b + 1)*q^3 + q^4 - q^5 + (b - 3)*q^6 + 2*q^7 + O(q^8)
*] *]

```

## Labels

It is possible to obtain the galois-conjugacy class of a newform by giving a descriptive label as an argument to `Newforms`. The format of the label is as follows:

`[G0N or G1N][Level]k[Weight][Isogeny Class]`.

Some example labels are `"G0N11k2A", "G0N1k12A", "G1N17k2B"`, and `"G1N9k3B"`. If the string `"G0N"` or `"G1N"` is omitted, then the default is `"G0N"`. Thus the following are also valid: `"11k2A", "1k12A", "37k4A"`. If `k[Weight]` is omitted, then the default is weight $2$, so the following are valid and all refer to weight $2$ modular forms on some $\Gamma_0(N)$: `"11A", "37A", "65B"`. In order, possibilities for the isogeny class are as follows:

`A, B, C, ..., Y, Z, AA, BB, CC, ..., ZZ, AAA, BBB, CCC, ...`.

This is essentially the notation used in [[Cremona, 1997](../../references.md#cite-cremona-algs)] for isogeny classes, though sometimes for levels $\leq 450$ the ordering differs from that in [[Cremona, 1997](../../references.md#cite-cremona-algs)].

Suppose $s$ is a valid label, and let $M$ be the space of modular forms that contains `ModularForm(s)`. Then `ModularForm(s)` is by definition `Newforms(M)[i]` where the isogeny class in the label $s$ is the $i$th isogeny class. For example `C` corresponds to the $3$rd isogeny class and `BB` corresponds to the $28$th.

### `Newforms(label): MonStgElt -> ModFrmElt`

The Galois-conjugacy class(es) of newforms described by the string $label$. See the introduction for a description of the notation used for the label.

### `Example: Newform Labeling (ex-5d3f25)`

We give many examples of constructing newforms using labels.

```magma
> Newforms("11A");
[*
q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 + O(q^8)
*]
>  Newforms("G0N11k2A");
[*
q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 + O(q^8)
*]
>  Newforms("G0N1k12A");
[*
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 +
O(q^8)
*]
>  Newforms("G1N17k2B");
[*
q + (-a^3 + a^2 - 1)*q^2 + (a^3 - a^2 - a - 1)*q^3 + (2*a^3 - a^2 +
2*a)*q^4 + (-a^3 - a^2)*q^5 + (-a^3 + a^2 - a + 1)*q^6 + (-a^3 + a^2 +
a - 1)*q^7 + O(q^8),
q + (-b^3 + b^2 - 1)*q^2 + (b^3 - b^2 - b - 1)*q^3 + (2*b^3 - b^2 +
2*b)*q^4 + (-b^3 - b^2)*q^5 + (-b^3 + b^2 - b + 1)*q^6 + (-b^3 + b^2 +
b - 1)*q^7 + O(q^8),
q + (-c^3 + c^2 - 1)*q^2 + (c^3 - c^2 - c - 1)*q^3 + (2*c^3 - c^2 +
2*c)*q^4 + (-c^3 - c^2)*q^5 + (-c^3 + c^2 - c + 1)*q^6 + (-c^3 + c^2 +
c - 1)*q^7 + O(q^8),
q + (-d^3 + d^2 - 1)*q^2 + (d^3 - d^2 - d - 1)*q^3 + (2*d^3 - d^2 +
2*d)*q^4 + (-d^3 - d^2)*q^5 + (-d^3 + d^2 - d + 1)*q^6 + (-d^3 + d^2 +
d - 1)*q^7 + O(q^8)
*]
> Newforms("G1N9k3B");
[*
1/3*(-5*zeta_6 - 2) + q + (4*zeta_6 + 1)*q^2 + q^3 + (20*zeta_6 -
15)*q^4 + (-25*zeta_6 + 26)*q^5 + (4*zeta_6 + 1)*q^6 + (-49*zeta_6 +
1)*q^7 + O(q^8),
1/3*(5*zeta_6 - 7) + q + (-4*zeta_6 + 5)*q^2 + q^3 + (-20*zeta_6 +
5)*q^4 + (25*zeta_6 + 1)*q^5 + (-4*zeta_6 + 5)*q^6 + (49*zeta_6 -
48)*q^7 + O(q^8)
*]
> Newforms("11k2A");
[*
q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 + O(q^8)
*]
> Newforms("11A");
[*
q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 + O(q^8)
*]
> Newforms("1k12A");
[*
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 +
O(q^8)
*]
> Newforms("37k4A");
[*
q + a*q^2 + 1/8*(-a^3 - 9*a^2 - 26*a - 22)*q^3 + (a^2 - 8)*q^4 +
1/8*(13*a^3 + 85*a^2 + 50*a - 186)*q^5 + 1/8*(-3*a^3 - 27*a^2 - 38*a +
6)*q^6 + 1/4*(-19*a^3 - 119*a^2 - 30*a + 170)*q^7 + O(q^8),
q + b*q^2 + 1/8*(-b^3 - 9*b^2 - 26*b - 22)*q^3 + (b^2 - 8)*q^4 +
1/8*(13*b^3 + 85*b^2 + 50*b - 186)*q^5 + 1/8*(-3*b^3 - 27*b^2 - 38*b +
6)*q^6 + 1/4*(-19*b^3 - 119*b^2 - 30*b + 170)*q^7 + O(q^8),
q + c*q^2 + 1/8*(-c^3 - 9*c^2 - 26*c - 22)*q^3 + (c^2 - 8)*q^4 +
1/8*(13*c^3 + 85*c^2 + 50*c - 186)*q^5 + 1/8*(-3*c^3 - 27*c^2 - 38*c +
6)*q^6 + 1/4*(-19*c^3 - 119*c^2 - 30*c + 170)*q^7 + O(q^8),
q + d*q^2 + 1/8*(-d^3 - 9*d^2 - 26*d - 22)*q^3 + (d^2 - 8)*q^4 +
1/8*(13*d^3 + 85*d^2 + 50*d - 186)*q^5 + 1/8*(-3*d^3 - 27*d^2 - 38*d +
6)*q^6 + 1/4*(-19*d^3 - 119*d^2 - 30*d + 170)*q^7 + O(q^8)
*]
> Newforms("37k2");
[* [*
q - 2*q^2 - 3*q^3 + 2*q^4 - 2*q^5 + 6*q^6 - q^7 + O(q^8)
*], [*
q + q^3 - 2*q^4 - q^7 + O(q^8)
*], [*
3/2 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + 12*q^6 + 8*q^7 + O(q^8)
*] *]

```