# Properties

 Label 63.1.d.a Level $63$ Weight $1$ Character orbit 63.d Self dual yes Analytic conductor $0.031$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -3, -7, 21 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [63,1,Mod(55,63)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(63, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("63.55");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$63 = 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 63.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$0.0314410957959$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{-3}, \sqrt{-7})$$ Artin image: $D_4$ Artin field: Galois closure of 4.0.189.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{4} - q^{7}+O(q^{10})$$ q - q^4 - q^7 $$q - q^{4} - q^{7} + q^{16} + q^{25} + q^{28} - 2 q^{37} - 2 q^{43} + q^{49} - q^{64} + 2 q^{67} + 2 q^{79}+O(q^{100})$$ q - q^4 - q^7 + q^16 + q^25 + q^28 - 2 * q^37 - 2 * q^43 + q^49 - q^64 + 2 * q^67 + 2 * q^79

## Expression as an eta quotient

$$f(z) = \eta(3z)\eta(21z)=q\prod_{n=1}^\infty(1 - q^{3n})^{}(1 - q^{21n})^{}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$$.

 $$n$$ $$10$$ $$29$$ $$\chi(n)$$ $$-1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 0
0 0 −1.00000 0 0 −1.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
21.c even 2 1 RM by $$\Q(\sqrt{21})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.1.d.a 1
3.b odd 2 1 CM 63.1.d.a 1
4.b odd 2 1 1008.1.f.a 1
5.b even 2 1 1575.1.h.b 1
5.c odd 4 2 1575.1.e.b 2
7.b odd 2 1 CM 63.1.d.a 1
7.c even 3 2 441.1.m.a 2
7.d odd 6 2 441.1.m.a 2
9.c even 3 2 567.1.l.b 2
9.d odd 6 2 567.1.l.b 2
12.b even 2 1 1008.1.f.a 1
15.d odd 2 1 1575.1.h.b 1
15.e even 4 2 1575.1.e.b 2
21.c even 2 1 RM 63.1.d.a 1
21.g even 6 2 441.1.m.a 2
21.h odd 6 2 441.1.m.a 2
28.d even 2 1 1008.1.f.a 1
35.c odd 2 1 1575.1.h.b 1
35.f even 4 2 1575.1.e.b 2
63.g even 3 2 3969.1.k.b 2
63.h even 3 2 3969.1.t.c 2
63.i even 6 2 3969.1.t.c 2
63.j odd 6 2 3969.1.t.c 2
63.k odd 6 2 3969.1.k.b 2
63.l odd 6 2 567.1.l.b 2
63.n odd 6 2 3969.1.k.b 2
63.o even 6 2 567.1.l.b 2
63.s even 6 2 3969.1.k.b 2
63.t odd 6 2 3969.1.t.c 2
84.h odd 2 1 1008.1.f.a 1
105.g even 2 1 1575.1.h.b 1
105.k odd 4 2 1575.1.e.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.1.d.a 1 1.a even 1 1 trivial
63.1.d.a 1 3.b odd 2 1 CM
63.1.d.a 1 7.b odd 2 1 CM
63.1.d.a 1 21.c even 2 1 RM
441.1.m.a 2 7.c even 3 2
441.1.m.a 2 7.d odd 6 2
441.1.m.a 2 21.g even 6 2
441.1.m.a 2 21.h odd 6 2
567.1.l.b 2 9.c even 3 2
567.1.l.b 2 9.d odd 6 2
567.1.l.b 2 63.l odd 6 2
567.1.l.b 2 63.o even 6 2
1008.1.f.a 1 4.b odd 2 1
1008.1.f.a 1 12.b even 2 1
1008.1.f.a 1 28.d even 2 1
1008.1.f.a 1 84.h odd 2 1
1575.1.e.b 2 5.c odd 4 2
1575.1.e.b 2 15.e even 4 2
1575.1.e.b 2 35.f even 4 2
1575.1.e.b 2 105.k odd 4 2
1575.1.h.b 1 5.b even 2 1
1575.1.h.b 1 15.d odd 2 1
1575.1.h.b 1 35.c odd 2 1
1575.1.h.b 1 105.g even 2 1
3969.1.k.b 2 63.g even 3 2
3969.1.k.b 2 63.k odd 6 2
3969.1.k.b 2 63.n odd 6 2
3969.1.k.b 2 63.s even 6 2
3969.1.t.c 2 63.h even 3 2
3969.1.t.c 2 63.i even 6 2
3969.1.t.c 2 63.j odd 6 2
3969.1.t.c 2 63.t odd 6 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(63, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 1$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T$$
$37$ $$T + 2$$
$41$ $$T$$
$43$ $$T + 2$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T$$
$67$ $$T - 2$$
$71$ $$T$$
$73$ $$T$$
$79$ $$T - 2$$
$83$ $$T$$
$89$ $$T$$
$97$ $$T$$