# Properties

 Label 624.1.l.a Level $624$ Weight $1$ Character orbit 624.l Self dual yes Analytic conductor $0.311$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -3, -39, 13 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,1,Mod(545,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("624.545");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$624 = 2^{4} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 624.l (of order $$2$$, degree $$1$$, not 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: $$0.311416567883$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{-3}, \sqrt{13})$$ Artin image: $D_4$ Artin field: Galois closure of 4.0.1872.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{9}+O(q^{10})$$ q + q^3 + q^9 $$q + q^{3} + q^{9} - q^{13} - q^{25} + q^{27} - q^{39} - 2 q^{43} + q^{49} - 2 q^{61} - q^{75} + 2 q^{79} + q^{81}+O(q^{100})$$ q + q^3 + q^9 - q^13 - q^25 + q^27 - q^39 - 2 * q^43 + q^49 - 2 * q^61 - q^75 + 2 * q^79 + q^81

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$$.

 $$n$$ $$79$$ $$145$$ $$209$$ $$469$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
545.1
 0
0 1.00000 0 0 0 0 0 1.00000 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 $$\Q(\sqrt{-3})$$
13.b even 2 1 RM by $$\Q(\sqrt{13})$$
39.d odd 2 1 CM by $$\Q(\sqrt{-39})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.1.l.a 1
3.b odd 2 1 CM 624.1.l.a 1
4.b odd 2 1 39.1.d.a 1
8.b even 2 1 2496.1.l.a 1
8.d odd 2 1 2496.1.l.b 1
12.b even 2 1 39.1.d.a 1
13.b even 2 1 RM 624.1.l.a 1
20.d odd 2 1 975.1.g.a 1
20.e even 4 2 975.1.e.a 2
24.f even 2 1 2496.1.l.b 1
24.h odd 2 1 2496.1.l.a 1
28.d even 2 1 1911.1.h.a 1
28.f even 6 2 1911.1.w.a 2
28.g odd 6 2 1911.1.w.b 2
36.f odd 6 2 1053.1.n.b 2
36.h even 6 2 1053.1.n.b 2
39.d odd 2 1 CM 624.1.l.a 1
52.b odd 2 1 39.1.d.a 1
52.f even 4 2 507.1.c.a 1
52.i odd 6 2 507.1.h.a 2
52.j odd 6 2 507.1.h.a 2
52.l even 12 4 507.1.i.a 2
60.h even 2 1 975.1.g.a 1
60.l odd 4 2 975.1.e.a 2
84.h odd 2 1 1911.1.h.a 1
84.j odd 6 2 1911.1.w.a 2
84.n even 6 2 1911.1.w.b 2
104.e even 2 1 2496.1.l.a 1
104.h odd 2 1 2496.1.l.b 1
156.h even 2 1 39.1.d.a 1
156.l odd 4 2 507.1.c.a 1
156.p even 6 2 507.1.h.a 2
156.r even 6 2 507.1.h.a 2
156.v odd 12 4 507.1.i.a 2
260.g odd 2 1 975.1.g.a 1
260.p even 4 2 975.1.e.a 2
312.b odd 2 1 2496.1.l.a 1
312.h even 2 1 2496.1.l.b 1
364.h even 2 1 1911.1.h.a 1
364.x even 6 2 1911.1.w.a 2
364.bl odd 6 2 1911.1.w.b 2
468.x even 6 2 1053.1.n.b 2
468.bg odd 6 2 1053.1.n.b 2
780.d even 2 1 975.1.g.a 1
780.w odd 4 2 975.1.e.a 2
1092.d odd 2 1 1911.1.h.a 1
1092.by even 6 2 1911.1.w.b 2
1092.ct odd 6 2 1911.1.w.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.1.d.a 1 4.b odd 2 1
39.1.d.a 1 12.b even 2 1
39.1.d.a 1 52.b odd 2 1
39.1.d.a 1 156.h even 2 1
507.1.c.a 1 52.f even 4 2
507.1.c.a 1 156.l odd 4 2
507.1.h.a 2 52.i odd 6 2
507.1.h.a 2 52.j odd 6 2
507.1.h.a 2 156.p even 6 2
507.1.h.a 2 156.r even 6 2
507.1.i.a 2 52.l even 12 4
507.1.i.a 2 156.v odd 12 4
624.1.l.a 1 1.a even 1 1 trivial
624.1.l.a 1 3.b odd 2 1 CM
624.1.l.a 1 13.b even 2 1 RM
624.1.l.a 1 39.d odd 2 1 CM
975.1.e.a 2 20.e even 4 2
975.1.e.a 2 60.l odd 4 2
975.1.e.a 2 260.p even 4 2
975.1.e.a 2 780.w odd 4 2
975.1.g.a 1 20.d odd 2 1
975.1.g.a 1 60.h even 2 1
975.1.g.a 1 260.g odd 2 1
975.1.g.a 1 780.d even 2 1
1053.1.n.b 2 36.f odd 6 2
1053.1.n.b 2 36.h even 6 2
1053.1.n.b 2 468.x even 6 2
1053.1.n.b 2 468.bg odd 6 2
1911.1.h.a 1 28.d even 2 1
1911.1.h.a 1 84.h odd 2 1
1911.1.h.a 1 364.h even 2 1
1911.1.h.a 1 1092.d odd 2 1
1911.1.w.a 2 28.f even 6 2
1911.1.w.a 2 84.j odd 6 2
1911.1.w.a 2 364.x even 6 2
1911.1.w.a 2 1092.ct odd 6 2
1911.1.w.b 2 28.g odd 6 2
1911.1.w.b 2 84.n even 6 2
1911.1.w.b 2 364.bl odd 6 2
1911.1.w.b 2 1092.by even 6 2
2496.1.l.a 1 8.b even 2 1
2496.1.l.a 1 24.h odd 2 1
2496.1.l.a 1 104.e even 2 1
2496.1.l.a 1 312.b odd 2 1
2496.1.l.b 1 8.d odd 2 1
2496.1.l.b 1 24.f even 2 1
2496.1.l.b 1 104.h odd 2 1
2496.1.l.b 1 312.h even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(624, [\chi])$$.

## Hecke characteristic polynomials

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