# Properties

 Label 676.1.c.b Level $676$ Weight $1$ Character orbit 676.c Self dual yes Analytic conductor $0.337$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -4 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [676,1,Mod(339,676)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(676, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("676.339");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$676 = 2^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 676.c (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: $$0.337367948540$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 52) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.676.1 Artin image: $S_3$ Artin field: Galois closure of 3.1.676.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - q^{5} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^4 - q^5 + q^8 + q^9 $$q + q^{2} + q^{4} - q^{5} + q^{8} + q^{9} - q^{10} + q^{16} - q^{17} + q^{18} - q^{20} - q^{29} + q^{32} - q^{34} + q^{36} - q^{37} - q^{40} - q^{41} - q^{45} + q^{49} - q^{53} - q^{58} - q^{61} + q^{64} - q^{68} + q^{72} - q^{73} - q^{74} - q^{80} + q^{81} - q^{82} + q^{85} + 2 q^{89} - q^{90} + 2 q^{97} + q^{98}+O(q^{100})$$ q + q^2 + q^4 - q^5 + q^8 + q^9 - q^10 + q^16 - q^17 + q^18 - q^20 - q^29 + q^32 - q^34 + q^36 - q^37 - q^40 - q^41 - q^45 + q^49 - q^53 - q^58 - q^61 + q^64 - q^68 + q^72 - q^73 - q^74 - q^80 + q^81 - q^82 + q^85 + 2 * q^89 - q^90 + 2 * q^97 + q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/676\mathbb{Z}\right)^\times$$.

 $$n$$ $$339$$ $$509$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
339.1
 0
1.00000 0 1.00000 −1.00000 0 0 1.00000 1.00000 −1.00000
 $$n$$: 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 676.1.c.b 1
4.b odd 2 1 CM 676.1.c.b 1
13.b even 2 1 676.1.c.a 1
13.c even 3 2 52.1.j.a 2
13.d odd 4 2 676.1.b.a 2
13.e even 6 2 676.1.j.a 2
13.f odd 12 4 676.1.i.a 4
39.i odd 6 2 468.1.br.a 2
52.b odd 2 1 676.1.c.a 1
52.f even 4 2 676.1.b.a 2
52.i odd 6 2 676.1.j.a 2
52.j odd 6 2 52.1.j.a 2
52.l even 12 4 676.1.i.a 4
65.n even 6 2 1300.1.bc.a 2
65.q odd 12 4 1300.1.w.a 4
91.g even 3 2 2548.1.q.b 2
91.h even 3 2 2548.1.bi.b 2
91.m odd 6 2 2548.1.q.a 2
91.n odd 6 2 2548.1.bn.a 2
91.v odd 6 2 2548.1.bi.a 2
104.n odd 6 2 832.1.bb.a 2
104.r even 6 2 832.1.bb.a 2
156.p even 6 2 468.1.br.a 2
208.bg odd 12 4 3328.1.v.b 4
208.bj even 12 4 3328.1.v.b 4
260.v odd 6 2 1300.1.bc.a 2
260.bj even 12 4 1300.1.w.a 4
364.q odd 6 2 2548.1.q.b 2
364.v even 6 2 2548.1.bn.a 2
364.ba even 6 2 2548.1.bi.a 2
364.bi odd 6 2 2548.1.bi.b 2
364.br even 6 2 2548.1.q.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.1.j.a 2 13.c even 3 2
52.1.j.a 2 52.j odd 6 2
468.1.br.a 2 39.i odd 6 2
468.1.br.a 2 156.p even 6 2
676.1.b.a 2 13.d odd 4 2
676.1.b.a 2 52.f even 4 2
676.1.c.a 1 13.b even 2 1
676.1.c.a 1 52.b odd 2 1
676.1.c.b 1 1.a even 1 1 trivial
676.1.c.b 1 4.b odd 2 1 CM
676.1.i.a 4 13.f odd 12 4
676.1.i.a 4 52.l even 12 4
676.1.j.a 2 13.e even 6 2
676.1.j.a 2 52.i odd 6 2
832.1.bb.a 2 104.n odd 6 2
832.1.bb.a 2 104.r even 6 2
1300.1.w.a 4 65.q odd 12 4
1300.1.w.a 4 260.bj even 12 4
1300.1.bc.a 2 65.n even 6 2
1300.1.bc.a 2 260.v odd 6 2
2548.1.q.a 2 91.m odd 6 2
2548.1.q.a 2 364.br even 6 2
2548.1.q.b 2 91.g even 3 2
2548.1.q.b 2 364.q odd 6 2
2548.1.bi.a 2 91.v odd 6 2
2548.1.bi.a 2 364.ba even 6 2
2548.1.bi.b 2 91.h even 3 2
2548.1.bi.b 2 364.bi odd 6 2
2548.1.bn.a 2 91.n odd 6 2
2548.1.bn.a 2 364.v even 6 2
3328.1.v.b 4 208.bg odd 12 4
3328.1.v.b 4 208.bj even 12 4

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 1$$ acting on $$S_{1}^{\mathrm{new}}(676, [\chi])$$.

## Hecke characteristic polynomials

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