# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [676,1,Mod(675,676)]

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, 1]))

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

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

chi := DirichletCharacter("676.675");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$676 = 2^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 676.b (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: no Analytic conductor: $$0.337367948540$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ 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: $C_4\times S_3$ Artin field: Galois closure of 12.0.458793060873472.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
675.1
 1.00000i − 1.00000i
1.00000i 0 −1.00000 1.00000i 0 0 1.00000i 1.00000 1.00000
675.2 1.00000i 0 −1.00000 1.00000i 0 0 1.00000i 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 $$\Q(\sqrt{-1})$$
13.b even 2 1 inner
52.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 676.1.b.a 2
4.b odd 2 1 CM 676.1.b.a 2
13.b even 2 1 inner 676.1.b.a 2
13.c even 3 2 676.1.i.a 4
13.d odd 4 1 676.1.c.a 1
13.d odd 4 1 676.1.c.b 1
13.e even 6 2 676.1.i.a 4
13.f odd 12 2 52.1.j.a 2
13.f odd 12 2 676.1.j.a 2
39.k even 12 2 468.1.br.a 2
52.b odd 2 1 inner 676.1.b.a 2
52.f even 4 1 676.1.c.a 1
52.f even 4 1 676.1.c.b 1
52.i odd 6 2 676.1.i.a 4
52.j odd 6 2 676.1.i.a 4
52.l even 12 2 52.1.j.a 2
52.l even 12 2 676.1.j.a 2
65.o even 12 2 1300.1.w.a 4
65.s odd 12 2 1300.1.bc.a 2
65.t even 12 2 1300.1.w.a 4
91.w even 12 2 2548.1.q.a 2
91.x odd 12 2 2548.1.bi.b 2
91.ba even 12 2 2548.1.bi.a 2
91.bc even 12 2 2548.1.bn.a 2
91.bd odd 12 2 2548.1.q.b 2
104.u even 12 2 832.1.bb.a 2
104.x odd 12 2 832.1.bb.a 2
156.v odd 12 2 468.1.br.a 2
208.be odd 12 2 3328.1.v.b 4
208.bf even 12 2 3328.1.v.b 4
208.bk even 12 2 3328.1.v.b 4
208.bl odd 12 2 3328.1.v.b 4
260.bc even 12 2 1300.1.bc.a 2
260.be odd 12 2 1300.1.w.a 4
260.bl odd 12 2 1300.1.w.a 4
364.bt even 12 2 2548.1.q.b 2
364.bv odd 12 2 2548.1.bn.a 2
364.bz odd 12 2 2548.1.bi.a 2
364.ca even 12 2 2548.1.bi.b 2
364.cg odd 12 2 2548.1.q.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.1.j.a 2 13.f odd 12 2
52.1.j.a 2 52.l even 12 2
468.1.br.a 2 39.k even 12 2
468.1.br.a 2 156.v odd 12 2
676.1.b.a 2 1.a even 1 1 trivial
676.1.b.a 2 4.b odd 2 1 CM
676.1.b.a 2 13.b even 2 1 inner
676.1.b.a 2 52.b odd 2 1 inner
676.1.c.a 1 13.d odd 4 1
676.1.c.a 1 52.f even 4 1
676.1.c.b 1 13.d odd 4 1
676.1.c.b 1 52.f even 4 1
676.1.i.a 4 13.c even 3 2
676.1.i.a 4 13.e even 6 2
676.1.i.a 4 52.i odd 6 2
676.1.i.a 4 52.j odd 6 2
676.1.j.a 2 13.f odd 12 2
676.1.j.a 2 52.l even 12 2
832.1.bb.a 2 104.u even 12 2
832.1.bb.a 2 104.x odd 12 2
1300.1.w.a 4 65.o even 12 2
1300.1.w.a 4 65.t even 12 2
1300.1.w.a 4 260.be odd 12 2
1300.1.w.a 4 260.bl odd 12 2
1300.1.bc.a 2 65.s odd 12 2
1300.1.bc.a 2 260.bc even 12 2
2548.1.q.a 2 91.w even 12 2
2548.1.q.a 2 364.cg odd 12 2
2548.1.q.b 2 91.bd odd 12 2
2548.1.q.b 2 364.bt even 12 2
2548.1.bi.a 2 91.ba even 12 2
2548.1.bi.a 2 364.bz odd 12 2
2548.1.bi.b 2 91.x odd 12 2
2548.1.bi.b 2 364.ca even 12 2
2548.1.bn.a 2 91.bc even 12 2
2548.1.bn.a 2 364.bv odd 12 2
3328.1.v.b 4 208.be odd 12 2
3328.1.v.b 4 208.bf even 12 2
3328.1.v.b 4 208.bk even 12 2
3328.1.v.b 4 208.bl odd 12 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(676, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$(T - 1)^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$(T + 1)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 1$$
$41$ $$T^{2} + 1$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$(T + 1)^{2}$$
$59$ $$T^{2}$$
$61$ $$(T + 1)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 1$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} + 4$$
$97$ $$T^{2} + 4$$