# Properties

 Label 1156.1.d.a Level $1156$ Weight $1$ Character orbit 1156.d Analytic conductor $0.577$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -4 Inner twists $4$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1156,1,Mod(1155,1156)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1156, 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("1156.1155");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1156 = 2^{2} \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1156.d (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: $$0.576919154604$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 68) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.19652.1 Artin image: $\SD_{16}$ Artin field: Galois closure of 8.2.26261675072.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - \beta q^{5} + q^{8} - q^{9} +O(q^{10})$$ q + q^2 + q^4 - b * q^5 + q^8 - q^9 $$q + q^{2} + q^{4} - \beta q^{5} + q^{8} - q^{9} - \beta q^{10} + q^{16} - q^{18} - \beta q^{20} - q^{25} + \beta q^{29} + q^{32} - q^{36} + \beta q^{37} - \beta q^{40} - \beta q^{41} + \beta q^{45} - q^{49} - q^{50} + \beta q^{58} + \beta q^{61} + q^{64} - q^{72} + \beta q^{73} + \beta q^{74} - \beta q^{80} + q^{81} - \beta q^{82} + \beta q^{90} - \beta q^{97} - q^{98} +O(q^{100})$$ q + q^2 + q^4 - b * q^5 + q^8 - q^9 - b * q^10 + q^16 - q^18 - b * q^20 - q^25 + b * q^29 + q^32 - q^36 + b * q^37 - b * q^40 - b * q^41 + b * q^45 - q^49 - q^50 + b * q^58 + b * q^61 + q^64 - q^72 + b * q^73 + b * q^74 - b * q^80 + q^81 - b * q^82 + b * q^90 - b * q^97 - q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9} + 2 q^{16} - 2 q^{18} - 2 q^{25} + 2 q^{32} - 2 q^{36} - 2 q^{49} - 2 q^{50} + 2 q^{64} - 2 q^{72} + 2 q^{81} - 2 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 + 2 * q^16 - 2 * q^18 - 2 * q^25 + 2 * q^32 - 2 * q^36 - 2 * q^49 - 2 * q^50 + 2 * q^64 - 2 * q^72 + 2 * q^81 - 2 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1156\mathbb{Z}\right)^\times$$.

 $$n$$ $$579$$ $$581$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1155.1
 1.41421i − 1.41421i
1.00000 0 1.00000 1.41421i 0 0 1.00000 −1.00000 1.41421i
1155.2 1.00000 0 1.00000 1.41421i 0 0 1.00000 −1.00000 1.41421i
 $$n$$: 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 $$\Q(\sqrt{-1})$$
17.b even 2 1 inner
68.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1156.1.d.a 2
4.b odd 2 1 CM 1156.1.d.a 2
17.b even 2 1 inner 1156.1.d.a 2
17.c even 4 2 1156.1.c.b 2
17.d even 8 2 68.1.f.a 2
17.d even 8 2 1156.1.f.b 2
17.e odd 16 8 1156.1.g.b 8
51.g odd 8 2 612.1.l.a 2
68.d odd 2 1 inner 1156.1.d.a 2
68.f odd 4 2 1156.1.c.b 2
68.g odd 8 2 68.1.f.a 2
68.g odd 8 2 1156.1.f.b 2
68.i even 16 8 1156.1.g.b 8
85.k odd 8 2 1700.1.n.a 2
85.m even 8 2 1700.1.p.a 2
85.n odd 8 2 1700.1.n.b 2
119.l odd 8 2 3332.1.m.b 2
119.q even 24 4 3332.1.bc.c 4
119.r odd 24 4 3332.1.bc.b 4
136.o even 8 2 1088.1.p.a 2
136.p odd 8 2 1088.1.p.a 2
204.p even 8 2 612.1.l.a 2
340.w even 8 2 1700.1.n.b 2
340.z even 8 2 1700.1.n.a 2
340.ba odd 8 2 1700.1.p.a 2
476.w even 8 2 3332.1.m.b 2
476.bg odd 24 4 3332.1.bc.c 4
476.bj even 24 4 3332.1.bc.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.1.f.a 2 17.d even 8 2
68.1.f.a 2 68.g odd 8 2
612.1.l.a 2 51.g odd 8 2
612.1.l.a 2 204.p even 8 2
1088.1.p.a 2 136.o even 8 2
1088.1.p.a 2 136.p odd 8 2
1156.1.c.b 2 17.c even 4 2
1156.1.c.b 2 68.f odd 4 2
1156.1.d.a 2 1.a even 1 1 trivial
1156.1.d.a 2 4.b odd 2 1 CM
1156.1.d.a 2 17.b even 2 1 inner
1156.1.d.a 2 68.d odd 2 1 inner
1156.1.f.b 2 17.d even 8 2
1156.1.f.b 2 68.g odd 8 2
1156.1.g.b 8 17.e odd 16 8
1156.1.g.b 8 68.i even 16 8
1700.1.n.a 2 85.k odd 8 2
1700.1.n.a 2 340.z even 8 2
1700.1.n.b 2 85.n odd 8 2
1700.1.n.b 2 340.w even 8 2
1700.1.p.a 2 85.m even 8 2
1700.1.p.a 2 340.ba odd 8 2
3332.1.m.b 2 119.l odd 8 2
3332.1.m.b 2 476.w even 8 2
3332.1.bc.b 4 119.r odd 24 4
3332.1.bc.b 4 476.bj even 24 4
3332.1.bc.c 4 119.q even 24 4
3332.1.bc.c 4 476.bg odd 24 4

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1156, [\chi])$$.

## Hecke characteristic polynomials

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