# Properties

 Label 1156.1.g.a Level $1156$ Weight $1$ Character orbit 1156.g Analytic conductor $0.577$ Analytic rank $0$ Dimension $4$ Projective image $D_{2}$ CM/RM discs -4, -68, 17 Inner twists $16$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1156,1,Mod(155,1156)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1156, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([4, 7]))

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

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

chi := DirichletCharacter("1156.155");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1156 = 2^{2} \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1156.g (of order $$8$$, degree $$4$$, 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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 68) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(i, \sqrt{17})$$ Artin image: $\OD_{32}$ Artin field: Galois closure of 16.8.732780301186512843008.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{8}^{3} q^{2} - \zeta_{8}^{2} q^{4} + \zeta_{8} q^{8} - \zeta_{8} q^{9} +O(q^{10})$$ q + z^3 * q^2 - z^2 * q^4 + z * q^8 - z * q^9 $$q + \zeta_{8}^{3} q^{2} - \zeta_{8}^{2} q^{4} + \zeta_{8} q^{8} - \zeta_{8} q^{9} - 2 \zeta_{8}^{2} q^{13} - q^{16} + q^{18} - \zeta_{8} q^{25} + 2 \zeta_{8} q^{26} - \zeta_{8}^{3} q^{32} + \zeta_{8}^{3} q^{36} - \zeta_{8}^{3} q^{49} + q^{50} - 2 q^{52} - 2 \zeta_{8}^{3} q^{53} + \zeta_{8}^{2} q^{64} - \zeta_{8}^{2} q^{72} + \zeta_{8}^{2} q^{81} + 2 \zeta_{8}^{2} q^{89} + \zeta_{8}^{2} q^{98} +O(q^{100})$$ q + z^3 * q^2 - z^2 * q^4 + z * q^8 - z * q^9 - 2*z^2 * q^13 - q^16 + q^18 - z * q^25 + 2*z * q^26 - z^3 * q^32 + z^3 * q^36 - z^3 * q^49 + q^50 - 2 * q^52 - 2*z^3 * q^53 + z^2 * q^64 - z^2 * q^72 + z^2 * q^81 + 2*z^2 * q^89 + z^2 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 4 q^{16} + 4 q^{18} + 4 q^{50} - 8 q^{52}+O(q^{100})$$ 4 * q - 4 * q^16 + 4 * q^18 + 4 * q^50 - 8 * q^52

## Character values

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

 $$n$$ $$579$$ $$581$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{8}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
155.1
 −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i
0.707107 + 0.707107i 0 1.00000i 0 0 0 −0.707107 + 0.707107i 0.707107 0.707107i 0
179.1 0.707107 0.707107i 0 1.00000i 0 0 0 −0.707107 0.707107i 0.707107 + 0.707107i 0
399.1 −0.707107 + 0.707107i 0 1.00000i 0 0 0 0.707107 + 0.707107i −0.707107 0.707107i 0
423.1 −0.707107 0.707107i 0 1.00000i 0 0 0 0.707107 0.707107i −0.707107 + 0.707107i 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
17.b even 2 1 RM by $$\Q(\sqrt{17})$$
68.d odd 2 1 CM by $$\Q(\sqrt{-17})$$
17.c even 4 2 inner
17.d even 8 4 inner
68.f odd 4 2 inner
68.g odd 8 4 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1156.1.g.a 4
4.b odd 2 1 CM 1156.1.g.a 4
17.b even 2 1 RM 1156.1.g.a 4
17.c even 4 2 inner 1156.1.g.a 4
17.d even 8 4 inner 1156.1.g.a 4
17.e odd 16 2 68.1.d.a 1
17.e odd 16 2 1156.1.c.a 1
17.e odd 16 4 1156.1.f.a 2
51.i even 16 2 612.1.e.a 1
68.d odd 2 1 CM 1156.1.g.a 4
68.f odd 4 2 inner 1156.1.g.a 4
68.g odd 8 4 inner 1156.1.g.a 4
68.i even 16 2 68.1.d.a 1
68.i even 16 2 1156.1.c.a 1
68.i even 16 4 1156.1.f.a 2
85.o even 16 2 1700.1.d.b 2
85.p odd 16 2 1700.1.h.d 1
85.r even 16 2 1700.1.d.b 2
119.p even 16 2 3332.1.g.a 1
119.s even 48 4 3332.1.o.d 2
119.t odd 48 4 3332.1.o.c 2
136.q odd 16 2 1088.1.g.a 1
136.s even 16 2 1088.1.g.a 1
204.t odd 16 2 612.1.e.a 1
340.bc odd 16 2 1700.1.d.b 2
340.bg even 16 2 1700.1.h.d 1
340.bj odd 16 2 1700.1.d.b 2
476.bf odd 16 2 3332.1.g.a 1
476.bk odd 48 4 3332.1.o.d 2
476.bm even 48 4 3332.1.o.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.1.d.a 1 17.e odd 16 2
68.1.d.a 1 68.i even 16 2
612.1.e.a 1 51.i even 16 2
612.1.e.a 1 204.t odd 16 2
1088.1.g.a 1 136.q odd 16 2
1088.1.g.a 1 136.s even 16 2
1156.1.c.a 1 17.e odd 16 2
1156.1.c.a 1 68.i even 16 2
1156.1.f.a 2 17.e odd 16 4
1156.1.f.a 2 68.i even 16 4
1156.1.g.a 4 1.a even 1 1 trivial
1156.1.g.a 4 4.b odd 2 1 CM
1156.1.g.a 4 17.b even 2 1 RM
1156.1.g.a 4 17.c even 4 2 inner
1156.1.g.a 4 17.d even 8 4 inner
1156.1.g.a 4 68.d odd 2 1 CM
1156.1.g.a 4 68.f odd 4 2 inner
1156.1.g.a 4 68.g odd 8 4 inner
1700.1.d.b 2 85.o even 16 2
1700.1.d.b 2 85.r even 16 2
1700.1.d.b 2 340.bc odd 16 2
1700.1.d.b 2 340.bj odd 16 2
1700.1.h.d 1 85.p odd 16 2
1700.1.h.d 1 340.bg even 16 2
3332.1.g.a 1 119.p even 16 2
3332.1.g.a 1 476.bf odd 16 2
3332.1.o.c 2 119.t odd 48 4
3332.1.o.c 2 476.bm even 48 4
3332.1.o.d 2 119.s even 48 4
3332.1.o.d 2 476.bk odd 48 4

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{1}^{\mathrm{new}}(1156, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$(T^{2} + 4)^{2}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4} + 16$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$(T^{2} + 4)^{2}$$
$97$ $$T^{4}$$