# Properties

 Label 1156.1.g.b Level $1156$ Weight $1$ Character orbit 1156.g Analytic conductor $0.577$ Analytic rank $0$ Dimension $8$ Projective image $D_{4}$ CM discriminant -4 Inner twists $16$

# Related objects

## Newspace parameters

 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

 Self dual: no Analytic conductor: $$0.576919154604$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{8})$$ Coefficient field: $$\Q(\zeta_{16})$$ Defining polynomial: $$x^{8} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 68) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.19652.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{16}^{6} q^{2} -\zeta_{16}^{4} q^{4} + ( -\zeta_{16}^{3} - \zeta_{16}^{7} ) q^{5} + \zeta_{16}^{2} q^{8} + \zeta_{16}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{16}^{6} q^{2} -\zeta_{16}^{4} q^{4} + ( -\zeta_{16}^{3} - \zeta_{16}^{7} ) q^{5} + \zeta_{16}^{2} q^{8} + \zeta_{16}^{2} q^{9} + ( \zeta_{16} + \zeta_{16}^{5} ) q^{10} - q^{16} - q^{18} + ( -\zeta_{16}^{3} + \zeta_{16}^{7} ) q^{20} -\zeta_{16}^{2} q^{25} + ( -\zeta_{16}^{3} - \zeta_{16}^{7} ) q^{29} -\zeta_{16}^{6} q^{32} -\zeta_{16}^{6} q^{36} + ( \zeta_{16}^{3} - \zeta_{16}^{7} ) q^{37} + ( \zeta_{16} - \zeta_{16}^{5} ) q^{40} + ( -\zeta_{16} - \zeta_{16}^{5} ) q^{41} + ( \zeta_{16} - \zeta_{16}^{5} ) q^{45} + \zeta_{16}^{6} q^{49} + q^{50} + ( \zeta_{16} + \zeta_{16}^{5} ) q^{58} + ( -\zeta_{16} - \zeta_{16}^{5} ) q^{61} + \zeta_{16}^{4} q^{64} + \zeta_{16}^{4} q^{72} + ( \zeta_{16}^{3} + \zeta_{16}^{7} ) q^{73} + ( -\zeta_{16} + \zeta_{16}^{5} ) q^{74} + ( \zeta_{16}^{3} + \zeta_{16}^{7} ) q^{80} + \zeta_{16}^{4} q^{81} + ( \zeta_{16}^{3} - \zeta_{16}^{7} ) q^{82} + ( \zeta_{16}^{3} + \zeta_{16}^{7} ) q^{90} + ( \zeta_{16}^{3} + \zeta_{16}^{7} ) q^{97} -\zeta_{16}^{4} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + O(q^{10})$$ $$8 q - 8 q^{16} - 8 q^{18} + 8 q^{50} + O(q^{100})$$

## Character values

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

 $$n$$ $$579$$ $$581$$ $$\chi(n)$$ $$-1$$ $$\zeta_{16}^{2}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
155.1
 −0.923880 + 0.382683i 0.923880 − 0.382683i −0.923880 − 0.382683i 0.923880 + 0.382683i −0.382683 + 0.923880i 0.382683 − 0.923880i −0.382683 − 0.923880i 0.382683 + 0.923880i
−0.707107 0.707107i 0 1.00000i −0.541196 1.30656i 0 0 0.707107 0.707107i 0.707107 0.707107i −0.541196 + 1.30656i
155.2 −0.707107 0.707107i 0 1.00000i 0.541196 + 1.30656i 0 0 0.707107 0.707107i 0.707107 0.707107i 0.541196 1.30656i
179.1 −0.707107 + 0.707107i 0 1.00000i −0.541196 + 1.30656i 0 0 0.707107 + 0.707107i 0.707107 + 0.707107i −0.541196 1.30656i
179.2 −0.707107 + 0.707107i 0 1.00000i 0.541196 1.30656i 0 0 0.707107 + 0.707107i 0.707107 + 0.707107i 0.541196 + 1.30656i
399.1 0.707107 0.707107i 0 1.00000i −1.30656 0.541196i 0 0 −0.707107 0.707107i −0.707107 0.707107i −1.30656 + 0.541196i
399.2 0.707107 0.707107i 0 1.00000i 1.30656 + 0.541196i 0 0 −0.707107 0.707107i −0.707107 0.707107i 1.30656 0.541196i
423.1 0.707107 + 0.707107i 0 1.00000i −1.30656 + 0.541196i 0 0 −0.707107 + 0.707107i −0.707107 + 0.707107i −1.30656 0.541196i
423.2 0.707107 + 0.707107i 0 1.00000i 1.30656 0.541196i 0 0 −0.707107 + 0.707107i −0.707107 + 0.707107i 1.30656 + 0.541196i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 423.2 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
17.c even 4 2 inner
17.d even 8 4 inner
68.d odd 2 1 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.b 8
4.b odd 2 1 CM 1156.1.g.b 8
17.b even 2 1 inner 1156.1.g.b 8
17.c even 4 2 inner 1156.1.g.b 8
17.d even 8 4 inner 1156.1.g.b 8
17.e odd 16 2 68.1.f.a 2
17.e odd 16 2 1156.1.c.b 2
17.e odd 16 2 1156.1.d.a 2
17.e odd 16 2 1156.1.f.b 2
51.i even 16 2 612.1.l.a 2
68.d odd 2 1 inner 1156.1.g.b 8
68.f odd 4 2 inner 1156.1.g.b 8
68.g odd 8 4 inner 1156.1.g.b 8
68.i even 16 2 68.1.f.a 2
68.i even 16 2 1156.1.c.b 2
68.i even 16 2 1156.1.d.a 2
68.i even 16 2 1156.1.f.b 2
85.o even 16 1 1700.1.n.a 2
85.o even 16 1 1700.1.n.b 2
85.p odd 16 2 1700.1.p.a 2
85.r even 16 1 1700.1.n.a 2
85.r even 16 1 1700.1.n.b 2
119.p even 16 2 3332.1.m.b 2
119.s even 48 4 3332.1.bc.b 4
119.t odd 48 4 3332.1.bc.c 4
136.q odd 16 2 1088.1.p.a 2
136.s even 16 2 1088.1.p.a 2
204.t odd 16 2 612.1.l.a 2
340.bc odd 16 1 1700.1.n.a 2
340.bc odd 16 1 1700.1.n.b 2
340.bg even 16 2 1700.1.p.a 2
340.bj odd 16 1 1700.1.n.a 2
340.bj odd 16 1 1700.1.n.b 2
476.bf odd 16 2 3332.1.m.b 2
476.bk odd 48 4 3332.1.bc.b 4
476.bm even 48 4 3332.1.bc.c 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 16$$ acting on $$S_{1}^{\mathrm{new}}(1156, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$16 + T^{8}$$
$7$ $$T^{8}$$
$11$ $$T^{8}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$T^{8}$$
$29$ $$16 + T^{8}$$
$31$ $$T^{8}$$
$37$ $$16 + T^{8}$$
$41$ $$16 + T^{8}$$
$43$ $$T^{8}$$
$47$ $$T^{8}$$
$53$ $$T^{8}$$
$59$ $$T^{8}$$
$61$ $$16 + T^{8}$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$16 + T^{8}$$
$79$ $$T^{8}$$
$83$ $$T^{8}$$
$89$ $$T^{8}$$
$97$ $$16 + T^{8}$$