# Properties

 Label 1183.1.bd.a Level $1183$ Weight $1$ Character orbit 1183.bd Analytic conductor $0.590$ Analytic rank $0$ Dimension $8$ Projective image $A_{4}$ CM/RM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1183 = 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1183.bd (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.590393909945$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$A_{4}$$ Projective field: Galois closure of 4.0.8281.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{11} q^{2} + q^{3} -\zeta_{24}^{7} q^{5} -\zeta_{24}^{11} q^{6} -\zeta_{24}^{5} q^{7} + \zeta_{24}^{9} q^{8} +O(q^{10})$$ $$q -\zeta_{24}^{11} q^{2} + q^{3} -\zeta_{24}^{7} q^{5} -\zeta_{24}^{11} q^{6} -\zeta_{24}^{5} q^{7} + \zeta_{24}^{9} q^{8} -\zeta_{24}^{6} q^{10} -\zeta_{24}^{9} q^{11} -\zeta_{24}^{4} q^{14} -\zeta_{24}^{7} q^{15} + \zeta_{24}^{8} q^{16} + \zeta_{24}^{10} q^{17} + \zeta_{24}^{3} q^{19} -\zeta_{24}^{5} q^{21} -\zeta_{24}^{8} q^{22} + \zeta_{24}^{2} q^{23} + \zeta_{24}^{9} q^{24} - q^{27} -\zeta_{24}^{6} q^{30} + \zeta_{24}^{11} q^{31} -\zeta_{24}^{9} q^{33} + \zeta_{24}^{9} q^{34} - q^{35} + \zeta_{24}^{5} q^{37} + \zeta_{24}^{2} q^{38} + \zeta_{24}^{4} q^{40} -\zeta_{24}^{4} q^{42} + \zeta_{24} q^{46} + \zeta_{24} q^{47} + \zeta_{24}^{8} q^{48} + \zeta_{24}^{10} q^{49} + \zeta_{24}^{10} q^{51} + \zeta_{24}^{8} q^{53} + \zeta_{24}^{11} q^{54} -\zeta_{24}^{4} q^{55} + \zeta_{24}^{2} q^{56} + \zeta_{24}^{3} q^{57} -\zeta_{24} q^{59} + q^{61} + \zeta_{24}^{10} q^{62} -\zeta_{24}^{6} q^{64} -\zeta_{24}^{8} q^{66} -\zeta_{24}^{3} q^{67} + \zeta_{24}^{2} q^{69} + \zeta_{24}^{11} q^{70} -\zeta_{24}^{5} q^{73} + \zeta_{24}^{4} q^{74} -\zeta_{24}^{2} q^{77} + \zeta_{24}^{4} q^{79} + \zeta_{24}^{3} q^{80} - q^{81} + \zeta_{24}^{5} q^{85} + \zeta_{24}^{6} q^{88} -\zeta_{24}^{5} q^{89} + \zeta_{24}^{11} q^{93} + q^{94} -\zeta_{24}^{10} q^{95} + 2 \zeta_{24}^{11} q^{97} + \zeta_{24}^{9} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{3} + O(q^{10})$$ $$8q + 8q^{3} - 4q^{14} - 4q^{16} + 4q^{22} - 8q^{27} - 8q^{35} + 4q^{40} - 4q^{42} - 4q^{48} - 4q^{53} - 4q^{55} + 8q^{61} + 4q^{66} + 4q^{74} + 4q^{79} - 8q^{81} + 8q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times$$.

 $$n$$ $$339$$ $$1016$$ $$\chi(n)$$ $$-\zeta_{24}^{4}$$ $$-\zeta_{24}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
249.1
 −0.258819 + 0.965926i 0.258819 − 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i −0.965926 − 0.258819i 0.965926 + 0.258819i −0.258819 − 0.965926i 0.258819 + 0.965926i
−0.258819 0.965926i 1.00000 0 −0.965926 0.258819i −0.258819 0.965926i 0.965926 0.258819i −0.707107 0.707107i 0 1.00000i
249.2 0.258819 + 0.965926i 1.00000 0 0.965926 + 0.258819i 0.258819 + 0.965926i −0.965926 + 0.258819i 0.707107 + 0.707107i 0 1.00000i
695.1 −0.965926 0.258819i 1.00000 0 −0.258819 0.965926i −0.965926 0.258819i 0.258819 0.965926i 0.707107 + 0.707107i 0 1.00000i
695.2 0.965926 + 0.258819i 1.00000 0 0.258819 + 0.965926i 0.965926 + 0.258819i −0.258819 + 0.965926i −0.707107 0.707107i 0 1.00000i
1103.1 −0.965926 + 0.258819i 1.00000 0 −0.258819 + 0.965926i −0.965926 + 0.258819i 0.258819 + 0.965926i 0.707107 0.707107i 0 1.00000i
1103.2 0.965926 0.258819i 1.00000 0 0.258819 0.965926i 0.965926 0.258819i −0.258819 0.965926i −0.707107 + 0.707107i 0 1.00000i
1164.1 −0.258819 + 0.965926i 1.00000 0 −0.965926 + 0.258819i −0.258819 + 0.965926i 0.965926 + 0.258819i −0.707107 + 0.707107i 0 1.00000i
1164.2 0.258819 0.965926i 1.00000 0 0.965926 0.258819i 0.258819 0.965926i −0.965926 0.258819i 0.707107 0.707107i 0 1.00000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1164.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.d odd 4 2 inner
91.g even 3 1 inner
91.u even 6 1 inner
91.bd odd 12 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.1.bd.a 8
7.c even 3 1 1183.1.x.a 8
13.b even 2 1 inner 1183.1.bd.a 8
13.c even 3 1 1183.1.x.a 8
13.c even 3 1 1183.1.z.a 8
13.d odd 4 2 inner 1183.1.bd.a 8
13.e even 6 1 1183.1.x.a 8
13.e even 6 1 1183.1.z.a 8
13.f odd 12 2 1183.1.x.a 8
13.f odd 12 2 1183.1.z.a 8
91.g even 3 1 inner 1183.1.bd.a 8
91.h even 3 1 1183.1.z.a 8
91.k even 6 1 1183.1.z.a 8
91.r even 6 1 1183.1.x.a 8
91.u even 6 1 inner 1183.1.bd.a 8
91.x odd 12 2 1183.1.z.a 8
91.z odd 12 2 1183.1.x.a 8
91.bd odd 12 2 inner 1183.1.bd.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1183.1.x.a 8 7.c even 3 1
1183.1.x.a 8 13.c even 3 1
1183.1.x.a 8 13.e even 6 1
1183.1.x.a 8 13.f odd 12 2
1183.1.x.a 8 91.r even 6 1
1183.1.x.a 8 91.z odd 12 2
1183.1.z.a 8 13.c even 3 1
1183.1.z.a 8 13.e even 6 1
1183.1.z.a 8 13.f odd 12 2
1183.1.z.a 8 91.h even 3 1
1183.1.z.a 8 91.k even 6 1
1183.1.z.a 8 91.x odd 12 2
1183.1.bd.a 8 1.a even 1 1 trivial
1183.1.bd.a 8 13.b even 2 1 inner
1183.1.bd.a 8 13.d odd 4 2 inner
1183.1.bd.a 8 91.g even 3 1 inner
1183.1.bd.a 8 91.u even 6 1 inner
1183.1.bd.a 8 91.bd odd 12 2 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1183, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{4} + T^{8}$$
$3$ $$( -1 + T )^{8}$$
$5$ $$1 - T^{4} + T^{8}$$
$7$ $$1 - T^{4} + T^{8}$$
$11$ $$( 1 + T^{4} )^{2}$$
$13$ $$T^{8}$$
$17$ $$( 1 - T^{2} + T^{4} )^{2}$$
$19$ $$( 1 + T^{4} )^{2}$$
$23$ $$( 1 - T^{2} + T^{4} )^{2}$$
$29$ $$T^{8}$$
$31$ $$1 - T^{4} + T^{8}$$
$37$ $$1 - T^{4} + T^{8}$$
$41$ $$T^{8}$$
$43$ $$T^{8}$$
$47$ $$1 - T^{4} + T^{8}$$
$53$ $$( 1 + T + T^{2} )^{4}$$
$59$ $$1 - T^{4} + T^{8}$$
$61$ $$( -1 + T )^{8}$$
$67$ $$( 1 + T^{4} )^{2}$$
$71$ $$T^{8}$$
$73$ $$1 - T^{4} + T^{8}$$
$79$ $$( 1 - T + T^{2} )^{4}$$
$83$ $$T^{8}$$
$89$ $$1 - T^{4} + T^{8}$$
$97$ $$256 - 16 T^{4} + T^{8}$$