Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1183.z (of order \(12\), degree \(4\), 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
Artin image: $C_8.A_4$
Artin field: Galois closure of 32.0.515207889456069254383620937908064472169867121.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

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

Character values

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

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(\zeta_{24}^{8}\) \(\zeta_{24}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
268.1
−0.965926 0.258819i
0.965926 + 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.258819 0.965926i
0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.965926 + 0.258819i −0.500000 0.866025i 0 0.965926 0.258819i 0.707107 + 0.707107i 0.707107 0.707107i 0.707107 0.707107i 0 −0.866025 + 0.500000i
268.2 0.965926 0.258819i −0.500000 0.866025i 0 −0.965926 + 0.258819i −0.707107 0.707107i −0.707107 + 0.707107i −0.707107 + 0.707107i 0 −0.866025 + 0.500000i
408.1 −0.258819 0.965926i −0.500000 0.866025i 0 0.258819 + 0.965926i −0.707107 + 0.707107i −0.707107 0.707107i −0.707107 0.707107i 0 0.866025 0.500000i
408.2 0.258819 + 0.965926i −0.500000 0.866025i 0 −0.258819 0.965926i 0.707107 0.707107i 0.707107 + 0.707107i 0.707107 + 0.707107i 0 0.866025 0.500000i
606.1 −0.258819 + 0.965926i −0.500000 + 0.866025i 0 0.258819 0.965926i −0.707107 0.707107i −0.707107 + 0.707107i −0.707107 + 0.707107i 0 0.866025 + 0.500000i
606.2 0.258819 0.965926i −0.500000 + 0.866025i 0 −0.258819 + 0.965926i 0.707107 + 0.707107i 0.707107 0.707107i 0.707107 0.707107i 0 0.866025 + 0.500000i
746.1 −0.965926 0.258819i −0.500000 + 0.866025i 0 0.965926 + 0.258819i 0.707107 0.707107i 0.707107 + 0.707107i 0.707107 + 0.707107i 0 −0.866025 0.500000i
746.2 0.965926 + 0.258819i −0.500000 + 0.866025i 0 −0.965926 0.258819i −0.707107 + 0.707107i −0.707107 0.707107i −0.707107 0.707107i 0 −0.866025 0.500000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 746.2
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

Hecke characteristic polynomials

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