Properties

Label 1296.1.x.a
Level $1296$
Weight $1$
Character orbit 1296.x
Analytic conductor $0.647$
Analytic rank $0$
Dimension $8$
Projective image $S_{4}$
CM/RM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1296.x (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.646788256372\)
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: no (minimal twist has level 432)
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.55296.2

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{24} q^{2} + \zeta_{24}^{2} q^{4} + \zeta_{24}^{5} q^{5} -\zeta_{24}^{10} q^{7} -\zeta_{24}^{3} q^{8} +O(q^{10})\) \( q -\zeta_{24} q^{2} + \zeta_{24}^{2} q^{4} + \zeta_{24}^{5} q^{5} -\zeta_{24}^{10} q^{7} -\zeta_{24}^{3} q^{8} -\zeta_{24}^{6} q^{10} + \zeta_{24} q^{11} + ( -\zeta_{24}^{2} - \zeta_{24}^{8} ) q^{13} + \zeta_{24}^{11} q^{14} + \zeta_{24}^{4} q^{16} + ( -\zeta_{24}^{3} - \zeta_{24}^{9} ) q^{17} + \zeta_{24}^{7} q^{20} -\zeta_{24}^{2} q^{22} + ( \zeta_{24}^{3} + \zeta_{24}^{9} ) q^{26} + q^{28} + \zeta_{24}^{8} q^{31} -\zeta_{24}^{5} q^{32} + ( \zeta_{24}^{4} + \zeta_{24}^{10} ) q^{34} + \zeta_{24}^{3} q^{35} -\zeta_{24}^{8} q^{40} + ( \zeta_{24}^{4} + \zeta_{24}^{10} ) q^{43} + \zeta_{24}^{3} q^{44} + ( \zeta_{24} - \zeta_{24}^{7} ) q^{47} + ( -\zeta_{24}^{4} - \zeta_{24}^{10} ) q^{52} + \zeta_{24}^{9} q^{53} + \zeta_{24}^{6} q^{55} -\zeta_{24} q^{56} -\zeta_{24}^{9} q^{62} + \zeta_{24}^{6} q^{64} + ( \zeta_{24} - \zeta_{24}^{7} ) q^{65} + ( \zeta_{24}^{2} + \zeta_{24}^{8} ) q^{67} + ( -\zeta_{24}^{5} - \zeta_{24}^{11} ) q^{68} -\zeta_{24}^{4} q^{70} + ( -\zeta_{24}^{3} + \zeta_{24}^{9} ) q^{71} -\zeta_{24}^{6} q^{73} -\zeta_{24}^{11} q^{77} + \zeta_{24}^{9} q^{80} + \zeta_{24}^{7} q^{83} + ( \zeta_{24}^{2} - \zeta_{24}^{8} ) q^{85} + ( -\zeta_{24}^{5} - \zeta_{24}^{11} ) q^{86} -\zeta_{24}^{4} q^{88} + ( \zeta_{24}^{3} - \zeta_{24}^{9} ) q^{89} + ( -1 - \zeta_{24}^{6} ) q^{91} + ( -\zeta_{24}^{2} + \zeta_{24}^{8} ) q^{94} -\zeta_{24}^{4} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + O(q^{10}) \) \( 8 q + 4 q^{13} + 4 q^{16} + 8 q^{28} - 4 q^{31} + 4 q^{34} + 4 q^{40} + 4 q^{43} - 4 q^{52} - 4 q^{67} - 4 q^{70} + 4 q^{85} - 4 q^{88} - 8 q^{91} - 4 q^{94} - 4 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(\zeta_{24}^{6}\) \(1\) \(-\zeta_{24}^{8}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
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
−0.258819 0.965926i
−0.965926 0.258819i 0 0.866025 + 0.500000i 0.258819 + 0.965926i 0 0.866025 0.500000i −0.707107 0.707107i 0 1.00000i
53.2 0.965926 + 0.258819i 0 0.866025 + 0.500000i −0.258819 0.965926i 0 0.866025 0.500000i 0.707107 + 0.707107i 0 1.00000i
269.1 −0.965926 + 0.258819i 0 0.866025 0.500000i 0.258819 0.965926i 0 0.866025 + 0.500000i −0.707107 + 0.707107i 0 1.00000i
269.2 0.965926 0.258819i 0 0.866025 0.500000i −0.258819 + 0.965926i 0 0.866025 + 0.500000i 0.707107 0.707107i 0 1.00000i
701.1 −0.258819 + 0.965926i 0 −0.866025 0.500000i 0.965926 0.258819i 0 −0.866025 + 0.500000i 0.707107 0.707107i 0 1.00000i
701.2 0.258819 0.965926i 0 −0.866025 0.500000i −0.965926 + 0.258819i 0 −0.866025 + 0.500000i −0.707107 + 0.707107i 0 1.00000i
917.1 −0.258819 0.965926i 0 −0.866025 + 0.500000i 0.965926 + 0.258819i 0 −0.866025 0.500000i 0.707107 + 0.707107i 0 1.00000i
917.2 0.258819 + 0.965926i 0 −0.866025 + 0.500000i −0.965926 0.258819i 0 −0.866025 0.500000i −0.707107 0.707107i 0 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 917.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner
144.w odd 12 1 inner
144.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.1.x.a 8
3.b odd 2 1 inner 1296.1.x.a 8
9.c even 3 1 432.1.j.a 4
9.c even 3 1 inner 1296.1.x.a 8
9.d odd 6 1 432.1.j.a 4
9.d odd 6 1 inner 1296.1.x.a 8
16.e even 4 1 inner 1296.1.x.a 8
36.f odd 6 1 1728.1.j.a 4
36.h even 6 1 1728.1.j.a 4
48.i odd 4 1 inner 1296.1.x.a 8
72.j odd 6 1 3456.1.j.b 4
72.l even 6 1 3456.1.j.a 4
72.n even 6 1 3456.1.j.b 4
72.p odd 6 1 3456.1.j.a 4
144.u even 12 1 1728.1.j.a 4
144.u even 12 1 3456.1.j.a 4
144.v odd 12 1 1728.1.j.a 4
144.v odd 12 1 3456.1.j.a 4
144.w odd 12 1 432.1.j.a 4
144.w odd 12 1 inner 1296.1.x.a 8
144.w odd 12 1 3456.1.j.b 4
144.x even 12 1 432.1.j.a 4
144.x even 12 1 inner 1296.1.x.a 8
144.x even 12 1 3456.1.j.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.1.j.a 4 9.c even 3 1
432.1.j.a 4 9.d odd 6 1
432.1.j.a 4 144.w odd 12 1
432.1.j.a 4 144.x even 12 1
1296.1.x.a 8 1.a even 1 1 trivial
1296.1.x.a 8 3.b odd 2 1 inner
1296.1.x.a 8 9.c even 3 1 inner
1296.1.x.a 8 9.d odd 6 1 inner
1296.1.x.a 8 16.e even 4 1 inner
1296.1.x.a 8 48.i odd 4 1 inner
1296.1.x.a 8 144.w odd 12 1 inner
1296.1.x.a 8 144.x even 12 1 inner
1728.1.j.a 4 36.f odd 6 1
1728.1.j.a 4 36.h even 6 1
1728.1.j.a 4 144.u even 12 1
1728.1.j.a 4 144.v odd 12 1
3456.1.j.a 4 72.l even 6 1
3456.1.j.a 4 72.p odd 6 1
3456.1.j.a 4 144.u even 12 1
3456.1.j.a 4 144.v odd 12 1
3456.1.j.b 4 72.j odd 6 1
3456.1.j.b 4 72.n even 6 1
3456.1.j.b 4 144.w odd 12 1
3456.1.j.b 4 144.x even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1296, [\chi])\).

Hecke characteristic polynomials

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