Properties

Label 2960.1.ez.a
Level $2960$
Weight $1$
Character orbit 2960.ez
Analytic conductor $1.477$
Analytic rank $0$
Dimension $8$
Projective image $S_{4}$
CM/RM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2960.ez (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.47723243739\)
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: \(S_{4}\)
Projective field: Galois closure of 4.0.14018560.7

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{24}^{7} q^{2} + \zeta_{24}^{5} q^{3} -\zeta_{24}^{2} q^{4} + \zeta_{24}^{5} q^{5} + q^{6} + ( -\zeta_{24}^{5} + \zeta_{24}^{11} ) q^{7} + \zeta_{24}^{9} q^{8} +O(q^{10})\) \( q -\zeta_{24}^{7} q^{2} + \zeta_{24}^{5} q^{3} -\zeta_{24}^{2} q^{4} + \zeta_{24}^{5} q^{5} + q^{6} + ( -\zeta_{24}^{5} + \zeta_{24}^{11} ) q^{7} + \zeta_{24}^{9} q^{8} + q^{10} + ( 1 - \zeta_{24}^{6} ) q^{11} -\zeta_{24}^{7} q^{12} -\zeta_{24}^{11} q^{13} + ( -1 + \zeta_{24}^{6} ) q^{14} + \zeta_{24}^{10} q^{15} + \zeta_{24}^{4} q^{16} + ( -\zeta_{24} + \zeta_{24}^{7} ) q^{17} + ( \zeta_{24}^{2} - \zeta_{24}^{8} ) q^{19} -\zeta_{24}^{7} q^{20} + ( -\zeta_{24}^{4} - \zeta_{24}^{10} ) q^{21} + ( -\zeta_{24} - \zeta_{24}^{7} ) q^{22} + ( -\zeta_{24}^{3} - \zeta_{24}^{9} ) q^{23} -\zeta_{24}^{2} q^{24} + \zeta_{24}^{10} q^{25} -\zeta_{24}^{6} q^{26} + \zeta_{24}^{3} q^{27} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{28} + \zeta_{24}^{5} q^{30} + \zeta_{24}^{6} q^{31} -\zeta_{24}^{11} q^{32} + ( \zeta_{24}^{5} - \zeta_{24}^{11} ) q^{33} + ( \zeta_{24}^{2} + \zeta_{24}^{8} ) q^{34} + ( -\zeta_{24}^{4} - \zeta_{24}^{10} ) q^{35} -\zeta_{24}^{5} q^{37} + ( -\zeta_{24}^{3} - \zeta_{24}^{9} ) q^{38} + \zeta_{24}^{4} q^{39} -\zeta_{24}^{2} q^{40} + \zeta_{24}^{2} q^{41} + ( -\zeta_{24}^{5} + \zeta_{24}^{11} ) q^{42} + \zeta_{24}^{3} q^{43} + ( -\zeta_{24}^{2} + \zeta_{24}^{8} ) q^{44} + ( -\zeta_{24}^{4} + \zeta_{24}^{10} ) q^{46} + \zeta_{24}^{9} q^{48} + \zeta_{24}^{4} q^{49} + \zeta_{24}^{5} q^{50} + ( -1 - \zeta_{24}^{6} ) q^{51} -\zeta_{24} q^{52} + \zeta_{24} q^{53} -\zeta_{24}^{10} q^{54} + ( \zeta_{24}^{5} - \zeta_{24}^{11} ) q^{55} + ( \zeta_{24}^{2} - \zeta_{24}^{8} ) q^{56} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{57} + q^{60} + ( \zeta_{24}^{2} + \zeta_{24}^{8} ) q^{61} + \zeta_{24} q^{62} -\zeta_{24}^{6} q^{64} + \zeta_{24}^{4} q^{65} + ( 1 - \zeta_{24}^{6} ) q^{66} + ( \zeta_{24}^{3} - \zeta_{24}^{9} ) q^{68} + ( \zeta_{24}^{2} - \zeta_{24}^{8} ) q^{69} + ( -\zeta_{24}^{5} + \zeta_{24}^{11} ) q^{70} -2 \zeta_{24}^{8} q^{71} - q^{74} -\zeta_{24}^{3} q^{75} + ( -\zeta_{24}^{4} + \zeta_{24}^{10} ) q^{76} + 2 \zeta_{24}^{11} q^{77} -\zeta_{24}^{11} q^{78} -2 \zeta_{24}^{2} q^{79} + \zeta_{24}^{9} q^{80} + \zeta_{24}^{8} q^{81} -\zeta_{24}^{9} q^{82} + ( -1 + \zeta_{24}^{6} ) q^{84} + ( -1 - \zeta_{24}^{6} ) q^{85} -\zeta_{24}^{10} q^{86} + ( \zeta_{24}^{3} + \zeta_{24}^{9} ) q^{88} + ( -\zeta_{24}^{4} + \zeta_{24}^{10} ) q^{91} + ( \zeta_{24}^{5} + \zeta_{24}^{11} ) q^{92} + \zeta_{24}^{11} q^{93} + ( \zeta_{24} + \zeta_{24}^{7} ) q^{95} + \zeta_{24}^{4} q^{96} -\zeta_{24}^{11} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{6} + O(q^{10}) \) \( 8 q + 8 q^{6} + 8 q^{10} + 8 q^{11} - 8 q^{14} + 4 q^{16} + 4 q^{19} - 4 q^{21} - 4 q^{34} - 4 q^{35} + 4 q^{39} - 4 q^{44} - 4 q^{46} + 4 q^{49} - 8 q^{51} + 4 q^{56} + 8 q^{60} - 4 q^{61} + 4 q^{65} + 8 q^{66} + 4 q^{69} + 8 q^{71} - 8 q^{74} - 4 q^{76} - 4 q^{81} - 8 q^{84} - 8 q^{85} - 4 q^{91} + 4 q^{96} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2960\mathbb{Z}\right)^\times\).

\(n\) \(741\) \(1777\) \(2481\) \(2591\)
\(\chi(n)\) \(\zeta_{24}^{6}\) \(-1\) \(\zeta_{24}^{8}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1099.1
−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
−0.258819 + 0.965926i −0.258819 0.965926i −0.866025 0.500000i −0.258819 0.965926i 1.00000 1.22474 + 0.707107i 0.707107 0.707107i 0 1.00000
1099.2 0.258819 0.965926i 0.258819 + 0.965926i −0.866025 0.500000i 0.258819 + 0.965926i 1.00000 −1.22474 0.707107i −0.707107 + 0.707107i 0 1.00000
1379.1 −0.258819 0.965926i −0.258819 + 0.965926i −0.866025 + 0.500000i −0.258819 + 0.965926i 1.00000 1.22474 0.707107i 0.707107 + 0.707107i 0 1.00000
1379.2 0.258819 + 0.965926i 0.258819 0.965926i −0.866025 + 0.500000i 0.258819 0.965926i 1.00000 −1.22474 + 0.707107i −0.707107 0.707107i 0 1.00000
2579.1 −0.965926 0.258819i −0.965926 + 0.258819i 0.866025 + 0.500000i −0.965926 + 0.258819i 1.00000 1.22474 + 0.707107i −0.707107 0.707107i 0 1.00000
2579.2 0.965926 + 0.258819i 0.965926 0.258819i 0.866025 + 0.500000i 0.965926 0.258819i 1.00000 −1.22474 0.707107i 0.707107 + 0.707107i 0 1.00000
2859.1 −0.965926 + 0.258819i −0.965926 0.258819i 0.866025 0.500000i −0.965926 0.258819i 1.00000 1.22474 0.707107i −0.707107 + 0.707107i 0 1.00000
2859.2 0.965926 0.258819i 0.965926 + 0.258819i 0.866025 0.500000i 0.965926 + 0.258819i 1.00000 −1.22474 + 0.707107i 0.707107 0.707107i 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2859.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
16.f odd 4 1 inner
37.c even 3 1 inner
80.k odd 4 1 inner
185.n even 6 1 inner
592.bi odd 12 1 inner
2960.ez odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2960.1.ez.a 8
5.b even 2 1 inner 2960.1.ez.a 8
16.f odd 4 1 inner 2960.1.ez.a 8
37.c even 3 1 inner 2960.1.ez.a 8
80.k odd 4 1 inner 2960.1.ez.a 8
185.n even 6 1 inner 2960.1.ez.a 8
592.bi odd 12 1 inner 2960.1.ez.a 8
2960.ez odd 12 1 inner 2960.1.ez.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2960.1.ez.a 8 1.a even 1 1 trivial
2960.1.ez.a 8 5.b even 2 1 inner
2960.1.ez.a 8 16.f odd 4 1 inner
2960.1.ez.a 8 37.c even 3 1 inner
2960.1.ez.a 8 80.k odd 4 1 inner
2960.1.ez.a 8 185.n even 6 1 inner
2960.1.ez.a 8 592.bi odd 12 1 inner
2960.1.ez.a 8 2960.ez odd 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2960, [\chi])\).

Hecke characteristic polynomials

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