Properties

Label 2960.1.gf.a
Level $2960$
Weight $1$
Character orbit 2960.gf
Analytic conductor $1.477$
Analytic rank $0$
Dimension $12$
Projective image $D_{36}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2960.gf (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.47723243739\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{36})\)
Defining polynomial: \(x^{12} - x^{6} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{36}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{36} + \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{36}^{2} q^{5} -\zeta_{36}^{17} q^{9} +O(q^{10})\) \( q -\zeta_{36}^{2} q^{5} -\zeta_{36}^{17} q^{9} -\zeta_{36} q^{13} + ( -\zeta_{36} - \zeta_{36}^{15} ) q^{17} + \zeta_{36}^{4} q^{25} + ( -\zeta_{36}^{7} - \zeta_{36}^{14} ) q^{29} -\zeta_{36}^{10} q^{37} + ( \zeta_{36}^{8} - \zeta_{36}^{12} ) q^{41} -\zeta_{36} q^{45} -\zeta_{36}^{5} q^{49} + ( -\zeta_{36}^{5} - \zeta_{36}^{8} ) q^{53} + ( 1 + \zeta_{36}^{11} ) q^{61} + \zeta_{36}^{3} q^{65} + ( -\zeta_{36}^{3} + \zeta_{36}^{6} ) q^{73} -\zeta_{36}^{16} q^{81} + ( \zeta_{36}^{3} + \zeta_{36}^{17} ) q^{85} + ( -\zeta_{36}^{4} - \zeta_{36}^{9} ) q^{89} + ( -\zeta_{36}^{2} + \zeta_{36}^{4} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + O(q^{10}) \) \( 12 q + 6 q^{41} + 12 q^{61} + 6 q^{73} + O(q^{100}) \)

Character values

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

\(n\) \(741\) \(1777\) \(2481\) \(2591\)
\(\chi(n)\) \(1\) \(-\zeta_{36}^{9}\) \(-\zeta_{36}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
383.1
0.342020 0.939693i
−0.984808 + 0.173648i
0.642788 0.766044i
−0.984808 0.173648i
0.342020 + 0.939693i
0.642788 + 0.766044i
−0.642788 0.766044i
−0.342020 0.939693i
0.984808 + 0.173648i
−0.642788 + 0.766044i
0.984808 0.173648i
−0.342020 + 0.939693i
0 0 0 0.766044 + 0.642788i 0 0 0 0.342020 + 0.939693i 0
463.1 0 0 0 −0.939693 + 0.342020i 0 0 0 −0.984808 0.173648i 0
607.1 0 0 0 0.173648 + 0.984808i 0 0 0 0.642788 + 0.766044i 0
927.1 0 0 0 −0.939693 0.342020i 0 0 0 −0.984808 + 0.173648i 0
1167.1 0 0 0 0.766044 0.642788i 0 0 0 0.342020 0.939693i 0
1263.1 0 0 0 0.173648 0.984808i 0 0 0 0.642788 0.766044i 0
1327.1 0 0 0 0.173648 0.984808i 0 0 0 −0.642788 + 0.766044i 0
1423.1 0 0 0 0.766044 0.642788i 0 0 0 −0.342020 + 0.939693i 0
1663.1 0 0 0 −0.939693 0.342020i 0 0 0 0.984808 0.173648i 0
1983.1 0 0 0 0.173648 + 0.984808i 0 0 0 −0.642788 0.766044i 0
2127.1 0 0 0 −0.939693 + 0.342020i 0 0 0 0.984808 + 0.173648i 0
2207.1 0 0 0 0.766044 + 0.642788i 0 0 0 −0.342020 0.939693i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2207.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
185.bc even 36 1 inner
740.bw odd 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2960.1.gf.a 12
4.b odd 2 1 CM 2960.1.gf.a 12
5.c odd 4 1 2960.1.go.a yes 12
20.e even 4 1 2960.1.go.a yes 12
37.i odd 36 1 2960.1.go.a yes 12
148.q even 36 1 2960.1.go.a yes 12
185.bc even 36 1 inner 2960.1.gf.a 12
740.bw odd 36 1 inner 2960.1.gf.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2960.1.gf.a 12 1.a even 1 1 trivial
2960.1.gf.a 12 4.b odd 2 1 CM
2960.1.gf.a 12 185.bc even 36 1 inner
2960.1.gf.a 12 740.bw odd 36 1 inner
2960.1.go.a yes 12 5.c odd 4 1
2960.1.go.a yes 12 20.e even 4 1
2960.1.go.a yes 12 37.i odd 36 1
2960.1.go.a yes 12 148.q even 36 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( T^{12} \)
$5$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$7$ \( T^{12} \)
$11$ \( T^{12} \)
$13$ \( 1 - T^{6} + T^{12} \)
$17$ \( 9 + 27 T^{2} + 9 T^{4} - 24 T^{6} + 18 T^{8} - 3 T^{10} + T^{12} \)
$19$ \( T^{12} \)
$23$ \( T^{12} \)
$29$ \( 1 - 6 T + 45 T^{2} - 110 T^{3} + 105 T^{4} - 36 T^{5} + 2 T^{6} + 6 T^{7} - 9 T^{8} + 2 T^{9} + T^{12} \)
$31$ \( T^{12} \)
$37$ \( ( 1 + T^{3} + T^{6} )^{2} \)
$41$ \( ( 3 - 9 T + 9 T^{2} - 6 T^{3} + 6 T^{4} - 3 T^{5} + T^{6} )^{2} \)
$43$ \( T^{12} \)
$47$ \( T^{12} \)
$53$ \( 1 - 4 T^{3} + 53 T^{6} - 14 T^{9} + T^{12} \)
$59$ \( T^{12} \)
$61$ \( 1 - 6 T + 51 T^{2} - 200 T^{3} + 480 T^{4} - 786 T^{5} + 923 T^{6} - 792 T^{7} + 495 T^{8} - 220 T^{9} + 66 T^{10} - 12 T^{11} + T^{12} \)
$67$ \( T^{12} \)
$71$ \( T^{12} \)
$73$ \( ( 1 + 2 T + 2 T^{2} - 2 T^{3} + T^{4} )^{3} \)
$79$ \( T^{12} \)
$83$ \( T^{12} \)
$89$ \( 1 + 12 T + 39 T^{2} + 14 T^{3} - 12 T^{4} + 12 T^{5} + 23 T^{6} + 15 T^{8} - 2 T^{9} + 6 T^{10} + T^{12} \)
$97$ \( ( 1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6} )^{2} \)
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