Properties

Label 2960.1.fm.a
Level $2960$
Weight $1$
Character orbit 2960.fm
Analytic conductor $1.477$
Analytic rank $0$
Dimension $4$
Projective image $S_{4}$
CM/RM no
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.fm (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.47723243739\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.0.350464000.5

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} -\zeta_{12}^{5} q^{3} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{5} q^{5} + q^{6} + ( -\zeta_{12}^{2} + \zeta_{12}^{5} ) q^{7} + \zeta_{12}^{3} q^{8} +O(q^{10})\) \( q + \zeta_{12} q^{2} -\zeta_{12}^{5} q^{3} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{5} q^{5} + q^{6} + ( -\zeta_{12}^{2} + \zeta_{12}^{5} ) q^{7} + \zeta_{12}^{3} q^{8} - q^{10} + ( 1 + \zeta_{12}^{3} ) q^{11} + \zeta_{12} q^{12} + \zeta_{12}^{5} q^{13} + ( -1 - \zeta_{12}^{3} ) q^{14} + \zeta_{12}^{4} q^{15} + \zeta_{12}^{4} q^{16} -\zeta_{12} q^{20} + ( -\zeta_{12} + \zeta_{12}^{4} ) q^{21} + ( \zeta_{12} + \zeta_{12}^{4} ) q^{22} + ( 1 + \zeta_{12}^{3} ) q^{23} + \zeta_{12}^{2} q^{24} -\zeta_{12}^{4} q^{25} - q^{26} + \zeta_{12}^{3} q^{27} + ( -\zeta_{12} - \zeta_{12}^{4} ) q^{28} + \zeta_{12}^{5} q^{30} - q^{31} + \zeta_{12}^{5} q^{32} + ( \zeta_{12}^{2} - \zeta_{12}^{5} ) q^{33} + ( \zeta_{12} - \zeta_{12}^{4} ) q^{35} -\zeta_{12}^{5} q^{37} + \zeta_{12}^{4} q^{39} -\zeta_{12}^{2} q^{40} -\zeta_{12}^{5} q^{41} + ( -\zeta_{12}^{2} + \zeta_{12}^{5} ) q^{42} - q^{43} + ( \zeta_{12}^{2} + \zeta_{12}^{5} ) q^{44} + ( \zeta_{12} + \zeta_{12}^{4} ) q^{46} + \zeta_{12}^{3} q^{48} + \zeta_{12} q^{49} -\zeta_{12}^{5} q^{50} -\zeta_{12} q^{52} + \zeta_{12}^{4} q^{53} + \zeta_{12}^{4} q^{54} + ( -\zeta_{12}^{2} + \zeta_{12}^{5} ) q^{55} + ( -\zeta_{12}^{2} - \zeta_{12}^{5} ) q^{56} + ( -\zeta_{12} - \zeta_{12}^{4} ) q^{59} - q^{60} + ( \zeta_{12}^{2} - \zeta_{12}^{5} ) q^{61} -\zeta_{12} q^{62} - q^{64} -\zeta_{12}^{4} q^{65} + ( 1 + \zeta_{12}^{3} ) q^{66} + ( \zeta_{12}^{2} - \zeta_{12}^{5} ) q^{69} + ( \zeta_{12}^{2} - \zeta_{12}^{5} ) q^{70} -2 \zeta_{12}^{5} q^{71} + q^{74} -\zeta_{12}^{3} q^{75} -2 \zeta_{12}^{2} q^{77} + \zeta_{12}^{5} q^{78} -\zeta_{12}^{3} q^{80} + \zeta_{12}^{2} q^{81} + q^{82} + ( -1 - \zeta_{12}^{3} ) q^{84} -\zeta_{12} q^{86} + ( -1 + \zeta_{12}^{3} ) q^{88} + ( \zeta_{12} - \zeta_{12}^{4} ) q^{91} + ( \zeta_{12}^{2} + \zeta_{12}^{5} ) q^{92} + \zeta_{12}^{5} q^{93} + \zeta_{12}^{4} q^{96} + \zeta_{12}^{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + 4q^{6} - 2q^{7} + O(q^{10}) \) \( 4q + 2q^{4} + 4q^{6} - 2q^{7} - 4q^{10} + 4q^{11} - 4q^{14} - 2q^{15} - 2q^{16} - 2q^{21} - 2q^{22} + 4q^{23} + 2q^{24} + 2q^{25} - 4q^{26} + 2q^{28} - 4q^{31} + 2q^{33} + 2q^{35} - 2q^{39} - 2q^{40} - 2q^{42} - 4q^{43} + 2q^{44} - 2q^{46} - 2q^{53} - 2q^{54} - 2q^{55} - 2q^{56} + 2q^{59} - 4q^{60} + 2q^{61} - 4q^{64} + 2q^{65} + 4q^{66} + 2q^{69} + 2q^{70} + 4q^{74} - 4q^{77} + 2q^{81} + 4q^{82} - 4q^{84} - 4q^{88} + 2q^{91} + 2q^{92} - 2q^{96} + 2q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(741\) \(1777\) \(2481\) \(2591\)
\(\chi(n)\) \(\zeta_{12}^{3}\) \(\zeta_{12}^{3}\) \(-\zeta_{12}^{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1157.1
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000 −1.36603 0.366025i 1.00000i 0 −1.00000
1453.1 −0.866025 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 0.866025 0.500000i 1.00000 0.366025 1.36603i 1.00000i 0 −1.00000
2357.1 −0.866025 + 0.500000i −0.866025 0.500000i 0.500000 0.866025i 0.866025 + 0.500000i 1.00000 0.366025 + 1.36603i 1.00000i 0 −1.00000
2653.1 0.866025 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i −0.866025 0.500000i 1.00000 −1.36603 + 0.366025i 1.00000i 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner
80.i odd 4 1 inner
2960.fm 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.fm.a yes 4
5.c odd 4 1 2960.1.dn.a 4
16.e even 4 1 2960.1.dn.a 4
37.c even 3 1 inner 2960.1.fm.a yes 4
80.i odd 4 1 inner 2960.1.fm.a yes 4
185.s odd 12 1 2960.1.dn.a 4
592.bj even 12 1 2960.1.dn.a 4
2960.fm odd 12 1 inner 2960.1.fm.a yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2960.1.dn.a 4 5.c odd 4 1
2960.1.dn.a 4 16.e even 4 1
2960.1.dn.a 4 185.s odd 12 1
2960.1.dn.a 4 592.bj even 12 1
2960.1.fm.a yes 4 1.a even 1 1 trivial
2960.1.fm.a yes 4 37.c even 3 1 inner
2960.1.fm.a yes 4 80.i odd 4 1 inner
2960.1.fm.a yes 4 2960.fm odd 12 1 inner

Hecke kernels

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

Hecke characteristic polynomials

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