Properties

Label 2960.1.et.a
Level $2960$
Weight $1$
Character orbit 2960.et
Analytic conductor $1.477$
Analytic rank $0$
Dimension $4$
Projective image $D_{12}$
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.et (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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{12}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1087.1
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 0 0 −0.500000 + 0.866025i 0 0 0 −0.866025 0.500000i 0
1503.1 0 0 0 −0.500000 + 0.866025i 0 0 0 0.866025 + 0.500000i 0
2767.1 0 0 0 −0.500000 0.866025i 0 0 0 0.866025 0.500000i 0
2783.1 0 0 0 −0.500000 0.866025i 0 0 0 −0.866025 + 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
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.u even 12 1 inner
740.bm 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.et.a yes 4
4.b odd 2 1 CM 2960.1.et.a yes 4
5.c odd 4 1 2960.1.eo.a 4
20.e even 4 1 2960.1.eo.a 4
37.g odd 12 1 2960.1.eo.a 4
148.l even 12 1 2960.1.eo.a 4
185.u even 12 1 inner 2960.1.et.a yes 4
740.bm odd 12 1 inner 2960.1.et.a yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2960.1.eo.a 4 5.c odd 4 1
2960.1.eo.a 4 20.e even 4 1
2960.1.eo.a 4 37.g odd 12 1
2960.1.eo.a 4 148.l even 12 1
2960.1.et.a yes 4 1.a even 1 1 trivial
2960.1.et.a yes 4 4.b odd 2 1 CM
2960.1.et.a yes 4 185.u even 12 1 inner
2960.1.et.a yes 4 740.bm odd 12 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 1 + T + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( T^{4} \)
$13$ \( 16 - 4 T^{2} + T^{4} \)
$17$ \( 9 + 3 T^{2} + T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( 1 - 2 T + 2 T^{2} + 2 T^{3} + T^{4} \)
$31$ \( T^{4} \)
$37$ \( ( 1 + T + T^{2} )^{2} \)
$41$ \( ( 3 + 3 T + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} \)
$59$ \( T^{4} \)
$61$ \( 1 - 2 T + 5 T^{2} - 4 T^{3} + T^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( 2 + 2 T + T^{2} )^{2} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( 1 - 4 T + 5 T^{2} - 2 T^{3} + T^{4} \)
$97$ \( ( 1 + T )^{4} \)
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