Properties

Label 2960.1.fx.b
Level $2960$
Weight $1$
Character orbit 2960.fx
Analytic conductor $1.477$
Analytic rank $0$
Dimension $6$
Projective image $D_{18}$
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.fx (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.47723243739\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{18}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{18} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{18}^{6} q^{5} + \zeta_{18}^{7} q^{9} +O(q^{10})\) \( q -\zeta_{18}^{6} q^{5} + \zeta_{18}^{7} q^{9} + ( -\zeta_{18}^{5} - \zeta_{18}^{8} ) q^{13} + ( \zeta_{18}^{2} + \zeta_{18}^{3} ) q^{17} -\zeta_{18}^{3} q^{25} + ( -\zeta_{18} - \zeta_{18}^{5} ) q^{29} + \zeta_{18}^{2} q^{37} + ( -\zeta_{18}^{6} + \zeta_{18}^{7} ) q^{41} + \zeta_{18}^{4} q^{45} + \zeta_{18} q^{49} + ( \zeta_{18} - \zeta_{18}^{7} ) q^{53} + ( -1 - \zeta_{18}^{4} ) q^{61} + ( -\zeta_{18}^{2} - \zeta_{18}^{5} ) q^{65} + ( -\zeta_{18}^{3} - \zeta_{18}^{6} ) q^{73} -\zeta_{18}^{5} q^{81} + ( 1 - \zeta_{18}^{8} ) q^{85} + ( 1 + \zeta_{18}^{8} ) q^{89} + ( -\zeta_{18}^{4} + \zeta_{18}^{8} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 3q^{5} + O(q^{10}) \) \( 6q + 3q^{5} + 3q^{17} - 3q^{25} + 3q^{41} - 6q^{61} + 6q^{85} + 6q^{89} + O(q^{100}) \)

Character values

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

\(n\) \(741\) \(1777\) \(2481\) \(2591\)
\(\chi(n)\) \(1\) \(-1\) \(\zeta_{18}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
719.1
0.939693 + 0.342020i
−0.766044 0.642788i
−0.173648 0.984808i
−0.173648 + 0.984808i
0.939693 0.342020i
−0.766044 + 0.642788i
0 0 0 0.500000 0.866025i 0 0 0 −0.766044 + 0.642788i 0
959.1 0 0 0 0.500000 + 0.866025i 0 0 0 −0.173648 + 0.984808i 0
1119.1 0 0 0 0.500000 0.866025i 0 0 0 0.939693 + 0.342020i 0
1439.1 0 0 0 0.500000 + 0.866025i 0 0 0 0.939693 0.342020i 0
2079.1 0 0 0 0.500000 + 0.866025i 0 0 0 −0.766044 0.642788i 0
2639.1 0 0 0 0.500000 0.866025i 0 0 0 −0.173648 0.984808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2639.1
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.x even 18 1 inner
740.bs odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2960.1.fx.b yes 6
4.b odd 2 1 CM 2960.1.fx.b yes 6
5.b even 2 1 2960.1.fx.a 6
20.d odd 2 1 2960.1.fx.a 6
37.f even 9 1 2960.1.fx.a 6
148.p odd 18 1 2960.1.fx.a 6
185.x even 18 1 inner 2960.1.fx.b yes 6
740.bs odd 18 1 inner 2960.1.fx.b yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2960.1.fx.a 6 5.b even 2 1
2960.1.fx.a 6 20.d odd 2 1
2960.1.fx.a 6 37.f even 9 1
2960.1.fx.a 6 148.p odd 18 1
2960.1.fx.b yes 6 1.a even 1 1 trivial
2960.1.fx.b yes 6 4.b odd 2 1 CM
2960.1.fx.b yes 6 185.x even 18 1 inner
2960.1.fx.b yes 6 740.bs odd 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{6} + 9 T_{13}^{3} + 27 \) acting on \(S_{1}^{\mathrm{new}}(2960, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( T^{6} \)
$5$ \( ( 1 - T + T^{2} )^{3} \)
$7$ \( T^{6} \)
$11$ \( T^{6} \)
$13$ \( 27 + 9 T^{3} + T^{6} \)
$17$ \( 3 - 9 T + 9 T^{2} - 6 T^{3} + 6 T^{4} - 3 T^{5} + T^{6} \)
$19$ \( T^{6} \)
$23$ \( T^{6} \)
$29$ \( 1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6} \)
$31$ \( T^{6} \)
$37$ \( 1 + T^{3} + T^{6} \)
$41$ \( 1 + 3 T + 3 T^{2} - 8 T^{3} + 6 T^{4} - 3 T^{5} + T^{6} \)
$43$ \( T^{6} \)
$47$ \( T^{6} \)
$53$ \( 27 - 9 T^{3} + T^{6} \)
$59$ \( T^{6} \)
$61$ \( 1 + 3 T + 12 T^{2} + 19 T^{3} + 15 T^{4} + 6 T^{5} + T^{6} \)
$67$ \( T^{6} \)
$71$ \( T^{6} \)
$73$ \( ( 3 + T^{2} )^{3} \)
$79$ \( T^{6} \)
$83$ \( T^{6} \)
$89$ \( 1 - 3 T + 12 T^{2} - 19 T^{3} + 15 T^{4} - 6 T^{5} + T^{6} \)
$97$ \( 3 + 9 T + 9 T^{2} - 3 T^{4} + T^{6} \)
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