Properties

Label 2960.2.a.v
Level $2960$
Weight $2$
Character orbit 2960.a
Self dual yes
Analytic conductor $23.636$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2960.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.6357189983\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.286164.1
Defining polynomial: \(x^{4} - x^{3} - 11 x^{2} - 3 x + 12\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 740)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{3} + q^{5} + ( 1 + \beta_{1} ) q^{7} + ( 3 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{3} + q^{5} + ( 1 + \beta_{1} ) q^{7} + ( 3 + \beta_{2} ) q^{9} + ( -2 + \beta_{2} - \beta_{3} ) q^{11} + ( -2 - \beta_{3} ) q^{13} + ( -1 + \beta_{1} ) q^{15} + ( 2 - \beta_{3} ) q^{17} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{19} + ( 4 + 2 \beta_{1} + \beta_{2} ) q^{21} -2 \beta_{1} q^{23} + q^{25} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{27} + 2 \beta_{1} q^{29} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{31} + ( -2 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{33} + ( 1 + \beta_{1} ) q^{35} + q^{37} + ( 2 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{39} + ( 8 - \beta_{2} + \beta_{3} ) q^{41} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{43} + ( 3 + \beta_{2} ) q^{45} + ( 1 + \beta_{1} - 2 \beta_{3} ) q^{47} + ( -1 + 4 \beta_{1} + \beta_{2} ) q^{49} + ( -2 - 2 \beta_{2} + 2 \beta_{3} ) q^{51} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{53} + ( -2 + \beta_{2} - \beta_{3} ) q^{55} + ( 4 - 2 \beta_{2} + 3 \beta_{3} ) q^{57} + ( 3 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{59} + 2 q^{61} + ( 1 + 5 \beta_{1} + \beta_{2} + \beta_{3} ) q^{63} + ( -2 - \beta_{3} ) q^{65} + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{67} + ( -10 - 2 \beta_{1} - 2 \beta_{2} ) q^{69} + ( 4 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{71} + ( 4 \beta_{1} + \beta_{2} - \beta_{3} ) q^{73} + ( -1 + \beta_{1} ) q^{75} + ( -4 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{77} + ( 5 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{79} + ( 5 + 2 \beta_{2} - 3 \beta_{3} ) q^{81} + ( -7 - \beta_{1} - 2 \beta_{2} ) q^{83} + ( 2 - \beta_{3} ) q^{85} + ( 10 + 2 \beta_{1} + 2 \beta_{2} ) q^{87} + ( -2 - 2 \beta_{3} ) q^{89} + ( -2 - 4 \beta_{1} - 2 \beta_{2} ) q^{91} + ( 2 + 6 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{93} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{95} + ( 4 - 2 \beta_{1} - 2 \beta_{2} ) q^{97} + ( 2 - 2 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 3q^{3} + 4q^{5} + 5q^{7} + 13q^{9} + O(q^{10}) \) \( 4q - 3q^{3} + 4q^{5} + 5q^{7} + 13q^{9} - 5q^{11} - 6q^{13} - 3q^{15} + 10q^{17} + 19q^{21} - 2q^{23} + 4q^{25} - 9q^{27} + 2q^{29} + 4q^{31} - 11q^{33} + 5q^{35} + 4q^{37} - 2q^{39} + 29q^{41} - 4q^{43} + 13q^{45} + 9q^{47} + q^{49} - 14q^{51} - 3q^{53} - 5q^{55} + 8q^{57} + 10q^{59} + 8q^{61} + 8q^{63} - 6q^{65} - 14q^{67} - 44q^{69} + 17q^{71} + 7q^{73} - 3q^{75} - 21q^{77} + 24q^{79} + 28q^{81} - 31q^{83} + 10q^{85} + 44q^{87} - 4q^{89} - 14q^{91} + 18q^{93} + 12q^{97} + 22q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 11 x^{2} - 3 x + 12\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 5 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 7 \nu + 2 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 2 \beta_{2} + 11 \beta_{1} + 8\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.24418
−1.51951
0.914132
3.84956
0 −3.24418 0 1.00000 0 −1.24418 0 7.52471 0
1.2 0 −2.51951 0 1.00000 0 −0.519509 0 3.34793 0
1.3 0 −0.0858680 0 1.00000 0 1.91413 0 −2.99263 0
1.4 0 2.84956 0 1.00000 0 4.84956 0 5.11999 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(37\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2960.2.a.v 4
4.b odd 2 1 740.2.a.f 4
12.b even 2 1 6660.2.a.r 4
20.d odd 2 1 3700.2.a.j 4
20.e even 4 2 3700.2.d.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.a.f 4 4.b odd 2 1
2960.2.a.v 4 1.a even 1 1 trivial
3700.2.a.j 4 20.d odd 2 1
3700.2.d.i 8 20.e even 4 2
6660.2.a.r 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2960))\):

\( T_{3}^{4} + 3 T_{3}^{3} - 8 T_{3}^{2} - 24 T_{3} - 2 \)
\( T_{7}^{4} - 5 T_{7}^{3} - 2 T_{7}^{2} + 12 T_{7} + 6 \)
\( T_{13}^{4} + 6 T_{13}^{3} - 20 T_{13}^{2} - 84 T_{13} + 160 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( -2 - 24 T - 8 T^{2} + 3 T^{3} + T^{4} \)
$5$ \( ( -1 + T )^{4} \)
$7$ \( 6 + 12 T - 2 T^{2} - 5 T^{3} + T^{4} \)
$11$ \( -240 - 192 T - 32 T^{2} + 5 T^{3} + T^{4} \)
$13$ \( 160 - 84 T - 20 T^{2} + 6 T^{3} + T^{4} \)
$17$ \( 48 + 108 T + 4 T^{2} - 10 T^{3} + T^{4} \)
$19$ \( 64 + 54 T - 38 T^{2} + T^{4} \)
$23$ \( 192 + 24 T - 44 T^{2} + 2 T^{3} + T^{4} \)
$29$ \( 192 - 24 T - 44 T^{2} - 2 T^{3} + T^{4} \)
$31$ \( 148 + 350 T - 90 T^{2} - 4 T^{3} + T^{4} \)
$37$ \( ( -1 + T )^{4} \)
$41$ \( -168 - 828 T + 274 T^{2} - 29 T^{3} + T^{4} \)
$43$ \( 1296 - 288 T - 128 T^{2} + 4 T^{3} + T^{4} \)
$47$ \( 54 + 720 T - 102 T^{2} - 9 T^{3} + T^{4} \)
$53$ \( 72 + 36 T - 54 T^{2} + 3 T^{3} + T^{4} \)
$59$ \( -1968 + 846 T - 62 T^{2} - 10 T^{3} + T^{4} \)
$61$ \( ( -2 + T )^{4} \)
$67$ \( -8 - 22 T + 6 T^{2} + 14 T^{3} + T^{4} \)
$71$ \( -288 + 240 T - 8 T^{2} - 17 T^{3} + T^{4} \)
$73$ \( 360 + 348 T - 146 T^{2} - 7 T^{3} + T^{4} \)
$79$ \( -332 - 354 T + 178 T^{2} - 24 T^{3} + T^{4} \)
$83$ \( -5238 - 288 T + 244 T^{2} + 31 T^{3} + T^{4} \)
$89$ \( 3504 - 240 T - 128 T^{2} + 4 T^{3} + T^{4} \)
$97$ \( 400 + 672 T - 80 T^{2} - 12 T^{3} + T^{4} \)
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