Properties

Label 2960.2.a.u
Level $2960$
Weight $2$
Character orbit 2960.a
Self dual yes
Analytic conductor $23.636$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.892.1
Defining polynomial: \(x^{3} - x^{2} - 8 x + 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - q^{5} + \beta_{1} q^{7} + ( 3 + 2 \beta_{1} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} - q^{5} + \beta_{1} q^{7} + ( 3 + 2 \beta_{1} ) q^{9} + ( -4 + \beta_{1} - \beta_{2} ) q^{11} + 2 \beta_{2} q^{13} -\beta_{2} q^{15} + ( \beta_{1} - \beta_{2} ) q^{17} + \beta_{2} q^{19} + ( 2 + 2 \beta_{2} ) q^{21} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{23} + q^{25} + ( 4 + 4 \beta_{2} ) q^{27} + ( -2 + \beta_{1} + \beta_{2} ) q^{29} -3 \beta_{1} q^{31} + ( -4 - 2 \beta_{1} - 2 \beta_{2} ) q^{33} -\beta_{1} q^{35} - q^{37} + ( 12 + 4 \beta_{1} ) q^{39} + ( -2 - 3 \beta_{1} + \beta_{2} ) q^{41} + ( 4 - \beta_{1} - \beta_{2} ) q^{43} + ( -3 - 2 \beta_{1} ) q^{45} + ( -2 \beta_{1} + \beta_{2} ) q^{47} + ( -1 - \beta_{1} + \beta_{2} ) q^{49} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{51} + ( 6 + 3 \beta_{1} - \beta_{2} ) q^{53} + ( 4 - \beta_{1} + \beta_{2} ) q^{55} + ( 6 + 2 \beta_{1} ) q^{57} + ( -4 - 2 \beta_{1} + \beta_{2} ) q^{59} + ( 2 + \beta_{1} + \beta_{2} ) q^{61} + ( 12 + \beta_{1} + 2 \beta_{2} ) q^{63} -2 \beta_{2} q^{65} + ( -2 \beta_{1} + \beta_{2} ) q^{67} + ( -8 - 4 \beta_{1} + 4 \beta_{2} ) q^{69} + ( -8 + 2 \beta_{1} + 2 \beta_{2} ) q^{71} + ( -6 + 2 \beta_{1} - 2 \beta_{2} ) q^{73} + \beta_{2} q^{75} + ( 4 - 5 \beta_{1} - \beta_{2} ) q^{77} + ( 8 + 2 \beta_{1} + \beta_{2} ) q^{79} + ( 15 + 2 \beta_{1} + 4 \beta_{2} ) q^{81} + ( -2 \beta_{1} - 3 \beta_{2} ) q^{83} + ( -\beta_{1} + \beta_{2} ) q^{85} + ( 8 + 2 \beta_{1} ) q^{87} + 6 q^{89} + ( 4 + 4 \beta_{2} ) q^{91} + ( -6 - 6 \beta_{2} ) q^{93} -\beta_{2} q^{95} + ( -6 - 3 \beta_{1} + \beta_{2} ) q^{97} + ( -4 - 7 \beta_{1} - 5 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{5} + q^{7} + 11q^{9} + O(q^{10}) \) \( 3q - 3q^{5} + q^{7} + 11q^{9} - 11q^{11} + q^{17} + 6q^{21} + 2q^{23} + 3q^{25} + 12q^{27} - 5q^{29} - 3q^{31} - 14q^{33} - q^{35} - 3q^{37} + 40q^{39} - 9q^{41} + 11q^{43} - 11q^{45} - 2q^{47} - 4q^{49} - 14q^{51} + 21q^{53} + 11q^{55} + 20q^{57} - 14q^{59} + 7q^{61} + 37q^{63} - 2q^{67} - 28q^{69} - 22q^{71} - 16q^{73} + 7q^{77} + 26q^{79} + 47q^{81} - 2q^{83} - q^{85} + 26q^{87} + 18q^{89} + 12q^{91} - 18q^{93} - 21q^{97} - 19q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 8 x + 10\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + \nu - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - \beta_{1} + 6\)

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
1.31955
−2.91729
2.59774
0 −2.93923 0 −1.00000 0 1.31955 0 5.63910 0
1.2 0 −0.406728 0 −1.00000 0 −2.91729 0 −2.83457 0
1.3 0 3.34596 0 −1.00000 0 2.59774 0 8.19547 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.u 3
4.b odd 2 1 370.2.a.g 3
12.b even 2 1 3330.2.a.bg 3
20.d odd 2 1 1850.2.a.z 3
20.e even 4 2 1850.2.b.o 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.a.g 3 4.b odd 2 1
1850.2.a.z 3 20.d odd 2 1
1850.2.b.o 6 20.e even 4 2
2960.2.a.u 3 1.a even 1 1 trivial
3330.2.a.bg 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2960))\):

\( T_{3}^{3} - 10 T_{3} - 4 \)
\( T_{7}^{3} - T_{7}^{2} - 8 T_{7} + 10 \)
\( T_{13}^{3} - 40 T_{13} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -4 - 10 T + T^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( 10 - 8 T - T^{2} + T^{3} \)
$11$ \( -8 + 28 T + 11 T^{2} + T^{3} \)
$13$ \( -32 - 40 T + T^{3} \)
$17$ \( -8 - 12 T - T^{2} + T^{3} \)
$19$ \( -4 - 10 T + T^{3} \)
$23$ \( -64 - 48 T - 2 T^{2} + T^{3} \)
$29$ \( -76 - 16 T + 5 T^{2} + T^{3} \)
$31$ \( -270 - 72 T + 3 T^{2} + T^{3} \)
$37$ \( ( 1 + T )^{3} \)
$41$ \( -364 - 40 T + 9 T^{2} + T^{3} \)
$43$ \( 80 + 16 T - 11 T^{2} + T^{3} \)
$47$ \( -56 - 30 T + 2 T^{2} + T^{3} \)
$53$ \( 316 + 80 T - 21 T^{2} + T^{3} \)
$59$ \( -80 + 34 T + 14 T^{2} + T^{3} \)
$61$ \( 4 - 8 T - 7 T^{2} + T^{3} \)
$67$ \( -56 - 30 T + 2 T^{2} + T^{3} \)
$71$ \( -640 + 64 T + 22 T^{2} + T^{3} \)
$73$ \( -208 + 36 T + 16 T^{2} + T^{3} \)
$79$ \( -224 + 170 T - 26 T^{2} + T^{3} \)
$83$ \( 664 - 158 T + 2 T^{2} + T^{3} \)
$89$ \( ( -6 + T )^{3} \)
$97$ \( -316 + 80 T + 21 T^{2} + T^{3} \)
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