Properties

Label 2960.2.a.y
Level $2960$
Weight $2$
Character orbit 2960.a
Self dual yes
Analytic conductor $23.636$
Analytic rank $0$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.6397264.1
Defining polynomial: \(x^{5} - 10 x^{3} - 2 x^{2} + 14 x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1480)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} - q^{5} + ( -1 - \beta_{1} + \beta_{3} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} - q^{5} + ( -1 - \beta_{1} + \beta_{3} ) q^{7} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{9} + ( -1 - \beta_{4} ) q^{11} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{13} -\beta_{1} q^{15} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{17} + ( \beta_{1} + 2 \beta_{2} ) q^{19} + ( -2 - 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{21} + ( -2 + 2 \beta_{1} ) q^{23} + q^{25} + ( 2 + 2 \beta_{1} - \beta_{3} + \beta_{4} ) q^{27} + ( 5 + 2 \beta_{1} - 2 \beta_{2} + \beta_{4} ) q^{29} + ( -3 + \beta_{1} + \beta_{3} ) q^{31} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{33} + ( 1 + \beta_{1} - \beta_{3} ) q^{35} + q^{37} + ( -2 - 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{39} + ( 1 + 2 \beta_{3} - \beta_{4} ) q^{41} + ( 3 - 2 \beta_{3} - \beta_{4} ) q^{43} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{45} + ( 3 \beta_{1} + \beta_{3} + \beta_{4} ) q^{47} + ( 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{49} + ( 4 + 4 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{51} + ( 5 - 2 \beta_{1} - \beta_{4} ) q^{53} + ( 1 + \beta_{4} ) q^{55} + ( 4 + 5 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{57} + ( -4 - \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{59} + ( 5 + 2 \beta_{2} + \beta_{4} ) q^{61} + ( -7 - 5 \beta_{1} - 2 \beta_{2} - \beta_{4} ) q^{63} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{65} + ( 2 - \beta_{1} - 2 \beta_{2} ) q^{67} + ( 8 + 2 \beta_{2} - 2 \beta_{3} ) q^{69} + ( -6 - 2 \beta_{1} - 2 \beta_{2} ) q^{71} + ( 4 - 2 \beta_{2} ) q^{73} + \beta_{1} q^{75} + ( 3 + 2 \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{77} + ( -4 - 3 \beta_{1} + \beta_{3} - \beta_{4} ) q^{79} + ( 1 + 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{81} + ( 2 + 3 \beta_{1} + 2 \beta_{2} ) q^{83} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} ) q^{85} + ( 6 + 4 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{87} + ( 6 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{89} + ( 6 + 3 \beta_{3} + \beta_{4} ) q^{91} + ( 6 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{93} + ( -\beta_{1} - 2 \beta_{2} ) q^{95} + ( -1 + 4 \beta_{2} - \beta_{4} ) q^{97} + ( -5 - 4 \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 5q^{5} - 3q^{7} + 5q^{9} + O(q^{10}) \) \( 5q - 5q^{5} - 3q^{7} + 5q^{9} - 3q^{11} + 4q^{13} + 7q^{17} + 4q^{19} - 10q^{21} - 10q^{23} + 5q^{25} + 6q^{27} + 19q^{29} - 13q^{31} + 6q^{33} + 3q^{35} + 5q^{37} - 14q^{39} + 11q^{41} + 13q^{43} - 5q^{45} + 2q^{49} + 12q^{51} + 27q^{53} + 3q^{55} + 12q^{57} - 16q^{59} + 27q^{61} - 37q^{63} - 4q^{65} + 6q^{67} + 40q^{69} - 34q^{71} + 16q^{73} + 9q^{77} - 16q^{79} + 9q^{81} + 14q^{83} - 7q^{85} + 42q^{87} + 38q^{89} + 34q^{91} + 30q^{93} - 4q^{95} + 5q^{97} - 23q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 10 x^{3} - 2 x^{2} + 14 x - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} - 8 \nu^{2} - 4 \nu + 4 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} - 10 \nu^{2} - 2 \nu + 12 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} + 2 \nu^{3} - 10 \nu^{2} - 18 \nu + 8 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{4} - \beta_{3} + 8 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(-8 \beta_{3} + 10 \beta_{2} + 12 \beta_{1} + 28\)

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
−2.65435
−1.64390
0.325094
0.925120
3.04803
0 −2.65435 0 −1.00000 0 −0.0991320 0 4.04557 0
1.2 0 −1.64390 0 −1.00000 0 −1.57272 0 −0.297599 0
1.3 0 0.325094 0 −1.00000 0 3.82696 0 −2.89431 0
1.4 0 0.925120 0 −1.00000 0 −0.763238 0 −2.14415 0
1.5 0 3.04803 0 −1.00000 0 −4.39187 0 6.29050 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.y 5
4.b odd 2 1 1480.2.a.i 5
20.d odd 2 1 7400.2.a.p 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1480.2.a.i 5 4.b odd 2 1
2960.2.a.y 5 1.a even 1 1 trivial
7400.2.a.p 5 20.d odd 2 1

Hecke kernels

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

\( T_{3}^{5} - 10 T_{3}^{3} - 2 T_{3}^{2} + 14 T_{3} - 4 \)
\( T_{7}^{5} + 3 T_{7}^{4} - 14 T_{7}^{3} - 40 T_{7}^{2} - 24 T_{7} - 2 \)
\( T_{13}^{5} - 4 T_{13}^{4} - 28 T_{13}^{3} + 44 T_{13}^{2} + 292 T_{13} + 272 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( -4 + 14 T - 2 T^{2} - 10 T^{3} + T^{5} \)
$5$ \( ( 1 + T )^{5} \)
$7$ \( -2 - 24 T - 40 T^{2} - 14 T^{3} + 3 T^{4} + T^{5} \)
$11$ \( 32 - 112 T - 144 T^{2} - 36 T^{3} + 3 T^{4} + T^{5} \)
$13$ \( 272 + 292 T + 44 T^{2} - 28 T^{3} - 4 T^{4} + T^{5} \)
$17$ \( -2188 + 208 T + 256 T^{2} - 32 T^{3} - 7 T^{4} + T^{5} \)
$19$ \( -2068 + 834 T + 186 T^{2} - 58 T^{3} - 4 T^{4} + T^{5} \)
$23$ \( -32 - 240 T - 176 T^{2} + 10 T^{4} + T^{5} \)
$29$ \( 18992 - 10016 T + 1472 T^{2} + 24 T^{3} - 19 T^{4} + T^{5} \)
$31$ \( -34 - 148 T - 104 T^{2} + 30 T^{3} + 13 T^{4} + T^{5} \)
$37$ \( ( -1 + T )^{5} \)
$41$ \( 1504 + 2256 T + 752 T^{2} - 88 T^{3} - 11 T^{4} + T^{5} \)
$43$ \( 3056 - 2880 T + 760 T^{2} - 16 T^{3} - 13 T^{4} + T^{5} \)
$47$ \( 7112 + 3610 T - 18 T^{2} - 146 T^{3} + T^{5} \)
$53$ \( 4688 - 1584 T - 424 T^{2} + 224 T^{3} - 27 T^{4} + T^{5} \)
$59$ \( -13088 - 7634 T - 1310 T^{2} - 6 T^{3} + 16 T^{4} + T^{5} \)
$61$ \( 42928 - 15840 T + 1216 T^{2} + 164 T^{3} - 27 T^{4} + T^{5} \)
$67$ \( 152 + 834 T + 178 T^{2} - 50 T^{3} - 6 T^{4} + T^{5} \)
$71$ \( -2432 + 448 T + 1440 T^{2} + 376 T^{3} + 34 T^{4} + T^{5} \)
$73$ \( 128 - 688 T + 272 T^{2} + 40 T^{3} - 16 T^{4} + T^{5} \)
$79$ \( 224 + 414 T - 334 T^{2} - 10 T^{3} + 16 T^{4} + T^{5} \)
$83$ \( -968 - 310 T + 562 T^{2} - 50 T^{3} - 14 T^{4} + T^{5} \)
$89$ \( 174496 - 46192 T + 2064 T^{2} + 360 T^{3} - 38 T^{4} + T^{5} \)
$97$ \( -11552 + 5296 T + 544 T^{2} - 228 T^{3} - 5 T^{4} + T^{5} \)
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