Properties

Label 2960.2.a.z
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: 5.5.935504.1
Defining polynomial: \( x^{5} - x^{4} - 8x^{3} + 4x^{2} + 8x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - q^{5} - \beta_{3} q^{7} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - q^{5} - \beta_{3} q^{7} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{9} + (\beta_{4} + \beta_{2} - \beta_1 + 2) q^{11} + (\beta_{4} - \beta_{3} + 1) q^{13} - \beta_1 q^{15} + ( - \beta_{4} - \beta_{3} + 2 \beta_1 - 1) q^{17} + ( - 3 \beta_{4} - 2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{19} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{21} + ( - 2 \beta_1 + 2) q^{23} + q^{25} + (\beta_{4} + \beta_{3} + \beta_{2} + 2) q^{27} + ( - 2 \beta_{3} - 2) q^{29} + (3 \beta_{4} + \beta_{3} - 3 \beta_{2} - \beta_1 + 2) q^{31} + (\beta_{4} + 2 \beta_{3} + \beta_{2} + 3 \beta_1 - 2) q^{33} + \beta_{3} q^{35} + q^{37} + (\beta_{3} + 3 \beta_1) q^{39} + (\beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 - 2) q^{41} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{43} + (\beta_{3} - \beta_{2} - \beta_1) q^{45} + (\beta_{3} + 2 \beta_{2} + 2) q^{47} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{49} + ( - \beta_{3} + \beta_1 + 4) q^{51} + ( - 3 \beta_{4} + \beta_{2} + \beta_1 - 4) q^{53} + ( - \beta_{4} - \beta_{2} + \beta_1 - 2) q^{55} + (\beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 1) q^{57} + (3 \beta_{4} + \beta_{3} - 3 \beta_{2} + \beta_1 + 6) q^{59} + (4 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 2) q^{61} + ( - \beta_{4} + \beta_{2} - 2 \beta_1 + 4) q^{63} + ( - \beta_{4} + \beta_{3} - 1) q^{65} + ( - 3 \beta_{4} - 2 \beta_{3} + \beta_{2} + 4 \beta_1 + 2) q^{67} + (2 \beta_{3} - 2 \beta_{2} - 6) q^{69} + ( - 3 \beta_{4} - 2 \beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{71} + ( - 3 \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 + 4) q^{73} + \beta_1 q^{75} + ( - \beta_{4} - 2 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 2) q^{77} + ( - \beta_{4} - \beta_{3} - \beta_{2} + 3 \beta_1) q^{79} + (\beta_{4} + 3 \beta_{3} + 2) q^{81} + ( - 2 \beta_{4} - \beta_1 + 6) q^{83} + (\beta_{4} + \beta_{3} - 2 \beta_1 + 1) q^{85} + (2 \beta_{3} - 2 \beta_{2} - 2) q^{87} + (2 \beta_{4} - 4 \beta_{2} - 2 \beta_1 - 4) q^{89} + ( - 4 \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1 + 6) q^{91} + ( - 3 \beta_{4} - 3 \beta_{3} + 1) q^{93} + (3 \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{95} + (2 \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 6) q^{97} + ( - 2 \beta_{4} - 4 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{3} - 5 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + q^{3} - 5 q^{5} + q^{7} + 2 q^{9} + 7 q^{11} + 4 q^{13} - q^{15} + 10 q^{19} - 5 q^{21} + 8 q^{23} + 5 q^{25} + 7 q^{27} - 8 q^{29} + 2 q^{31} - 11 q^{33} - q^{35} + 5 q^{37} + 2 q^{39} - 13 q^{41} + 18 q^{43} - 2 q^{45} + 9 q^{47} - 6 q^{49} + 22 q^{51} - 13 q^{53} - 7 q^{55} + 4 q^{57} + 24 q^{59} - 2 q^{61} + 20 q^{63} - 4 q^{65} + 22 q^{67} - 32 q^{69} + 21 q^{71} + 29 q^{73} + q^{75} + 11 q^{77} + 6 q^{79} + 5 q^{81} + 33 q^{83} - 12 q^{87} - 26 q^{89} + 38 q^{91} + 14 q^{93} - 10 q^{95} + 26 q^{97} + 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 8x^{3} + 4x^{2} + 8x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} - \nu^{3} - 7\nu^{2} + 4\nu + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - \nu^{3} - 8\nu^{2} + 5\nu + 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\nu^{4} + 3\nu^{3} + 15\nu^{2} - 15\nu - 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + \beta_{2} + 6\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 6\beta_{3} + 9\beta_{2} + 9\beta _1 + 20 \) Copy content Toggle raw display

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
−2.44705
−0.550328
−0.355205
1.43118
2.92141
0 −2.44705 0 −1.00000 0 3.62974 0 2.98807 0
1.2 0 −0.550328 0 −1.00000 0 −1.08387 0 −2.69714 0
1.3 0 −0.355205 0 −1.00000 0 −3.27534 0 −2.87383 0
1.4 0 1.43118 0 −1.00000 0 1.96628 0 −0.951735 0
1.5 0 2.92141 0 −1.00000 0 −0.236809 0 5.53464 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.z 5
4.b odd 2 1 1480.2.a.h 5
20.d odd 2 1 7400.2.a.q 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1480.2.a.h 5 4.b odd 2 1
2960.2.a.z 5 1.a even 1 1 trivial
7400.2.a.q 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2960))\):

\( T_{3}^{5} - T_{3}^{4} - 8T_{3}^{3} + 4T_{3}^{2} + 8T_{3} + 2 \) Copy content Toggle raw display
\( T_{7}^{5} - T_{7}^{4} - 14T_{7}^{3} + 8T_{7}^{2} + 28T_{7} + 6 \) Copy content Toggle raw display
\( T_{13}^{5} - 4T_{13}^{4} - 28T_{13}^{3} + 172T_{13}^{2} - 284T_{13} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - T^{4} - 8 T^{3} + 4 T^{2} + 8 T + 2 \) Copy content Toggle raw display
$5$ \( (T + 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - T^{4} - 14 T^{3} + 8 T^{2} + \cdots + 6 \) Copy content Toggle raw display
$11$ \( T^{5} - 7 T^{4} - 20 T^{3} + 192 T^{2} + \cdots - 288 \) Copy content Toggle raw display
$13$ \( T^{5} - 4 T^{4} - 28 T^{3} + 172 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$17$ \( T^{5} - 28 T^{3} - 20 T^{2} + \cdots + 112 \) Copy content Toggle raw display
$19$ \( T^{5} - 10 T^{4} - 26 T^{3} + \cdots + 1544 \) Copy content Toggle raw display
$23$ \( T^{5} - 8 T^{4} - 8 T^{3} + 128 T^{2} + \cdots - 192 \) Copy content Toggle raw display
$29$ \( T^{5} + 8 T^{4} - 32 T^{3} - 240 T^{2} + \cdots + 896 \) Copy content Toggle raw display
$31$ \( T^{5} - 2 T^{4} - 130 T^{3} + \cdots + 7556 \) Copy content Toggle raw display
$37$ \( (T - 1)^{5} \) Copy content Toggle raw display
$41$ \( T^{5} + 13 T^{4} - 40 T^{3} + \cdots + 224 \) Copy content Toggle raw display
$43$ \( T^{5} - 18 T^{4} + 40 T^{3} + \cdots + 9056 \) Copy content Toggle raw display
$47$ \( T^{5} - 9 T^{4} - 54 T^{3} + 672 T^{2} + \cdots + 886 \) Copy content Toggle raw display
$53$ \( T^{5} + 13 T^{4} - 16 T^{3} + \cdots - 4656 \) Copy content Toggle raw display
$59$ \( T^{5} - 24 T^{4} + 38 T^{3} + \cdots - 44056 \) Copy content Toggle raw display
$61$ \( T^{5} + 2 T^{4} - 216 T^{3} + \cdots + 43424 \) Copy content Toggle raw display
$67$ \( T^{5} - 22 T^{4} + 90 T^{3} + \cdots + 7944 \) Copy content Toggle raw display
$71$ \( T^{5} - 21 T^{4} + 100 T^{3} + \cdots + 11072 \) Copy content Toggle raw display
$73$ \( T^{5} - 29 T^{4} + 264 T^{3} + \cdots + 7088 \) Copy content Toggle raw display
$79$ \( T^{5} - 6 T^{4} - 78 T^{3} + 18 T^{2} + \cdots + 692 \) Copy content Toggle raw display
$83$ \( T^{5} - 33 T^{4} + 368 T^{3} + \cdots + 6074 \) Copy content Toggle raw display
$89$ \( T^{5} + 26 T^{4} + 40 T^{3} + \cdots + 43936 \) Copy content Toggle raw display
$97$ \( T^{5} - 26 T^{4} + 120 T^{3} + \cdots + 9024 \) Copy content Toggle raw display
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