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: \( x^{5} - 10x^{3} - 2x^{2} + 14x - 4 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - q^{5} + (\beta_{3} - \beta_1 - 1) q^{7} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - q^{5} + (\beta_{3} - \beta_1 - 1) q^{7} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{9} + ( - \beta_{4} - 1) q^{11} + (\beta_{3} + \beta_{2} - \beta_1) q^{13} - \beta_1 q^{15} + ( - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{17} + (2 \beta_{2} + \beta_1) q^{19} + (\beta_{3} - \beta_{2} - 3 \beta_1 - 2) q^{21} + (2 \beta_1 - 2) q^{23} + q^{25} + (\beta_{4} - \beta_{3} + 2 \beta_1 + 2) q^{27} + (\beta_{4} - 2 \beta_{2} + 2 \beta_1 + 5) q^{29} + (\beta_{3} + \beta_1 - 3) q^{31} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{33} + ( - \beta_{3} + \beta_1 + 1) q^{35} + q^{37} + (\beta_{4} + \beta_{3} - 2 \beta_{2} - 2) q^{39} + ( - \beta_{4} + 2 \beta_{3} + 1) q^{41} + ( - \beta_{4} - 2 \beta_{3} + 3) q^{43} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{45} + (\beta_{4} + \beta_{3} + 3 \beta_1) q^{47} + (\beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_1) q^{49} + (\beta_{4} - \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 4) q^{51} + ( - \beta_{4} - 2 \beta_1 + 5) q^{53} + (\beta_{4} + 1) q^{55} + (2 \beta_{4} - \beta_{3} - \beta_{2} + 5 \beta_1 + 4) q^{57} + ( - \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1 - 4) q^{59} + (\beta_{4} + 2 \beta_{2} + 5) q^{61} + ( - \beta_{4} - 2 \beta_{2} - 5 \beta_1 - 7) q^{63} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{65} + ( - 2 \beta_{2} - \beta_1 + 2) q^{67} + ( - 2 \beta_{3} + 2 \beta_{2} + 8) q^{69} + ( - 2 \beta_{2} - 2 \beta_1 - 6) q^{71} + ( - 2 \beta_{2} + 4) q^{73} + \beta_1 q^{75} + (\beta_{4} - 2 \beta_{3} + 2 \beta_1 + 3) q^{77} + ( - \beta_{4} + \beta_{3} - 3 \beta_1 - 4) q^{79} + (\beta_{3} + \beta_{2} + 3 \beta_1 + 1) q^{81} + (2 \beta_{2} + 3 \beta_1 + 2) q^{83} + (\beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{85} + ( - 2 \beta_{4} - 2 \beta_{3} + 6 \beta_{2} + 4 \beta_1 + 6) q^{87} + ( - 2 \beta_{4} + 2 \beta_{2} + 2 \beta_1 + 6) q^{89} + (\beta_{4} + 3 \beta_{3} + 6) q^{91} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 + 6) q^{93} + ( - 2 \beta_{2} - \beta_1) q^{95} + ( - \beta_{4} + 4 \beta_{2} - 1) q^{97} + (\beta_{4} + 2 \beta_{3} - 4 \beta_1 - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{5} - 3 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{5} - 3 q^{7} + 5 q^{9} - 3 q^{11} + 4 q^{13} + 7 q^{17} + 4 q^{19} - 10 q^{21} - 10 q^{23} + 5 q^{25} + 6 q^{27} + 19 q^{29} - 13 q^{31} + 6 q^{33} + 3 q^{35} + 5 q^{37} - 14 q^{39} + 11 q^{41} + 13 q^{43} - 5 q^{45} + 2 q^{49} + 12 q^{51} + 27 q^{53} + 3 q^{55} + 12 q^{57} - 16 q^{59} + 27 q^{61} - 37 q^{63} - 4 q^{65} + 6 q^{67} + 40 q^{69} - 34 q^{71} + 16 q^{73} + 9 q^{77} - 16 q^{79} + 9 q^{81} + 14 q^{83} - 7 q^{85} + 42 q^{87} + 38 q^{89} + 34 q^{91} + 30 q^{93} - 4 q^{95} + 5 q^{97} - 23 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 8\nu^{2} - 4\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} - 10\nu^{2} - 2\nu + 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} + 2\nu^{3} - 10\nu^{2} - 18\nu + 8 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - \beta_{3} + 8\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{3} + 10\beta_{2} + 12\beta _1 + 28 \) 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
−2.65435
−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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2960))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - 10 T^{3} - 2 T^{2} + 14 T - 4 \) Copy content Toggle raw display
$5$ \( (T + 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} + 3 T^{4} - 14 T^{3} - 40 T^{2} + \cdots - 2 \) Copy content Toggle raw display
$11$ \( T^{5} + 3 T^{4} - 36 T^{3} - 144 T^{2} + \cdots + 32 \) Copy content Toggle raw display
$13$ \( T^{5} - 4 T^{4} - 28 T^{3} + 44 T^{2} + \cdots + 272 \) Copy content Toggle raw display
$17$ \( T^{5} - 7 T^{4} - 32 T^{3} + \cdots - 2188 \) Copy content Toggle raw display
$19$ \( T^{5} - 4 T^{4} - 58 T^{3} + \cdots - 2068 \) Copy content Toggle raw display
$23$ \( T^{5} + 10 T^{4} - 176 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$29$ \( T^{5} - 19 T^{4} + 24 T^{3} + \cdots + 18992 \) Copy content Toggle raw display
$31$ \( T^{5} + 13 T^{4} + 30 T^{3} - 104 T^{2} + \cdots - 34 \) Copy content Toggle raw display
$37$ \( (T - 1)^{5} \) Copy content Toggle raw display
$41$ \( T^{5} - 11 T^{4} - 88 T^{3} + \cdots + 1504 \) Copy content Toggle raw display
$43$ \( T^{5} - 13 T^{4} - 16 T^{3} + \cdots + 3056 \) Copy content Toggle raw display
$47$ \( T^{5} - 146 T^{3} - 18 T^{2} + \cdots + 7112 \) Copy content Toggle raw display
$53$ \( T^{5} - 27 T^{4} + 224 T^{3} + \cdots + 4688 \) Copy content Toggle raw display
$59$ \( T^{5} + 16 T^{4} - 6 T^{3} + \cdots - 13088 \) Copy content Toggle raw display
$61$ \( T^{5} - 27 T^{4} + 164 T^{3} + \cdots + 42928 \) Copy content Toggle raw display
$67$ \( T^{5} - 6 T^{4} - 50 T^{3} + 178 T^{2} + \cdots + 152 \) Copy content Toggle raw display
$71$ \( T^{5} + 34 T^{4} + 376 T^{3} + \cdots - 2432 \) Copy content Toggle raw display
$73$ \( T^{5} - 16 T^{4} + 40 T^{3} + \cdots + 128 \) Copy content Toggle raw display
$79$ \( T^{5} + 16 T^{4} - 10 T^{3} + \cdots + 224 \) Copy content Toggle raw display
$83$ \( T^{5} - 14 T^{4} - 50 T^{3} + \cdots - 968 \) Copy content Toggle raw display
$89$ \( T^{5} - 38 T^{4} + 360 T^{3} + \cdots + 174496 \) Copy content Toggle raw display
$97$ \( T^{5} - 5 T^{4} - 228 T^{3} + \cdots - 11552 \) Copy content Toggle raw display
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