Properties

Label 2960.2.a.s
Level $2960$
Weight $2$
Character orbit 2960.a
Self dual yes
Analytic conductor $23.636$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 6 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 4\)

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
3.12489
−0.363328
−1.76156
0 −3.12489 0 1.00000 0 3.12489 0 6.76491 0
1.2 0 0.363328 0 1.00000 0 −0.363328 0 −2.86799 0
1.3 0 1.76156 0 1.00000 0 −1.76156 0 0.103084 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.s 3
4.b odd 2 1 1480.2.a.f 3
20.d odd 2 1 7400.2.a.m 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1480.2.a.f 3 4.b odd 2 1
2960.2.a.s 3 1.a even 1 1 trivial
7400.2.a.m 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2960))\):

\( T_{3}^{3} + T_{3}^{2} - 6 T_{3} + 2 \)
\( T_{7}^{3} - T_{7}^{2} - 6 T_{7} - 2 \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( 2 - 6 T + T^{2} + T^{3} \)
$5$ \( ( -1 + T )^{3} \)
$7$ \( -2 - 6 T - T^{2} + T^{3} \)
$11$ \( -8 + 5 T^{2} + T^{3} \)
$13$ \( T^{3} \)
$17$ \( T^{3} \)
$19$ \( -64 + 2 T + 8 T^{2} + T^{3} \)
$23$ \( ( 4 + T )^{3} \)
$29$ \( -32 - 28 T + 4 T^{2} + T^{3} \)
$31$ \( -4 + 2 T + 6 T^{2} + T^{3} \)
$37$ \( ( -1 + T )^{3} \)
$41$ \( 20 - 24 T + T^{2} + T^{3} \)
$43$ \( -64 + 10 T^{2} + T^{3} \)
$47$ \( 134 - 118 T + 3 T^{2} + T^{3} \)
$53$ \( 452 - 32 T - 11 T^{2} + T^{3} \)
$59$ \( 232 - 62 T - 4 T^{2} + T^{3} \)
$61$ \( -40 - 68 T + 10 T^{2} + T^{3} \)
$67$ \( 40 + 194 T + 28 T^{2} + T^{3} \)
$71$ \( -32 + 32 T + 15 T^{2} + T^{3} \)
$73$ \( 764 - 120 T - 5 T^{2} + T^{3} \)
$79$ \( 212 - 94 T + 2 T^{2} + T^{3} \)
$83$ \( 106 - 230 T + 5 T^{2} + T^{3} \)
$89$ \( 200 - 148 T + 6 T^{2} + T^{3} \)
$97$ \( ( -6 + T )^{3} \)
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