Properties

Label 2960.2.a.t
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: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 740)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - q^{5} + ( -2 \beta_{1} + \beta_{2} ) q^{7} + ( -\beta_{1} - \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} - q^{5} + ( -2 \beta_{1} + \beta_{2} ) q^{7} + ( -\beta_{1} - \beta_{2} ) q^{9} + ( 2 - 2 \beta_{1} ) q^{11} + ( 3 - 3 \beta_{1} - \beta_{2} ) q^{13} -\beta_{2} q^{15} + ( 3 + \beta_{1} + 3 \beta_{2} ) q^{17} + ( -6 + \beta_{2} ) q^{19} + ( 5 - 3 \beta_{1} - \beta_{2} ) q^{21} + 2 \beta_{1} q^{23} + q^{25} + ( -2 - 2 \beta_{2} ) q^{27} + ( -2 + 4 \beta_{1} + 2 \beta_{2} ) q^{29} + 3 \beta_{2} q^{31} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{33} + ( 2 \beta_{1} - \beta_{2} ) q^{35} - q^{37} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{39} + ( -2 - 2 \beta_{2} ) q^{41} + ( -2 - 4 \beta_{1} - 2 \beta_{2} ) q^{43} + ( \beta_{1} + \beta_{2} ) q^{45} + ( -6 + 4 \beta_{1} - \beta_{2} ) q^{47} + ( 8 - \beta_{1} + 3 \beta_{2} ) q^{49} + ( 8 - 2 \beta_{1} ) q^{51} + ( 6 \beta_{1} + 2 \beta_{2} ) q^{53} + ( -2 + 2 \beta_{1} ) q^{55} + ( 3 - \beta_{1} - 7 \beta_{2} ) q^{57} + ( -4 + 4 \beta_{1} + 5 \beta_{2} ) q^{59} + ( 2 + 4 \beta_{1} + 8 \beta_{2} ) q^{61} + ( 4 \beta_{1} + 3 \beta_{2} ) q^{63} + ( -3 + 3 \beta_{1} + \beta_{2} ) q^{65} + ( -6 - 2 \beta_{1} - 7 \beta_{2} ) q^{67} + ( -2 + 2 \beta_{1} ) q^{69} + ( 10 + 2 \beta_{1} - 2 \beta_{2} ) q^{71} + 2 \beta_{1} q^{73} + \beta_{2} q^{75} + ( 10 - 2 \beta_{1} + 6 \beta_{2} ) q^{77} + ( 2 + 2 \beta_{1} - 3 \beta_{2} ) q^{79} + ( -6 + 5 \beta_{1} + 3 \beta_{2} ) q^{81} + ( 6 - 2 \beta_{1} + \beta_{2} ) q^{83} + ( -3 - \beta_{1} - 3 \beta_{2} ) q^{85} + ( 2 + 2 \beta_{1} - 4 \beta_{2} ) q^{87} + ( -6 \beta_{1} - 6 \beta_{2} ) q^{89} + ( 10 + 10 \beta_{2} ) q^{91} + ( 9 - 3 \beta_{1} - 3 \beta_{2} ) q^{93} + ( 6 - \beta_{2} ) q^{95} + ( 6 - 2 \beta_{2} ) q^{97} + ( 2 + 2 \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{5} - 2q^{7} - q^{9} + O(q^{10}) \) \( 3q - 3q^{5} - 2q^{7} - q^{9} + 4q^{11} + 6q^{13} + 10q^{17} - 18q^{19} + 12q^{21} + 2q^{23} + 3q^{25} - 6q^{27} - 2q^{29} + 4q^{33} + 2q^{35} - 3q^{37} - 2q^{39} - 6q^{41} - 10q^{43} + q^{45} - 14q^{47} + 23q^{49} + 22q^{51} + 6q^{53} - 4q^{55} + 8q^{57} - 8q^{59} + 10q^{61} + 4q^{63} - 6q^{65} - 20q^{67} - 4q^{69} + 32q^{71} + 2q^{73} + 28q^{77} + 8q^{79} - 13q^{81} + 16q^{83} - 10q^{85} + 8q^{87} - 6q^{89} + 30q^{91} + 24q^{93} + 18q^{95} + 18q^{97} + 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

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

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
0.311108
2.17009
−1.48119
0 −2.21432 0 −1.00000 0 −2.83654 0 1.90321 0
1.2 0 0.539189 0 −1.00000 0 −3.80098 0 −2.70928 0
1.3 0 1.67513 0 −1.00000 0 4.63752 0 −0.193937 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.t 3
4.b odd 2 1 740.2.a.e 3
12.b even 2 1 6660.2.a.q 3
20.d odd 2 1 3700.2.a.i 3
20.e even 4 2 3700.2.d.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.a.e 3 4.b odd 2 1
2960.2.a.t 3 1.a even 1 1 trivial
3700.2.a.i 3 20.d odd 2 1
3700.2.d.h 6 20.e even 4 2
6660.2.a.q 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2960))\):

\( T_{3}^{3} - 4 T_{3} + 2 \)
\( T_{7}^{3} + 2 T_{7}^{2} - 20 T_{7} - 50 \)
\( T_{13}^{3} - 6 T_{13}^{2} - 16 T_{13} + 100 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( 2 - 4 T + T^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( -50 - 20 T + 2 T^{2} + T^{3} \)
$11$ \( 16 - 8 T - 4 T^{2} + T^{3} \)
$13$ \( 100 - 16 T - 6 T^{2} + T^{3} \)
$17$ \( 148 - 10 T^{2} + T^{3} \)
$19$ \( 194 + 104 T + 18 T^{2} + T^{3} \)
$23$ \( 8 - 12 T - 2 T^{2} + T^{3} \)
$29$ \( -184 - 52 T + 2 T^{2} + T^{3} \)
$31$ \( 54 - 36 T + T^{3} \)
$37$ \( ( 1 + T )^{3} \)
$41$ \( -40 - 4 T + 6 T^{2} + T^{3} \)
$43$ \( 8 - 20 T + 10 T^{2} + T^{3} \)
$47$ \( -74 + 14 T^{2} + T^{3} \)
$53$ \( -200 - 100 T - 6 T^{2} + T^{3} \)
$59$ \( -158 - 92 T + 8 T^{2} + T^{3} \)
$61$ \( 2056 - 212 T - 10 T^{2} + T^{3} \)
$67$ \( -1850 - 48 T + 20 T^{2} + T^{3} \)
$71$ \( -736 + 304 T - 32 T^{2} + T^{3} \)
$73$ \( 8 - 12 T - 2 T^{2} + T^{3} \)
$79$ \( 262 - 40 T - 8 T^{2} + T^{3} \)
$83$ \( -74 + 64 T - 16 T^{2} + T^{3} \)
$89$ \( -216 - 180 T + 6 T^{2} + T^{3} \)
$97$ \( -136 + 92 T - 18 T^{2} + T^{3} \)
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