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: \( x^{3} - x^{2} - 3x + 1 \) 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 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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - q^{5} + (\beta_{2} - 2 \beta_1) q^{7} + ( - \beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} - q^{5} + (\beta_{2} - 2 \beta_1) q^{7} + ( - \beta_{2} - \beta_1) q^{9} + ( - 2 \beta_1 + 2) q^{11} + ( - \beta_{2} - 3 \beta_1 + 3) q^{13} - \beta_{2} q^{15} + (3 \beta_{2} + \beta_1 + 3) q^{17} + (\beta_{2} - 6) q^{19} + ( - \beta_{2} - 3 \beta_1 + 5) q^{21} + 2 \beta_1 q^{23} + q^{25} + ( - 2 \beta_{2} - 2) q^{27} + (2 \beta_{2} + 4 \beta_1 - 2) q^{29} + 3 \beta_{2} q^{31} + (2 \beta_{2} - 2 \beta_1 + 2) q^{33} + ( - \beta_{2} + 2 \beta_1) q^{35} - q^{37} + (4 \beta_{2} - 2 \beta_1) q^{39} + ( - 2 \beta_{2} - 2) q^{41} + ( - 2 \beta_{2} - 4 \beta_1 - 2) q^{43} + (\beta_{2} + \beta_1) q^{45} + ( - \beta_{2} + 4 \beta_1 - 6) q^{47} + (3 \beta_{2} - \beta_1 + 8) q^{49} + ( - 2 \beta_1 + 8) q^{51} + (2 \beta_{2} + 6 \beta_1) q^{53} + (2 \beta_1 - 2) q^{55} + ( - 7 \beta_{2} - \beta_1 + 3) q^{57} + (5 \beta_{2} + 4 \beta_1 - 4) q^{59} + (8 \beta_{2} + 4 \beta_1 + 2) q^{61} + (3 \beta_{2} + 4 \beta_1) q^{63} + (\beta_{2} + 3 \beta_1 - 3) q^{65} + ( - 7 \beta_{2} - 2 \beta_1 - 6) q^{67} + (2 \beta_1 - 2) q^{69} + ( - 2 \beta_{2} + 2 \beta_1 + 10) q^{71} + 2 \beta_1 q^{73} + \beta_{2} q^{75} + (6 \beta_{2} - 2 \beta_1 + 10) q^{77} + ( - 3 \beta_{2} + 2 \beta_1 + 2) q^{79} + (3 \beta_{2} + 5 \beta_1 - 6) q^{81} + (\beta_{2} - 2 \beta_1 + 6) q^{83} + ( - 3 \beta_{2} - \beta_1 - 3) q^{85} + ( - 4 \beta_{2} + 2 \beta_1 + 2) q^{87} + ( - 6 \beta_{2} - 6 \beta_1) q^{89} + (10 \beta_{2} + 10) q^{91} + ( - 3 \beta_{2} - 3 \beta_1 + 9) q^{93} + ( - \beta_{2} + 6) q^{95} + ( - 2 \beta_{2} + 6) q^{97} + (2 \beta_1 + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} - 2 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{5} - 2 q^{7} - q^{9} + 4 q^{11} + 6 q^{13} + 10 q^{17} - 18 q^{19} + 12 q^{21} + 2 q^{23} + 3 q^{25} - 6 q^{27} - 2 q^{29} + 4 q^{33} + 2 q^{35} - 3 q^{37} - 2 q^{39} - 6 q^{41} - 10 q^{43} + q^{45} - 14 q^{47} + 23 q^{49} + 22 q^{51} + 6 q^{53} - 4 q^{55} + 8 q^{57} - 8 q^{59} + 10 q^{61} + 4 q^{63} - 6 q^{65} - 20 q^{67} - 4 q^{69} + 32 q^{71} + 2 q^{73} + 28 q^{77} + 8 q^{79} - 13 q^{81} + 16 q^{83} - 10 q^{85} + 8 q^{87} - 6 q^{89} + 30 q^{91} + 24 q^{93} + 18 q^{95} + 18 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) 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
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2960))\):

\( T_{3}^{3} - 4T_{3} + 2 \) Copy content Toggle raw display
\( T_{7}^{3} + 2T_{7}^{2} - 20T_{7} - 50 \) Copy content Toggle raw display
\( T_{13}^{3} - 6T_{13}^{2} - 16T_{13} + 100 \) Copy content Toggle raw display

Hecke characteristic polynomials

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