Properties

Label 2960.2.a.q
Level $2960$
Weight $2$
Character orbit 2960.a
Self dual yes
Analytic conductor $23.636$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) 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 370)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{3} + q^{5} + ( - \beta + 3) q^{7} + (2 \beta + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{3} + q^{5} + ( - \beta + 3) q^{7} + (2 \beta + 1) q^{9} + ( - 2 \beta + 2) q^{11} + (2 \beta - 2) q^{13} + (\beta + 1) q^{15} + ( - 2 \beta + 2) q^{17} + (3 \beta - 1) q^{19} + 2 \beta q^{21} + 8 q^{23} + q^{25} + 4 q^{27} + ( - 4 \beta - 2) q^{29} + (\beta + 1) q^{31} - 4 q^{33} + ( - \beta + 3) q^{35} + q^{37} + 4 q^{39} - 2 q^{41} + 4 \beta q^{43} + (2 \beta + 1) q^{45} + ( - \beta + 3) q^{47} + ( - 6 \beta + 5) q^{49} - 4 q^{51} - 6 q^{53} + ( - 2 \beta + 2) q^{55} + (2 \beta + 8) q^{57} + ( - 3 \beta + 5) q^{59} + (4 \beta + 2) q^{61} + (5 \beta - 3) q^{63} + (2 \beta - 2) q^{65} + ( - 5 \beta - 5) q^{67} + (8 \beta + 8) q^{69} + (4 \beta + 4) q^{71} + (4 \beta + 6) q^{73} + (\beta + 1) q^{75} + ( - 8 \beta + 12) q^{77} + (\beta - 7) q^{79} + ( - 2 \beta + 1) q^{81} + ( - \beta + 7) q^{83} + ( - 2 \beta + 2) q^{85} + ( - 6 \beta - 14) q^{87} - 2 q^{89} + (8 \beta - 12) q^{91} + (2 \beta + 4) q^{93} + (3 \beta - 1) q^{95} - 2 q^{97} + (2 \beta - 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9} + 4 q^{11} - 4 q^{13} + 2 q^{15} + 4 q^{17} - 2 q^{19} + 16 q^{23} + 2 q^{25} + 8 q^{27} - 4 q^{29} + 2 q^{31} - 8 q^{33} + 6 q^{35} + 2 q^{37} + 8 q^{39} - 4 q^{41} + 2 q^{45} + 6 q^{47} + 10 q^{49} - 8 q^{51} - 12 q^{53} + 4 q^{55} + 16 q^{57} + 10 q^{59} + 4 q^{61} - 6 q^{63} - 4 q^{65} - 10 q^{67} + 16 q^{69} + 8 q^{71} + 12 q^{73} + 2 q^{75} + 24 q^{77} - 14 q^{79} + 2 q^{81} + 14 q^{83} + 4 q^{85} - 28 q^{87} - 4 q^{89} - 24 q^{91} + 8 q^{93} - 2 q^{95} - 4 q^{97} - 20 q^{99}+O(q^{100}) \) 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
−1.73205
1.73205
0 −0.732051 0 1.00000 0 4.73205 0 −2.46410 0
1.2 0 2.73205 0 1.00000 0 1.26795 0 4.46410 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.q 2
4.b odd 2 1 370.2.a.e 2
12.b even 2 1 3330.2.a.bd 2
20.d odd 2 1 1850.2.a.x 2
20.e even 4 2 1850.2.b.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.a.e 2 4.b odd 2 1
1850.2.a.x 2 20.d odd 2 1
1850.2.b.l 4 20.e even 4 2
2960.2.a.q 2 1.a even 1 1 trivial
3330.2.a.bd 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2960))\):

\( T_{3}^{2} - 2T_{3} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 6T_{7} + 6 \) Copy content Toggle raw display
\( T_{13}^{2} + 4T_{13} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 6T + 6 \) Copy content Toggle raw display
$11$ \( T^{2} - 4T - 8 \) Copy content Toggle raw display
$13$ \( T^{2} + 4T - 8 \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 8 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 26 \) Copy content Toggle raw display
$23$ \( (T - 8)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 4T - 44 \) Copy content Toggle raw display
$31$ \( T^{2} - 2T - 2 \) Copy content Toggle raw display
$37$ \( (T - 1)^{2} \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 48 \) Copy content Toggle raw display
$47$ \( T^{2} - 6T + 6 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 10T - 2 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T - 44 \) Copy content Toggle raw display
$67$ \( T^{2} + 10T - 50 \) Copy content Toggle raw display
$71$ \( T^{2} - 8T - 32 \) Copy content Toggle raw display
$73$ \( T^{2} - 12T - 12 \) Copy content Toggle raw display
$79$ \( T^{2} + 14T + 46 \) Copy content Toggle raw display
$83$ \( T^{2} - 14T + 46 \) Copy content Toggle raw display
$89$ \( (T + 2)^{2} \) Copy content Toggle raw display
$97$ \( (T + 2)^{2} \) Copy content Toggle raw display
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