Properties

Label 2960.2.a.o
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: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{3} + q^{5} + ( - \beta + 2) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{3} + q^{5} + ( - \beta + 2) q^{7} + q^{9} + \beta q^{11} + ( - 2 \beta + 2) q^{13} - 2 q^{15} + ( - \beta - 2) q^{17} + 2 q^{19} + (2 \beta - 4) q^{21} - 2 \beta q^{23} + q^{25} + 4 q^{27} + (3 \beta - 2) q^{29} + ( - \beta + 6) q^{31} - 2 \beta q^{33} + ( - \beta + 2) q^{35} + q^{37} + (4 \beta - 4) q^{39} + (\beta + 2) q^{41} + ( - \beta - 4) q^{43} + q^{45} + (2 \beta + 2) q^{47} + ( - 3 \beta + 5) q^{49} + (2 \beta + 4) q^{51} + (\beta - 2) q^{53} + \beta q^{55} - 4 q^{57} + ( - 2 \beta - 6) q^{59} + ( - \beta - 2) q^{61} + ( - \beta + 2) q^{63} + ( - 2 \beta + 2) q^{65} + ( - 2 \beta + 2) q^{67} + 4 \beta q^{69} + 2 \beta q^{71} + (2 \beta + 2) q^{73} - 2 q^{75} + (\beta - 8) q^{77} + (2 \beta - 2) q^{79} - 11 q^{81} + (2 \beta - 6) q^{83} + ( - \beta - 2) q^{85} + ( - 6 \beta + 4) q^{87} + 10 q^{89} + ( - 4 \beta + 20) q^{91} + (2 \beta - 12) q^{93} + 2 q^{95} + ( - 3 \beta + 10) q^{97} + \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 2 q^{5} + 3 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{3} + 2 q^{5} + 3 q^{7} + 2 q^{9} + q^{11} + 2 q^{13} - 4 q^{15} - 5 q^{17} + 4 q^{19} - 6 q^{21} - 2 q^{23} + 2 q^{25} + 8 q^{27} - q^{29} + 11 q^{31} - 2 q^{33} + 3 q^{35} + 2 q^{37} - 4 q^{39} + 5 q^{41} - 9 q^{43} + 2 q^{45} + 6 q^{47} + 7 q^{49} + 10 q^{51} - 3 q^{53} + q^{55} - 8 q^{57} - 14 q^{59} - 5 q^{61} + 3 q^{63} + 2 q^{65} + 2 q^{67} + 4 q^{69} + 2 q^{71} + 6 q^{73} - 4 q^{75} - 15 q^{77} - 2 q^{79} - 22 q^{81} - 10 q^{83} - 5 q^{85} + 2 q^{87} + 20 q^{89} + 36 q^{91} - 22 q^{93} + 4 q^{95} + 17 q^{97} + q^{99}+O(q^{100}) \) 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
3.37228
−2.37228
0 −2.00000 0 1.00000 0 −1.37228 0 1.00000 0
1.2 0 −2.00000 0 1.00000 0 4.37228 0 1.00000 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.o 2
4.b odd 2 1 370.2.a.f 2
12.b even 2 1 3330.2.a.bb 2
20.d odd 2 1 1850.2.a.q 2
20.e even 4 2 1850.2.b.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.a.f 2 4.b odd 2 1
1850.2.a.q 2 20.d odd 2 1
1850.2.b.m 4 20.e even 4 2
2960.2.a.o 2 1.a even 1 1 trivial
3330.2.a.bb 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2960))\):

\( T_{3} + 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 3T_{7} - 6 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 2)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$11$ \( T^{2} - T - 8 \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 32 \) Copy content Toggle raw display
$17$ \( T^{2} + 5T - 2 \) Copy content Toggle raw display
$19$ \( (T - 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 2T - 32 \) Copy content Toggle raw display
$29$ \( T^{2} + T - 74 \) Copy content Toggle raw display
$31$ \( T^{2} - 11T + 22 \) Copy content Toggle raw display
$37$ \( (T - 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 5T - 2 \) Copy content Toggle raw display
$43$ \( T^{2} + 9T + 12 \) Copy content Toggle raw display
$47$ \( T^{2} - 6T - 24 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T - 6 \) Copy content Toggle raw display
$59$ \( T^{2} + 14T + 16 \) Copy content Toggle raw display
$61$ \( T^{2} + 5T - 2 \) Copy content Toggle raw display
$67$ \( T^{2} - 2T - 32 \) Copy content Toggle raw display
$71$ \( T^{2} - 2T - 32 \) Copy content Toggle raw display
$73$ \( T^{2} - 6T - 24 \) Copy content Toggle raw display
$79$ \( T^{2} + 2T - 32 \) Copy content Toggle raw display
$83$ \( T^{2} + 10T - 8 \) Copy content Toggle raw display
$89$ \( (T - 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 17T - 2 \) Copy content Toggle raw display
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