Properties

Label 2960.2.a.w
Level $2960$
Weight $2$
Character orbit 2960.a
Self dual yes
Analytic conductor $23.636$
Analytic rank $1$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.973904.1
Defining polynomial: \( x^{5} - 2x^{4} - 8x^{3} + 6x^{2} + 19x + 6 \) 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 185)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} - q^{5} + (\beta_{4} - \beta_{2} - \beta_1 - 2) q^{7} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{3} - q^{5} + (\beta_{4} - \beta_{2} - \beta_1 - 2) q^{7} + (\beta_{2} + 1) q^{9} + (\beta_{3} + 1) q^{11} + ( - \beta_{3} - \beta_{2} + 1) q^{13} + ( - \beta_1 + 1) q^{15} + ( - 2 \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{17} + ( - \beta_{3} + 2 \beta_{2} + \beta_1) q^{19} + ( - 2 \beta_{4} + \beta_{2} - 2 \beta_1 + 2) q^{21} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{23} + q^{25} + (\beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1) q^{27} - 2 \beta_1 q^{29} + ( - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 1) q^{31} + (2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{33} + ( - \beta_{4} + \beta_{2} + \beta_1 + 2) q^{35} + q^{37} + ( - 3 \beta_{4} - \beta_{2} + 1) q^{39} + ( - \beta_{3} - 1) q^{41} + ( - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{43} + ( - \beta_{2} - 1) q^{45} + (\beta_{4} - 3 \beta_{2} - 3 \beta_1) q^{47} + ( - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 5) q^{49} + (3 \beta_{4} - 4 \beta_{3} + \beta_{2} - 1) q^{51} + (4 \beta_{4} - \beta_{3} - 2 \beta_1 - 1) q^{53} + ( - \beta_{3} - 1) q^{55} + (3 \beta_{3} - 3 \beta_{2} - 1) q^{57} + (\beta_{4} + \beta_{3} - \beta_{2} - 3 \beta_1 + 7) q^{59} + (2 \beta_{4} + 2 \beta_{2} - 4) q^{61} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 - 6) q^{63} + (\beta_{3} + \beta_{2} - 1) q^{65} + ( - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 - 4) q^{67} + ( - 2 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 4) q^{69} + (2 \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{71} + ( - 4 \beta_{4} - \beta_{3} + 4 \beta_1 + 1) q^{73} + (\beta_1 - 1) q^{75} + (2 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 3) q^{77} + (3 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 7) q^{79} + ( - \beta_{3} - \beta_{2} - 6) q^{81} + ( - 2 \beta_{4} - 2 \beta_{3} + \beta_1 - 5) q^{83} + (2 \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{85} + ( - 2 \beta_{2} - 2 \beta_1 - 6) q^{87} + (2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 2) q^{89} + (\beta_{4} + 4 \beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{91} + (4 \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 - 5) q^{93} + (\beta_{3} - 2 \beta_{2} - \beta_1) q^{95} + (2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 6) q^{97} + ( - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 3 q^{3} - 5 q^{5} - 11 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 3 q^{3} - 5 q^{5} - 11 q^{7} + 6 q^{9} + 5 q^{11} + 4 q^{13} + 3 q^{15} + 4 q^{19} + 3 q^{21} - 4 q^{23} + 5 q^{25} - 3 q^{27} - 4 q^{29} - 8 q^{31} + 5 q^{33} + 11 q^{35} + 5 q^{37} - 2 q^{39} - 5 q^{41} - 10 q^{43} - 6 q^{45} - 7 q^{47} + 22 q^{49} + 2 q^{51} - q^{53} - 5 q^{55} - 8 q^{57} + 30 q^{59} - 14 q^{61} - 30 q^{63} - 4 q^{65} - 24 q^{67} + 8 q^{69} + 7 q^{71} + 5 q^{73} - 3 q^{75} - 17 q^{77} - 28 q^{79} - 31 q^{81} - 27 q^{83} - 36 q^{87} + 6 q^{89} + 14 q^{91} - 22 q^{93} - 4 q^{95} - 26 q^{97} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 8x^{3} + 6x^{2} + 19x + 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{4} + 4\nu^{3} + 2\nu^{2} - 12\nu - 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 3\nu^{3} - 4\nu^{2} + 9\nu + 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + 2\beta_{2} + 7\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} + 3\beta_{3} + 10\beta_{2} + 20\beta _1 + 18 \) 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
−1.62871
−1.38679
−0.383115
2.10563
3.29298
0 −2.62871 0 −1.00000 0 −2.55244 0 3.91009 0
1.2 0 −2.38679 0 −1.00000 0 −4.78404 0 2.69675 0
1.3 0 −1.38311 0 −1.00000 0 2.62521 0 −1.08699 0
1.4 0 1.10563 0 −1.00000 0 −2.46164 0 −1.77758 0
1.5 0 2.29298 0 −1.00000 0 −3.82710 0 2.25774 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.w 5
4.b odd 2 1 185.2.a.e 5
12.b even 2 1 1665.2.a.p 5
20.d odd 2 1 925.2.a.f 5
20.e even 4 2 925.2.b.f 10
28.d even 2 1 9065.2.a.k 5
60.h even 2 1 8325.2.a.ch 5
148.b odd 2 1 6845.2.a.f 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.a.e 5 4.b odd 2 1
925.2.a.f 5 20.d odd 2 1
925.2.b.f 10 20.e even 4 2
1665.2.a.p 5 12.b even 2 1
2960.2.a.w 5 1.a even 1 1 trivial
6845.2.a.f 5 148.b odd 2 1
8325.2.a.ch 5 60.h even 2 1
9065.2.a.k 5 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2960))\):

\( T_{3}^{5} + 3T_{3}^{4} - 6T_{3}^{3} - 20T_{3}^{2} + 4T_{3} + 22 \) Copy content Toggle raw display
\( T_{7}^{5} + 11T_{7}^{4} + 32T_{7}^{3} - 32T_{7}^{2} - 268T_{7} - 302 \) Copy content Toggle raw display
\( T_{13}^{5} - 4T_{13}^{4} - 28T_{13}^{3} + 60T_{13}^{2} + 148T_{13} - 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} + 3 T^{4} - 6 T^{3} - 20 T^{2} + \cdots + 22 \) Copy content Toggle raw display
$5$ \( (T + 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} + 11 T^{4} + 32 T^{3} + \cdots - 302 \) Copy content Toggle raw display
$11$ \( T^{5} - 5 T^{4} - 8 T^{3} + 48 T^{2} + \cdots - 96 \) Copy content Toggle raw display
$13$ \( T^{5} - 4 T^{4} - 28 T^{3} + 60 T^{2} + \cdots - 256 \) Copy content Toggle raw display
$17$ \( T^{5} - 52 T^{3} + 12 T^{2} + \cdots + 192 \) Copy content Toggle raw display
$19$ \( T^{5} - 4 T^{4} - 38 T^{3} + 66 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$23$ \( T^{5} + 4 T^{4} - 72 T^{3} - 208 T^{2} + \cdots - 192 \) Copy content Toggle raw display
$29$ \( T^{5} + 4 T^{4} - 32 T^{3} - 48 T^{2} + \cdots - 192 \) Copy content Toggle raw display
$31$ \( T^{5} + 8 T^{4} - 30 T^{3} - 342 T^{2} + \cdots - 324 \) Copy content Toggle raw display
$37$ \( (T - 1)^{5} \) Copy content Toggle raw display
$41$ \( T^{5} + 5 T^{4} - 8 T^{3} - 48 T^{2} + \cdots + 96 \) Copy content Toggle raw display
$43$ \( T^{5} + 10 T^{4} - 80 T^{3} + \cdots + 2528 \) Copy content Toggle raw display
$47$ \( T^{5} + 7 T^{4} - 92 T^{3} - 592 T^{2} + \cdots + 978 \) Copy content Toggle raw display
$53$ \( T^{5} + T^{4} - 144 T^{3} - 40 T^{2} + \cdots + 528 \) Copy content Toggle raw display
$59$ \( T^{5} - 30 T^{4} + 302 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$61$ \( T^{5} + 14 T^{4} - 8 T^{3} + \cdots + 3296 \) Copy content Toggle raw display
$67$ \( T^{5} + 24 T^{4} + 94 T^{3} + \cdots - 10952 \) Copy content Toggle raw display
$71$ \( T^{5} - 7 T^{4} - 132 T^{3} + \cdots + 7104 \) Copy content Toggle raw display
$73$ \( T^{5} - 5 T^{4} - 192 T^{3} + \cdots + 368 \) Copy content Toggle raw display
$79$ \( T^{5} + 28 T^{4} + 134 T^{3} + \cdots + 19508 \) Copy content Toggle raw display
$83$ \( T^{5} + 27 T^{4} + 178 T^{3} + \cdots - 4818 \) Copy content Toggle raw display
$89$ \( T^{5} - 6 T^{4} - 248 T^{3} + \cdots + 22944 \) Copy content Toggle raw display
$97$ \( T^{5} + 26 T^{4} - 88 T^{3} + \cdots - 166976 \) Copy content Toggle raw display
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