Properties

Label 2960.2.a.ba
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: 5.5.368464.1
Defining polynomial: \(x^{5} - 2 x^{4} - 6 x^{3} + 6 x^{2} + 6 x - 4\)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + q^{5} + ( -1 - \beta_{4} ) q^{7} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{9} +O(q^{10})\) \( q + \beta_{3} q^{3} + q^{5} + ( -1 - \beta_{4} ) q^{7} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{9} + ( -1 - \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{11} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{13} + \beta_{3} q^{15} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{17} + ( -3 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{19} + ( -2 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{21} + ( -2 \beta_{2} - 2 \beta_{3} ) q^{23} + q^{25} + ( \beta_{1} + \beta_{3} + 2 \beta_{4} ) q^{27} + ( 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} ) q^{29} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{31} + ( -3 - \beta_{1} - 2 \beta_{3} - \beta_{4} ) q^{33} + ( -1 - \beta_{4} ) q^{35} - q^{37} + ( -1 - 2 \beta_{1} - \beta_{3} - \beta_{4} ) q^{39} + ( -1 - 3 \beta_{1} + \beta_{4} ) q^{41} + ( -4 + 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} ) q^{43} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{45} + ( -1 + 4 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{47} + ( -1 - \beta_{2} + \beta_{3} + 3 \beta_{4} ) q^{49} + ( -1 - 3 \beta_{3} - \beta_{4} ) q^{51} + ( -3 - \beta_{1} + \beta_{4} ) q^{53} + ( -1 - \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{55} + ( 1 + \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{57} + ( -4 + 3 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{59} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{61} + ( -1 - 5 \beta_{1} - 2 \beta_{2} - 7 \beta_{3} - \beta_{4} ) q^{63} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{65} + ( -1 + \beta_{1} + 4 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{67} + ( -4 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{69} + ( -3 - \beta_{1} + 4 \beta_{2} + 6 \beta_{3} - \beta_{4} ) q^{71} + ( -3 + 3 \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{73} + \beta_{3} q^{75} + ( -5 + 5 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} + \beta_{4} ) q^{77} + ( -8 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{79} + ( 4 - \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{81} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{83} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{85} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{87} + ( -4 \beta_{1} + 2 \beta_{2} - 4 \beta_{4} ) q^{89} + ( -1 + 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{91} + ( -1 - \beta_{1} + \beta_{2} - 5 \beta_{3} - 2 \beta_{4} ) q^{93} + ( -3 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{95} + ( -4 + 4 \beta_{1} + 2 \beta_{2} + 6 \beta_{4} ) q^{97} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + q^{3} + 5q^{5} - 7q^{7} + 2q^{9} + O(q^{10}) \) \( 5q + q^{3} + 5q^{5} - 7q^{7} + 2q^{9} - 7q^{11} + 2q^{13} + q^{15} - 8q^{17} - 14q^{19} - 9q^{21} - 2q^{23} + 5q^{25} + 7q^{27} + 2q^{29} - 8q^{31} - 21q^{33} - 7q^{35} - 5q^{37} - 12q^{39} - 9q^{41} - 14q^{43} + 2q^{45} + 5q^{47} + 2q^{49} - 10q^{51} - 15q^{53} - 7q^{55} + 4q^{57} - 12q^{59} + 12q^{61} - 24q^{63} + 2q^{65} + 2q^{67} - 30q^{69} - 13q^{71} - 5q^{73} + q^{75} - 7q^{77} - 36q^{79} + 21q^{81} + 9q^{83} - 8q^{85} + 6q^{87} - 16q^{89} + 4q^{91} - 16q^{93} - 14q^{95} - 18q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 6 x^{3} + 6 x^{2} + 6 x - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} - 2 \nu^{3} - 4 \nu^{2} + 2 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} - 2 \nu^{3} - 6 \nu^{2} + 4 \nu + 6 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{4} - 10 \nu^{2} - 4 \nu + 12 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{3} + \beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 6 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(2 \beta_{4} - 10 \beta_{3} + 10 \beta_{2} + 14 \beta_{1} + 18\)

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
3.14884
−1.09027
1.17837
−1.78948
0.552543
0 −2.51369 0 1.00000 0 −0.281955 0 3.31863 0
1.2 0 −0.744131 0 1.00000 0 −3.94357 0 −2.44627 0
1.3 0 0.518894 0 1.00000 0 1.33546 0 −2.73075 0
1.4 0 0.671838 0 1.00000 0 0.305070 0 −2.54863 0
1.5 0 3.06709 0 1.00000 0 −4.41500 0 6.40702 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.ba 5
4.b odd 2 1 185.2.a.d 5
12.b even 2 1 1665.2.a.q 5
20.d odd 2 1 925.2.a.h 5
20.e even 4 2 925.2.b.g 10
28.d even 2 1 9065.2.a.j 5
60.h even 2 1 8325.2.a.cc 5
148.b odd 2 1 6845.2.a.g 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.a.d 5 4.b odd 2 1
925.2.a.h 5 20.d odd 2 1
925.2.b.g 10 20.e even 4 2
1665.2.a.q 5 12.b even 2 1
2960.2.a.ba 5 1.a even 1 1 trivial
6845.2.a.g 5 148.b odd 2 1
8325.2.a.cc 5 60.h even 2 1
9065.2.a.j 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2960))\):

\( T_{3}^{5} - T_{3}^{4} - 8 T_{3}^{3} + 4 T_{3}^{2} + 4 T_{3} - 2 \)
\( T_{7}^{5} + 7 T_{7}^{4} + 6 T_{7}^{3} - 24 T_{7}^{2} + 2 \)
\( T_{13}^{5} - 2 T_{13}^{4} - 20 T_{13}^{3} + 20 T_{13}^{2} + 76 T_{13} - 88 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( -2 + 4 T + 4 T^{2} - 8 T^{3} - T^{4} + T^{5} \)
$5$ \( ( -1 + T )^{5} \)
$7$ \( 2 - 24 T^{2} + 6 T^{3} + 7 T^{4} + T^{5} \)
$11$ \( 32 - 176 T - 144 T^{2} - 12 T^{3} + 7 T^{4} + T^{5} \)
$13$ \( -88 + 76 T + 20 T^{2} - 20 T^{3} - 2 T^{4} + T^{5} \)
$17$ \( -32 - 92 T - 36 T^{2} + 12 T^{3} + 8 T^{4} + T^{5} \)
$19$ \( -2224 - 1782 T - 362 T^{2} + 26 T^{3} + 14 T^{4} + T^{5} \)
$23$ \( -1504 + 880 T - 56 T^{3} + 2 T^{4} + T^{5} \)
$29$ \( 32 - 176 T + 272 T^{2} - 80 T^{3} - 2 T^{4} + T^{5} \)
$31$ \( 3016 + 346 T - 314 T^{2} - 38 T^{3} + 8 T^{4} + T^{5} \)
$37$ \( ( 1 + T )^{5} \)
$41$ \( -928 + 1488 T - 304 T^{2} - 64 T^{3} + 9 T^{4} + T^{5} \)
$43$ \( 1312 - 432 T - 560 T^{2} - 16 T^{3} + 14 T^{4} + T^{5} \)
$47$ \( 3994 + 1740 T - 128 T^{2} - 114 T^{3} - 5 T^{4} + T^{5} \)
$53$ \( -16 - 112 T + 40 T^{2} + 64 T^{3} + 15 T^{4} + T^{5} \)
$59$ \( -5456 + 3842 T - 526 T^{2} - 94 T^{3} + 12 T^{4} + T^{5} \)
$61$ \( -512 + 240 T + 176 T^{2} - 32 T^{3} - 12 T^{4} + T^{5} \)
$67$ \( 7568 + 7166 T + 70 T^{2} - 174 T^{3} - 2 T^{4} + T^{5} \)
$71$ \( 291136 + 28352 T - 4176 T^{2} - 348 T^{3} + 13 T^{4} + T^{5} \)
$73$ \( 176 + 16 T - 336 T^{2} - 68 T^{3} + 5 T^{4} + T^{5} \)
$79$ \( -5912 + 1594 T + 1938 T^{2} + 430 T^{3} + 36 T^{4} + T^{5} \)
$83$ \( -314 - 1828 T + 1040 T^{2} - 104 T^{3} - 9 T^{4} + T^{5} \)
$89$ \( 1856 - 21008 T - 5136 T^{2} - 240 T^{3} + 16 T^{4} + T^{5} \)
$97$ \( -193408 + 44384 T + 496 T^{2} - 464 T^{3} + T^{5} \)
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