Properties

Label 7360.2.a.co
Level $7360$
Weight $2$
Character orbit 7360.a
Self dual yes
Analytic conductor $58.770$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7360.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(58.7698958877\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.13955077.1
Defining polynomial: \(x^{5} - 14 x^{3} - x^{2} + 32 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + q^{5} + ( -\beta_{3} - \beta_{4} ) q^{7} + ( 2 + \beta_{2} + \beta_{4} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + q^{5} + ( -\beta_{3} - \beta_{4} ) q^{7} + ( 2 + \beta_{2} + \beta_{4} ) q^{9} + ( -\beta_{1} + \beta_{2} ) q^{11} + ( -1 - \beta_{2} + \beta_{4} ) q^{13} -\beta_{1} q^{15} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{17} + ( -1 - 2 \beta_{1} - \beta_{4} ) q^{19} + ( -1 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{21} - q^{23} + q^{25} + ( -1 - 3 \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{27} + ( -1 + \beta_{2} - \beta_{3} ) q^{29} + ( 4 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} ) q^{31} + ( 3 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{33} + ( -\beta_{3} - \beta_{4} ) q^{35} + ( -3 + \beta_{3} ) q^{37} + ( 3 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{39} + ( 5 - \beta_{1} + \beta_{3} ) q^{41} + ( 2 + \beta_{2} + \beta_{4} ) q^{45} + ( -2 - \beta_{2} - 2 \beta_{3} ) q^{47} + ( 6 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{49} + ( -3 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{51} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{53} + ( -\beta_{1} + \beta_{2} ) q^{55} + ( 9 + 3 \beta_{1} + \beta_{2} + \beta_{4} ) q^{57} + ( 1 + \beta_{3} - 2 \beta_{4} ) q^{59} + ( \beta_{1} + \beta_{2} + 2 \beta_{4} ) q^{61} + ( -8 + \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{63} + ( -1 - \beta_{2} + \beta_{4} ) q^{65} + ( -1 + 2 \beta_{1} - \beta_{3} - 2 \beta_{4} ) q^{67} + \beta_{1} q^{69} + ( 1 - 3 \beta_{1} + \beta_{3} - 2 \beta_{4} ) q^{71} + ( -\beta_{2} - 2 \beta_{3} ) q^{73} -\beta_{1} q^{75} + ( -3 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{77} -2 \beta_{2} q^{79} + ( 10 - 3 \beta_{1} + 5 \beta_{2} + \beta_{4} ) q^{81} + ( 9 - \beta_{3} ) q^{83} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{85} + ( -2 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{87} + ( 2 - 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} ) q^{89} + ( -4 + 5 \beta_{1} + \beta_{2} + 4 \beta_{4} ) q^{91} + ( 7 - 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{93} + ( -1 - 2 \beta_{1} - \beta_{4} ) q^{95} + ( 6 + \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{97} + ( 11 - 4 \beta_{1} + 4 \beta_{2} + 3 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{5} - 2 q^{7} + 13 q^{9} + O(q^{10}) \) \( 5 q + 5 q^{5} - 2 q^{7} + 13 q^{9} + q^{11} - 4 q^{13} + 4 q^{17} - 7 q^{19} - 6 q^{21} - 5 q^{23} + 5 q^{25} - 3 q^{27} - 4 q^{29} + 19 q^{31} + 17 q^{33} - 2 q^{35} - 15 q^{37} + 19 q^{39} + 25 q^{41} + 13 q^{45} - 11 q^{47} + 25 q^{49} - 19 q^{51} - 3 q^{53} + q^{55} + 48 q^{57} + q^{59} + 5 q^{61} - 41 q^{63} - 4 q^{65} - 9 q^{67} + q^{71} - q^{73} - 18 q^{77} - 2 q^{79} + 57 q^{81} + 45 q^{83} + 4 q^{85} - 9 q^{87} + 6 q^{89} - 11 q^{91} + 39 q^{93} - 7 q^{95} + 25 q^{97} + 65 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 14 x^{3} - x^{2} + 32 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} - 10 \nu^{2} + 3 \nu + 4 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{4} + 4 \nu^{3} + 14 \nu^{2} - 39 \nu - 28 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{4} + 14 \nu^{2} - 3 \nu - 24 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(-\beta_{4} + 2 \beta_{3} + 9 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(10 \beta_{4} + 14 \beta_{2} - 3 \beta_{1} + 46\)

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
3.30649
1.93283
−0.568386
−1.31091
−3.36002
0 −3.30649 0 1.00000 0 −2.55040 0 7.93288 0
1.2 0 −1.93283 0 1.00000 0 2.38236 0 0.735829 0
1.3 0 0.568386 0 1.00000 0 4.73770 0 −2.67694 0
1.4 0 1.31091 0 1.00000 0 −4.66212 0 −1.28151 0
1.5 0 3.36002 0 1.00000 0 −1.90754 0 8.28974 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
\(2\) \(1\)
\(5\) \(-1\)
\(23\) \(1\)

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 7360.2.a.co 5
4.b odd 2 1 7360.2.a.cp 5
8.b even 2 1 920.2.a.j 5
8.d odd 2 1 1840.2.a.v 5
24.h odd 2 1 8280.2.a.bs 5
40.e odd 2 1 9200.2.a.cu 5
40.f even 2 1 4600.2.a.be 5
40.i odd 4 2 4600.2.e.u 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.j 5 8.b even 2 1
1840.2.a.v 5 8.d odd 2 1
4600.2.a.be 5 40.f even 2 1
4600.2.e.u 10 40.i odd 4 2
7360.2.a.co 5 1.a even 1 1 trivial
7360.2.a.cp 5 4.b odd 2 1
8280.2.a.bs 5 24.h odd 2 1
9200.2.a.cu 5 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7360))\):

\( T_{3}^{5} - 14 T_{3}^{3} + T_{3}^{2} + 32 T_{3} - 16 \)
\( T_{7}^{5} + 2 T_{7}^{4} - 28 T_{7}^{3} - 57 T_{7}^{2} + 128 T_{7} + 256 \)
\( T_{11}^{5} - T_{11}^{4} - 35 T_{11}^{3} + 28 T_{11}^{2} + 172 T_{11} - 64 \)
\( T_{13}^{5} + 4 T_{13}^{4} - 46 T_{13}^{3} - 75 T_{13}^{2} + 600 T_{13} - 500 \)
\( T_{17}^{5} - 4 T_{17}^{4} - 46 T_{17}^{3} + 157 T_{17}^{2} - 138 T_{17} + 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \)
$3$ \( -16 + 32 T + T^{2} - 14 T^{3} + T^{5} \)
$5$ \( ( -1 + T )^{5} \)
$7$ \( 256 + 128 T - 57 T^{2} - 28 T^{3} + 2 T^{4} + T^{5} \)
$11$ \( -64 + 172 T + 28 T^{2} - 35 T^{3} - T^{4} + T^{5} \)
$13$ \( -500 + 600 T - 75 T^{2} - 46 T^{3} + 4 T^{4} + T^{5} \)
$17$ \( 32 - 138 T + 157 T^{2} - 46 T^{3} - 4 T^{4} + T^{5} \)
$19$ \( -512 + 676 T - 180 T^{2} - 41 T^{3} + 7 T^{4} + T^{5} \)
$23$ \( ( 1 + T )^{5} \)
$29$ \( -8 + 64 T + 36 T^{2} - 41 T^{3} + 4 T^{4} + T^{5} \)
$31$ \( -128 - 53 T + 183 T^{2} + 72 T^{3} - 19 T^{4} + T^{5} \)
$37$ \( -64 - 176 T + 36 T^{2} + 64 T^{3} + 15 T^{4} + T^{5} \)
$41$ \( 2182 + 27 T - 653 T^{2} + 212 T^{3} - 25 T^{4} + T^{5} \)
$43$ \( T^{5} \)
$47$ \( 512 - 800 T - 1116 T^{2} - 110 T^{3} + 11 T^{4} + T^{5} \)
$53$ \( 20272 + 5808 T - 440 T^{2} - 160 T^{3} + 3 T^{4} + T^{5} \)
$59$ \( -13568 + 4560 T + 236 T^{2} - 142 T^{3} - T^{4} + T^{5} \)
$61$ \( -7664 + 2092 T + 568 T^{2} - 115 T^{3} - 5 T^{4} + T^{5} \)
$67$ \( -8192 + 4832 T - 432 T^{2} - 126 T^{3} + 9 T^{4} + T^{5} \)
$71$ \( -3968 + 8139 T + 407 T^{2} - 214 T^{3} - T^{4} + T^{5} \)
$73$ \( -1328 + 2072 T - 272 T^{2} - 158 T^{3} + T^{4} + T^{5} \)
$79$ \( -1024 + 2432 T + 128 T^{2} - 128 T^{3} + 2 T^{4} + T^{5} \)
$83$ \( -41216 + 26608 T - 6588 T^{2} + 784 T^{3} - 45 T^{4} + T^{5} \)
$89$ \( -8192 + 2560 T + 512 T^{2} - 136 T^{3} - 6 T^{4} + T^{5} \)
$97$ \( -49616 - 16908 T + 4164 T^{2} - 39 T^{3} - 25 T^{4} + T^{5} \)
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