Properties

Label 7360.2.a.by
Level $7360$
Weight $2$
Character orbit 7360.a
Self dual yes
Analytic conductor $58.770$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.2597.1
Defining polynomial: \(x^{3} - x^{2} - 9 x + 8\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} - q^{5} + ( 1 - \beta_{1} ) q^{7} + ( 3 + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} - q^{5} + ( 1 - \beta_{1} ) q^{7} + ( 3 + \beta_{2} ) q^{9} + ( -2 - \beta_{1} ) q^{11} + \beta_{2} q^{13} + \beta_{1} q^{15} + ( 3 + \beta_{2} ) q^{17} + ( 4 + \beta_{2} ) q^{19} + ( 6 - \beta_{1} + \beta_{2} ) q^{21} + q^{23} + q^{25} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{27} + ( -5 + \beta_{1} + \beta_{2} ) q^{29} + ( -3 + \beta_{1} ) q^{31} + ( 6 + 2 \beta_{1} + \beta_{2} ) q^{33} + ( -1 + \beta_{1} ) q^{35} + ( -3 + 3 \beta_{1} + \beta_{2} ) q^{37} + ( 2 - 3 \beta_{1} - \beta_{2} ) q^{39} + ( 3 - \beta_{2} ) q^{41} -8 q^{43} + ( -3 - \beta_{2} ) q^{45} + 2 \beta_{2} q^{47} + ( -2 \beta_{1} + \beta_{2} ) q^{49} + ( 2 - 6 \beta_{1} - \beta_{2} ) q^{51} + ( 1 - 3 \beta_{1} - \beta_{2} ) q^{53} + ( 2 + \beta_{1} ) q^{55} + ( 2 - 7 \beta_{1} - \beta_{2} ) q^{57} + ( 5 + \beta_{1} + \beta_{2} ) q^{59} + ( -4 + \beta_{1} - 2 \beta_{2} ) q^{61} + ( 5 - 6 \beta_{1} ) q^{63} -\beta_{2} q^{65} + ( -3 + 3 \beta_{1} + \beta_{2} ) q^{67} -\beta_{1} q^{69} + ( -7 - 2 \beta_{1} + \beta_{2} ) q^{71} + ( 6 - 4 \beta_{1} - 2 \beta_{2} ) q^{73} -\beta_{1} q^{75} + ( 4 + \beta_{1} + \beta_{2} ) q^{77} -4 \beta_{1} q^{79} + ( 7 + \beta_{1} + \beta_{2} ) q^{81} + ( 5 + \beta_{1} - \beta_{2} ) q^{83} + ( -3 - \beta_{2} ) q^{85} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{87} + ( -4 + 4 \beta_{1} ) q^{89} + ( 2 - 3 \beta_{1} ) q^{91} + ( -6 + 3 \beta_{1} - \beta_{2} ) q^{93} + ( -4 - \beta_{2} ) q^{95} + ( -2 + 3 \beta_{1} ) q^{97} + ( -4 - 6 \beta_{1} - 3 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{3} - 3q^{5} + 2q^{7} + 10q^{9} + O(q^{10}) \) \( 3q - q^{3} - 3q^{5} + 2q^{7} + 10q^{9} - 7q^{11} + q^{13} + q^{15} + 10q^{17} + 13q^{19} + 18q^{21} + 3q^{23} + 3q^{25} + 2q^{27} - 13q^{29} - 8q^{31} + 21q^{33} - 2q^{35} - 5q^{37} + 2q^{39} + 8q^{41} - 24q^{43} - 10q^{45} + 2q^{47} - q^{49} - q^{51} - q^{53} + 7q^{55} - 2q^{57} + 17q^{59} - 13q^{61} + 9q^{63} - q^{65} - 5q^{67} - q^{69} - 22q^{71} + 12q^{73} - q^{75} + 14q^{77} - 4q^{79} + 23q^{81} + 15q^{83} - 10q^{85} - 12q^{87} - 8q^{89} + 3q^{91} - 16q^{93} - 13q^{95} - 3q^{97} - 21q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 9 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 6\)

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
3.07912
0.878468
−2.95759
0 −3.07912 0 −1.00000 0 −2.07912 0 6.48097 0
1.2 0 −0.878468 0 −1.00000 0 0.121532 0 −2.22829 0
1.3 0 2.95759 0 −1.00000 0 3.95759 0 5.74732 0
\(n\): e.g. 2-40 or 990-1000
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.by 3
4.b odd 2 1 7360.2.a.cc 3
8.b even 2 1 920.2.a.h 3
8.d odd 2 1 1840.2.a.s 3
24.h odd 2 1 8280.2.a.bj 3
40.e odd 2 1 9200.2.a.ce 3
40.f even 2 1 4600.2.a.x 3
40.i odd 4 2 4600.2.e.p 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.h 3 8.b even 2 1
1840.2.a.s 3 8.d odd 2 1
4600.2.a.x 3 40.f even 2 1
4600.2.e.p 6 40.i odd 4 2
7360.2.a.by 3 1.a even 1 1 trivial
7360.2.a.cc 3 4.b odd 2 1
8280.2.a.bj 3 24.h odd 2 1
9200.2.a.ce 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7360))\):

\( T_{3}^{3} + T_{3}^{2} - 9 T_{3} - 8 \)
\( T_{7}^{3} - 2 T_{7}^{2} - 8 T_{7} + 1 \)
\( T_{11}^{3} + 7 T_{11}^{2} + 7 T_{11} - 14 \)
\( T_{13}^{3} - T_{13}^{2} - 23 T_{13} + 50 \)
\( T_{17}^{3} - 10 T_{17}^{2} + 10 T_{17} + 83 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( -8 - 9 T + T^{2} + T^{3} \)
$5$ \( ( 1 + T )^{3} \)
$7$ \( 1 - 8 T - 2 T^{2} + T^{3} \)
$11$ \( -14 + 7 T + 7 T^{2} + T^{3} \)
$13$ \( 50 - 23 T - T^{2} + T^{3} \)
$17$ \( 83 + 10 T - 10 T^{2} + T^{3} \)
$19$ \( 62 + 33 T - 13 T^{2} + T^{3} \)
$23$ \( ( -1 + T )^{3} \)
$29$ \( -76 + 26 T + 13 T^{2} + T^{3} \)
$31$ \( -1 + 12 T + 8 T^{2} + T^{3} \)
$37$ \( -496 - 92 T + 5 T^{2} + T^{3} \)
$41$ \( 1 - 2 T - 8 T^{2} + T^{3} \)
$43$ \( ( 8 + T )^{3} \)
$47$ \( 400 - 92 T - 2 T^{2} + T^{3} \)
$53$ \( 300 - 100 T + T^{2} + T^{3} \)
$59$ \( -36 + 66 T - 17 T^{2} + T^{3} \)
$61$ \( -720 - 51 T + 13 T^{2} + T^{3} \)
$67$ \( -496 - 92 T + 5 T^{2} + T^{3} \)
$71$ \( -225 + 96 T + 22 T^{2} + T^{3} \)
$73$ \( 2120 - 176 T - 12 T^{2} + T^{3} \)
$79$ \( -512 - 144 T + 4 T^{2} + T^{3} \)
$83$ \( 36 + 40 T - 15 T^{2} + T^{3} \)
$89$ \( -64 - 128 T + 8 T^{2} + T^{3} \)
$97$ \( 50 - 81 T + 3 T^{2} + T^{3} \)
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