Properties

Label 7360.2.a.bo
Level $7360$
Weight $2$
Character orbit 7360.a
Self dual yes
Analytic conductor $58.770$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 460)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.56155
2.56155
0 −1.56155 0 −1.00000 0 −2.56155 0 −0.561553 0
1.2 0 2.56155 0 −1.00000 0 1.56155 0 3.56155 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.bo 2
4.b odd 2 1 7360.2.a.bi 2
8.b even 2 1 1840.2.a.m 2
8.d odd 2 1 460.2.a.e 2
24.f even 2 1 4140.2.a.m 2
40.e odd 2 1 2300.2.a.i 2
40.f even 2 1 9200.2.a.bv 2
40.k even 4 2 2300.2.c.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.a.e 2 8.d odd 2 1
1840.2.a.m 2 8.b even 2 1
2300.2.a.i 2 40.e odd 2 1
2300.2.c.h 4 40.k even 4 2
4140.2.a.m 2 24.f even 2 1
7360.2.a.bi 2 4.b odd 2 1
7360.2.a.bo 2 1.a even 1 1 trivial
9200.2.a.bv 2 40.f even 2 1

Hecke kernels

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

\( T_{3}^{2} - T_{3} - 4 \)
\( T_{7}^{2} + T_{7} - 4 \)
\( T_{11} - 2 \)
\( T_{13}^{2} - 3 T_{13} - 2 \)
\( T_{17}^{2} - T_{17} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( -4 - T + T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( -4 + T + T^{2} \)
$11$ \( ( -2 + T )^{2} \)
$13$ \( -2 - 3 T + T^{2} \)
$17$ \( -4 - T + T^{2} \)
$19$ \( ( -6 + T )^{2} \)
$23$ \( ( 1 + T )^{2} \)
$29$ \( -13 + 4 T + T^{2} \)
$31$ \( -67 - 2 T + T^{2} \)
$37$ \( 2 - 5 T + T^{2} \)
$41$ \( -17 + T^{2} \)
$43$ \( T^{2} \)
$47$ \( -36 - 3 T + T^{2} \)
$53$ \( 2 - 5 T + T^{2} \)
$59$ \( -32 - 5 T + T^{2} \)
$61$ \( -8 - 6 T + T^{2} \)
$67$ \( 38 + 13 T + T^{2} \)
$71$ \( 19 + 12 T + T^{2} \)
$73$ \( 38 + 13 T + T^{2} \)
$79$ \( 64 + 18 T + T^{2} \)
$83$ \( -208 - T + T^{2} \)
$89$ \( -32 - 12 T + T^{2} \)
$97$ \( 64 + 18 T + T^{2} \)
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