Properties

Label 720.6.a.a
Level $720$
Weight $6$
Character orbit 720.a
Self dual yes
Analytic conductor $115.476$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 720.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(115.476350265\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 25q^{5} - 192q^{7} + O(q^{10}) \) \( q - 25q^{5} - 192q^{7} - 148q^{11} + 286q^{13} + 1678q^{17} - 1060q^{19} + 2976q^{23} + 625q^{25} + 3410q^{29} + 2448q^{31} + 4800q^{35} + 182q^{37} + 9398q^{41} + 1244q^{43} - 12088q^{47} + 20057q^{49} - 23846q^{53} + 3700q^{55} - 20020q^{59} + 32302q^{61} - 7150q^{65} - 60972q^{67} - 32648q^{71} - 38774q^{73} + 28416q^{77} + 33360q^{79} + 16716q^{83} - 41950q^{85} - 101370q^{89} - 54912q^{91} + 26500q^{95} - 119038q^{97} + O(q^{100}) \)

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
0
0 0 0 −25.0000 0 −192.000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)

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 720.6.a.a 1
3.b odd 2 1 80.6.a.e 1
4.b odd 2 1 45.6.a.b 1
12.b even 2 1 5.6.a.a 1
15.d odd 2 1 400.6.a.g 1
15.e even 4 2 400.6.c.j 2
20.d odd 2 1 225.6.a.f 1
20.e even 4 2 225.6.b.e 2
24.f even 2 1 320.6.a.j 1
24.h odd 2 1 320.6.a.g 1
60.h even 2 1 25.6.a.a 1
60.l odd 4 2 25.6.b.a 2
84.h odd 2 1 245.6.a.b 1
132.d odd 2 1 605.6.a.a 1
156.h even 2 1 845.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.a.a 1 12.b even 2 1
25.6.a.a 1 60.h even 2 1
25.6.b.a 2 60.l odd 4 2
45.6.a.b 1 4.b odd 2 1
80.6.a.e 1 3.b odd 2 1
225.6.a.f 1 20.d odd 2 1
225.6.b.e 2 20.e even 4 2
245.6.a.b 1 84.h odd 2 1
320.6.a.g 1 24.h odd 2 1
320.6.a.j 1 24.f even 2 1
400.6.a.g 1 15.d odd 2 1
400.6.c.j 2 15.e even 4 2
605.6.a.a 1 132.d odd 2 1
720.6.a.a 1 1.a even 1 1 trivial
845.6.a.b 1 156.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(720))\):

\( T_{7} + 192 \)
\( T_{11} + 148 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 25 + T \)
$7$ \( 192 + T \)
$11$ \( 148 + T \)
$13$ \( -286 + T \)
$17$ \( -1678 + T \)
$19$ \( 1060 + T \)
$23$ \( -2976 + T \)
$29$ \( -3410 + T \)
$31$ \( -2448 + T \)
$37$ \( -182 + T \)
$41$ \( -9398 + T \)
$43$ \( -1244 + T \)
$47$ \( 12088 + T \)
$53$ \( 23846 + T \)
$59$ \( 20020 + T \)
$61$ \( -32302 + T \)
$67$ \( 60972 + T \)
$71$ \( 32648 + T \)
$73$ \( 38774 + T \)
$79$ \( -33360 + T \)
$83$ \( -16716 + T \)
$89$ \( 101370 + T \)
$97$ \( 119038 + T \)
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