Properties

Label 720.4.a.o
Level $720$
Weight $4$
Character orbit 720.a
Self dual yes
Analytic conductor $42.481$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 720.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(42.4813752041\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 5 q^{5} + 30 q^{7} + O(q^{10}) \) \( q - 5 q^{5} + 30 q^{7} - 50 q^{11} - 20 q^{13} - 10 q^{17} + 44 q^{19} - 120 q^{23} + 25 q^{25} - 50 q^{29} - 108 q^{31} - 150 q^{35} - 40 q^{37} + 400 q^{41} - 280 q^{43} + 280 q^{47} + 557 q^{49} - 610 q^{53} + 250 q^{55} - 50 q^{59} - 518 q^{61} + 100 q^{65} + 180 q^{67} - 700 q^{71} - 410 q^{73} - 1500 q^{77} + 516 q^{79} - 660 q^{83} + 50 q^{85} - 1500 q^{89} - 600 q^{91} - 220 q^{95} - 1630 q^{97} + O(q^{100}) \)

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 −5.00000 0 30.0000 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 720.4.a.o 1
3.b odd 2 1 720.4.a.bc 1
4.b odd 2 1 45.4.a.e yes 1
12.b even 2 1 45.4.a.a 1
20.d odd 2 1 225.4.a.a 1
20.e even 4 2 225.4.b.b 2
28.d even 2 1 2205.4.a.t 1
36.f odd 6 2 405.4.e.b 2
36.h even 6 2 405.4.e.n 2
60.h even 2 1 225.4.a.h 1
60.l odd 4 2 225.4.b.a 2
84.h odd 2 1 2205.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.a.a 1 12.b even 2 1
45.4.a.e yes 1 4.b odd 2 1
225.4.a.a 1 20.d odd 2 1
225.4.a.h 1 60.h even 2 1
225.4.b.a 2 60.l odd 4 2
225.4.b.b 2 20.e even 4 2
405.4.e.b 2 36.f odd 6 2
405.4.e.n 2 36.h even 6 2
720.4.a.o 1 1.a even 1 1 trivial
720.4.a.bc 1 3.b odd 2 1
2205.4.a.a 1 84.h odd 2 1
2205.4.a.t 1 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(720))\):

\( T_{7} - 30 \)
\( T_{11} + 50 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 5 + T \)
$7$ \( -30 + T \)
$11$ \( 50 + T \)
$13$ \( 20 + T \)
$17$ \( 10 + T \)
$19$ \( -44 + T \)
$23$ \( 120 + T \)
$29$ \( 50 + T \)
$31$ \( 108 + T \)
$37$ \( 40 + T \)
$41$ \( -400 + T \)
$43$ \( 280 + T \)
$47$ \( -280 + T \)
$53$ \( 610 + T \)
$59$ \( 50 + T \)
$61$ \( 518 + T \)
$67$ \( -180 + T \)
$71$ \( 700 + T \)
$73$ \( 410 + T \)
$79$ \( -516 + T \)
$83$ \( 660 + T \)
$89$ \( 1500 + T \)
$97$ \( 1630 + T \)
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