Properties

Label 720.4.a.c
Level $720$
Weight $4$
Character orbit 720.a
Self dual yes
Analytic conductor $42.481$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 5 q^{5} - 32 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{5} - 32 q^{7} + 36 q^{11} - 10 q^{13} + 78 q^{17} - 140 q^{19} - 192 q^{23} + 25 q^{25} - 6 q^{29} + 16 q^{31} + 160 q^{35} - 34 q^{37} + 390 q^{41} + 52 q^{43} + 408 q^{47} + 681 q^{49} + 114 q^{53} - 180 q^{55} + 516 q^{59} - 58 q^{61} + 50 q^{65} + 892 q^{67} - 120 q^{71} - 646 q^{73} - 1152 q^{77} + 1168 q^{79} - 732 q^{83} - 390 q^{85} + 1590 q^{89} + 320 q^{91} + 700 q^{95} + 194 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 −32.0000 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.c 1
3.b odd 2 1 240.4.a.j 1
4.b odd 2 1 180.4.a.c 1
12.b even 2 1 60.4.a.b 1
15.d odd 2 1 1200.4.a.s 1
15.e even 4 2 1200.4.f.e 2
20.d odd 2 1 900.4.a.b 1
20.e even 4 2 900.4.d.b 2
24.f even 2 1 960.4.a.bb 1
24.h odd 2 1 960.4.a.a 1
36.f odd 6 2 1620.4.i.g 2
36.h even 6 2 1620.4.i.a 2
60.h even 2 1 300.4.a.e 1
60.l odd 4 2 300.4.d.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.a.b 1 12.b even 2 1
180.4.a.c 1 4.b odd 2 1
240.4.a.j 1 3.b odd 2 1
300.4.a.e 1 60.h even 2 1
300.4.d.d 2 60.l odd 4 2
720.4.a.c 1 1.a even 1 1 trivial
900.4.a.b 1 20.d odd 2 1
900.4.d.b 2 20.e even 4 2
960.4.a.a 1 24.h odd 2 1
960.4.a.bb 1 24.f even 2 1
1200.4.a.s 1 15.d odd 2 1
1200.4.f.e 2 15.e even 4 2
1620.4.i.a 2 36.h even 6 2
1620.4.i.g 2 36.f odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(720))\):

\( T_{7} + 32 \) Copy content Toggle raw display
\( T_{11} - 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 5 \) Copy content Toggle raw display
$7$ \( T + 32 \) Copy content Toggle raw display
$11$ \( T - 36 \) Copy content Toggle raw display
$13$ \( T + 10 \) Copy content Toggle raw display
$17$ \( T - 78 \) Copy content Toggle raw display
$19$ \( T + 140 \) Copy content Toggle raw display
$23$ \( T + 192 \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T - 16 \) Copy content Toggle raw display
$37$ \( T + 34 \) Copy content Toggle raw display
$41$ \( T - 390 \) Copy content Toggle raw display
$43$ \( T - 52 \) Copy content Toggle raw display
$47$ \( T - 408 \) Copy content Toggle raw display
$53$ \( T - 114 \) Copy content Toggle raw display
$59$ \( T - 516 \) Copy content Toggle raw display
$61$ \( T + 58 \) Copy content Toggle raw display
$67$ \( T - 892 \) Copy content Toggle raw display
$71$ \( T + 120 \) Copy content Toggle raw display
$73$ \( T + 646 \) Copy content Toggle raw display
$79$ \( T - 1168 \) Copy content Toggle raw display
$83$ \( T + 732 \) Copy content Toggle raw display
$89$ \( T - 1590 \) Copy content Toggle raw display
$97$ \( T - 194 \) Copy content Toggle raw display
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