Properties

Label 720.4.a.i
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 120)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 5 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{5} + 4 q^{11} + 54 q^{13} - 114 q^{17} - 44 q^{19} + 96 q^{23} + 25 q^{25} - 134 q^{29} + 272 q^{31} - 98 q^{37} + 6 q^{41} - 12 q^{43} - 200 q^{47} - 343 q^{49} - 654 q^{53} - 20 q^{55} + 36 q^{59} - 442 q^{61} - 270 q^{65} + 188 q^{67} - 632 q^{71} - 390 q^{73} - 688 q^{79} + 1188 q^{83} + 570 q^{85} + 694 q^{89} + 220 q^{95} - 1726 q^{97}+O(q^{100}) \) 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 0 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.i 1
3.b odd 2 1 240.4.a.k 1
4.b odd 2 1 360.4.a.c 1
12.b even 2 1 120.4.a.d 1
15.d odd 2 1 1200.4.a.j 1
15.e even 4 2 1200.4.f.l 2
20.d odd 2 1 1800.4.a.s 1
20.e even 4 2 1800.4.f.m 2
24.f even 2 1 960.4.a.x 1
24.h odd 2 1 960.4.a.e 1
60.h even 2 1 600.4.a.m 1
60.l odd 4 2 600.4.f.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.a.d 1 12.b even 2 1
240.4.a.k 1 3.b odd 2 1
360.4.a.c 1 4.b odd 2 1
600.4.a.m 1 60.h even 2 1
600.4.f.d 2 60.l odd 4 2
720.4.a.i 1 1.a even 1 1 trivial
960.4.a.e 1 24.h odd 2 1
960.4.a.x 1 24.f even 2 1
1200.4.a.j 1 15.d odd 2 1
1200.4.f.l 2 15.e even 4 2
1800.4.a.s 1 20.d odd 2 1
1800.4.f.m 2 20.e even 4 2

Hecke kernels

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

\( T_{7} \) Copy content Toggle raw display
\( T_{11} - 4 \) 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
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 4 \) Copy content Toggle raw display
$13$ \( T - 54 \) Copy content Toggle raw display
$17$ \( T + 114 \) Copy content Toggle raw display
$19$ \( T + 44 \) Copy content Toggle raw display
$23$ \( T - 96 \) Copy content Toggle raw display
$29$ \( T + 134 \) Copy content Toggle raw display
$31$ \( T - 272 \) Copy content Toggle raw display
$37$ \( T + 98 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T + 12 \) Copy content Toggle raw display
$47$ \( T + 200 \) Copy content Toggle raw display
$53$ \( T + 654 \) Copy content Toggle raw display
$59$ \( T - 36 \) Copy content Toggle raw display
$61$ \( T + 442 \) Copy content Toggle raw display
$67$ \( T - 188 \) Copy content Toggle raw display
$71$ \( T + 632 \) Copy content Toggle raw display
$73$ \( T + 390 \) Copy content Toggle raw display
$79$ \( T + 688 \) Copy content Toggle raw display
$83$ \( T - 1188 \) Copy content Toggle raw display
$89$ \( T - 694 \) Copy content Toggle raw display
$97$ \( T + 1726 \) Copy content Toggle raw display
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