Properties

Label 720.4.a.z
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 360)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 5 q^{5} + 18 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{5} + 18 q^{7} - 34 q^{11} + 12 q^{13} - 102 q^{17} - 164 q^{19} - 48 q^{23} + 25 q^{25} + 146 q^{29} - 100 q^{31} + 90 q^{35} + 328 q^{37} - 288 q^{41} - 120 q^{43} - 16 q^{47} - 19 q^{49} - 126 q^{53} - 170 q^{55} - 642 q^{59} + 602 q^{61} + 60 q^{65} - 436 q^{67} - 652 q^{71} + 1062 q^{73} - 612 q^{77} - 388 q^{79} + 444 q^{83} - 510 q^{85} - 820 q^{89} + 216 q^{91} - 820 q^{95} - 766 q^{97}+O(q^{100}) \) 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 18.0000 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.z 1
3.b odd 2 1 720.4.a.m 1
4.b odd 2 1 360.4.a.j yes 1
12.b even 2 1 360.4.a.a 1
20.d odd 2 1 1800.4.a.be 1
20.e even 4 2 1800.4.f.s 2
60.h even 2 1 1800.4.a.bc 1
60.l odd 4 2 1800.4.f.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.4.a.a 1 12.b even 2 1
360.4.a.j yes 1 4.b odd 2 1
720.4.a.m 1 3.b odd 2 1
720.4.a.z 1 1.a even 1 1 trivial
1800.4.a.bc 1 60.h even 2 1
1800.4.a.be 1 20.d odd 2 1
1800.4.f.e 2 60.l odd 4 2
1800.4.f.s 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(720))\):

\( T_{7} - 18 \) Copy content Toggle raw display
\( T_{11} + 34 \) 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
$7$ \( T - 18 \) Copy content Toggle raw display
$11$ \( T + 34 \) Copy content Toggle raw display
$13$ \( T - 12 \) Copy content Toggle raw display
$17$ \( T + 102 \) Copy content Toggle raw display
$19$ \( T + 164 \) Copy content Toggle raw display
$23$ \( T + 48 \) Copy content Toggle raw display
$29$ \( T - 146 \) Copy content Toggle raw display
$31$ \( T + 100 \) Copy content Toggle raw display
$37$ \( T - 328 \) Copy content Toggle raw display
$41$ \( T + 288 \) Copy content Toggle raw display
$43$ \( T + 120 \) Copy content Toggle raw display
$47$ \( T + 16 \) Copy content Toggle raw display
$53$ \( T + 126 \) Copy content Toggle raw display
$59$ \( T + 642 \) Copy content Toggle raw display
$61$ \( T - 602 \) Copy content Toggle raw display
$67$ \( T + 436 \) Copy content Toggle raw display
$71$ \( T + 652 \) Copy content Toggle raw display
$73$ \( T - 1062 \) Copy content Toggle raw display
$79$ \( T + 388 \) Copy content Toggle raw display
$83$ \( T - 444 \) Copy content Toggle raw display
$89$ \( T + 820 \) Copy content Toggle raw display
$97$ \( T + 766 \) Copy content Toggle raw display
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