Properties

Label 720.4.a.bd
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 40)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 5 q^{5} + 34 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{5} + 34 q^{7} + 16 q^{11} + 58 q^{13} + 70 q^{17} - 4 q^{19} - 134 q^{23} + 25 q^{25} + 242 q^{29} - 100 q^{31} + 170 q^{35} - 438 q^{37} + 138 q^{41} - 178 q^{43} + 22 q^{47} + 813 q^{49} - 162 q^{53} + 80 q^{55} - 268 q^{59} + 250 q^{61} + 290 q^{65} - 422 q^{67} - 852 q^{71} + 306 q^{73} + 544 q^{77} + 456 q^{79} + 434 q^{83} + 350 q^{85} + 726 q^{89} + 1972 q^{91} - 20 q^{95} + 1378 q^{97}+O(q^{100}) \) 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 34.0000 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.bd 1
3.b odd 2 1 80.4.a.e 1
4.b odd 2 1 360.4.a.h 1
12.b even 2 1 40.4.a.a 1
15.d odd 2 1 400.4.a.e 1
15.e even 4 2 400.4.c.f 2
20.d odd 2 1 1800.4.a.bi 1
20.e even 4 2 1800.4.f.j 2
24.f even 2 1 320.4.a.l 1
24.h odd 2 1 320.4.a.c 1
48.i odd 4 2 1280.4.d.a 2
48.k even 4 2 1280.4.d.p 2
60.h even 2 1 200.4.a.i 1
60.l odd 4 2 200.4.c.c 2
84.h odd 2 1 1960.4.a.h 1
120.i odd 2 1 1600.4.a.br 1
120.m even 2 1 1600.4.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.a.a 1 12.b even 2 1
80.4.a.e 1 3.b odd 2 1
200.4.a.i 1 60.h even 2 1
200.4.c.c 2 60.l odd 4 2
320.4.a.c 1 24.h odd 2 1
320.4.a.l 1 24.f even 2 1
360.4.a.h 1 4.b odd 2 1
400.4.a.e 1 15.d odd 2 1
400.4.c.f 2 15.e even 4 2
720.4.a.bd 1 1.a even 1 1 trivial
1280.4.d.a 2 48.i odd 4 2
1280.4.d.p 2 48.k even 4 2
1600.4.a.j 1 120.m even 2 1
1600.4.a.br 1 120.i odd 2 1
1800.4.a.bi 1 20.d odd 2 1
1800.4.f.j 2 20.e even 4 2
1960.4.a.h 1 84.h odd 2 1

Hecke kernels

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

\( T_{7} - 34 \) Copy content Toggle raw display
\( T_{11} - 16 \) 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
$7$ \( T - 34 \) Copy content Toggle raw display
$11$ \( T - 16 \) Copy content Toggle raw display
$13$ \( T - 58 \) Copy content Toggle raw display
$17$ \( T - 70 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T + 134 \) Copy content Toggle raw display
$29$ \( T - 242 \) Copy content Toggle raw display
$31$ \( T + 100 \) Copy content Toggle raw display
$37$ \( T + 438 \) Copy content Toggle raw display
$41$ \( T - 138 \) Copy content Toggle raw display
$43$ \( T + 178 \) Copy content Toggle raw display
$47$ \( T - 22 \) Copy content Toggle raw display
$53$ \( T + 162 \) Copy content Toggle raw display
$59$ \( T + 268 \) Copy content Toggle raw display
$61$ \( T - 250 \) Copy content Toggle raw display
$67$ \( T + 422 \) Copy content Toggle raw display
$71$ \( T + 852 \) Copy content Toggle raw display
$73$ \( T - 306 \) Copy content Toggle raw display
$79$ \( T - 456 \) Copy content Toggle raw display
$83$ \( T - 434 \) Copy content Toggle raw display
$89$ \( T - 726 \) Copy content Toggle raw display
$97$ \( T - 1378 \) Copy content Toggle raw display
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