Properties

Label 720.4.a.ba
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

\(f(q)\) \(=\) \( q + 5 q^{5} + 18 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{5} + 18 q^{7} - 16 q^{11} - 6 q^{13} + 6 q^{17} + 124 q^{19} + 42 q^{23} + 25 q^{25} - 142 q^{29} + 188 q^{31} + 90 q^{35} + 202 q^{37} - 54 q^{41} - 66 q^{43} + 38 q^{47} - 19 q^{49} - 738 q^{53} - 80 q^{55} + 564 q^{59} - 262 q^{61} - 30 q^{65} + 554 q^{67} + 140 q^{71} + 882 q^{73} - 288 q^{77} + 1160 q^{79} + 642 q^{83} + 30 q^{85} + 854 q^{89} - 108 q^{91} + 620 q^{95} - 478 q^{97}+O(q^{100}) \) 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.ba 1
3.b odd 2 1 80.4.a.a 1
4.b odd 2 1 360.4.a.i 1
12.b even 2 1 40.4.a.c 1
15.d odd 2 1 400.4.a.u 1
15.e even 4 2 400.4.c.a 2
20.d odd 2 1 1800.4.a.bd 1
20.e even 4 2 1800.4.f.n 2
24.f even 2 1 320.4.a.a 1
24.h odd 2 1 320.4.a.n 1
48.i odd 4 2 1280.4.d.b 2
48.k even 4 2 1280.4.d.o 2
60.h even 2 1 200.4.a.a 1
60.l odd 4 2 200.4.c.a 2
84.h odd 2 1 1960.4.a.a 1
120.i odd 2 1 1600.4.a.a 1
120.m even 2 1 1600.4.a.ca 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.a.c 1 12.b even 2 1
80.4.a.a 1 3.b odd 2 1
200.4.a.a 1 60.h even 2 1
200.4.c.a 2 60.l odd 4 2
320.4.a.a 1 24.f even 2 1
320.4.a.n 1 24.h odd 2 1
360.4.a.i 1 4.b odd 2 1
400.4.a.u 1 15.d odd 2 1
400.4.c.a 2 15.e even 4 2
720.4.a.ba 1 1.a even 1 1 trivial
1280.4.d.b 2 48.i odd 4 2
1280.4.d.o 2 48.k even 4 2
1600.4.a.a 1 120.i odd 2 1
1600.4.a.ca 1 120.m even 2 1
1800.4.a.bd 1 20.d odd 2 1
1800.4.f.n 2 20.e even 4 2
1960.4.a.a 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(720))\):

\( T_{7} - 18 \) 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
$7$ \( T - 18 \) Copy content Toggle raw display
$11$ \( T + 16 \) Copy content Toggle raw display
$13$ \( T + 6 \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T - 124 \) Copy content Toggle raw display
$23$ \( T - 42 \) Copy content Toggle raw display
$29$ \( T + 142 \) Copy content Toggle raw display
$31$ \( T - 188 \) Copy content Toggle raw display
$37$ \( T - 202 \) Copy content Toggle raw display
$41$ \( T + 54 \) Copy content Toggle raw display
$43$ \( T + 66 \) Copy content Toggle raw display
$47$ \( T - 38 \) Copy content Toggle raw display
$53$ \( T + 738 \) Copy content Toggle raw display
$59$ \( T - 564 \) Copy content Toggle raw display
$61$ \( T + 262 \) Copy content Toggle raw display
$67$ \( T - 554 \) Copy content Toggle raw display
$71$ \( T - 140 \) Copy content Toggle raw display
$73$ \( T - 882 \) Copy content Toggle raw display
$79$ \( T - 1160 \) Copy content Toggle raw display
$83$ \( T - 642 \) Copy content Toggle raw display
$89$ \( T - 854 \) Copy content Toggle raw display
$97$ \( T + 478 \) Copy content Toggle raw display
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