Properties

Label 720.4.a.j
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 10)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 5 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{5} + 4 q^{7} + 12 q^{11} - 58 q^{13} - 66 q^{17} + 100 q^{19} + 132 q^{23} + 25 q^{25} + 90 q^{29} - 152 q^{31} - 20 q^{35} - 34 q^{37} + 438 q^{41} - 32 q^{43} - 204 q^{47} - 327 q^{49} - 222 q^{53} - 60 q^{55} + 420 q^{59} + 902 q^{61} + 290 q^{65} + 1024 q^{67} + 432 q^{71} + 362 q^{73} + 48 q^{77} + 160 q^{79} + 72 q^{83} + 330 q^{85} - 810 q^{89} - 232 q^{91} - 500 q^{95} + 1106 q^{97}+O(q^{100}) \) 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 4.00000 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.j 1
3.b odd 2 1 80.4.a.f 1
4.b odd 2 1 90.4.a.a 1
12.b even 2 1 10.4.a.a 1
15.d odd 2 1 400.4.a.b 1
15.e even 4 2 400.4.c.c 2
20.d odd 2 1 450.4.a.q 1
20.e even 4 2 450.4.c.d 2
24.f even 2 1 320.4.a.m 1
24.h odd 2 1 320.4.a.b 1
36.f odd 6 2 810.4.e.w 2
36.h even 6 2 810.4.e.c 2
48.i odd 4 2 1280.4.d.g 2
48.k even 4 2 1280.4.d.j 2
60.h even 2 1 50.4.a.c 1
60.l odd 4 2 50.4.b.a 2
84.h odd 2 1 490.4.a.o 1
84.j odd 6 2 490.4.e.a 2
84.n even 6 2 490.4.e.i 2
120.i odd 2 1 1600.4.a.bx 1
120.m even 2 1 1600.4.a.d 1
132.d odd 2 1 1210.4.a.b 1
156.h even 2 1 1690.4.a.a 1
420.o odd 2 1 2450.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.a.a 1 12.b even 2 1
50.4.a.c 1 60.h even 2 1
50.4.b.a 2 60.l odd 4 2
80.4.a.f 1 3.b odd 2 1
90.4.a.a 1 4.b odd 2 1
320.4.a.b 1 24.h odd 2 1
320.4.a.m 1 24.f even 2 1
400.4.a.b 1 15.d odd 2 1
400.4.c.c 2 15.e even 4 2
450.4.a.q 1 20.d odd 2 1
450.4.c.d 2 20.e even 4 2
490.4.a.o 1 84.h odd 2 1
490.4.e.a 2 84.j odd 6 2
490.4.e.i 2 84.n even 6 2
720.4.a.j 1 1.a even 1 1 trivial
810.4.e.c 2 36.h even 6 2
810.4.e.w 2 36.f odd 6 2
1210.4.a.b 1 132.d odd 2 1
1280.4.d.g 2 48.i odd 4 2
1280.4.d.j 2 48.k even 4 2
1600.4.a.d 1 120.m even 2 1
1600.4.a.bx 1 120.i odd 2 1
1690.4.a.a 1 156.h even 2 1
2450.4.a.b 1 420.o odd 2 1

Hecke kernels

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

\( T_{7} - 4 \) Copy content Toggle raw display
\( T_{11} - 12 \) 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
$7$ \( T - 4 \) Copy content Toggle raw display
$11$ \( T - 12 \) Copy content Toggle raw display
$13$ \( T + 58 \) Copy content Toggle raw display
$17$ \( T + 66 \) Copy content Toggle raw display
$19$ \( T - 100 \) Copy content Toggle raw display
$23$ \( T - 132 \) Copy content Toggle raw display
$29$ \( T - 90 \) Copy content Toggle raw display
$31$ \( T + 152 \) Copy content Toggle raw display
$37$ \( T + 34 \) Copy content Toggle raw display
$41$ \( T - 438 \) Copy content Toggle raw display
$43$ \( T + 32 \) Copy content Toggle raw display
$47$ \( T + 204 \) Copy content Toggle raw display
$53$ \( T + 222 \) Copy content Toggle raw display
$59$ \( T - 420 \) Copy content Toggle raw display
$61$ \( T - 902 \) Copy content Toggle raw display
$67$ \( T - 1024 \) Copy content Toggle raw display
$71$ \( T - 432 \) Copy content Toggle raw display
$73$ \( T - 362 \) Copy content Toggle raw display
$79$ \( T - 160 \) Copy content Toggle raw display
$83$ \( T - 72 \) Copy content Toggle raw display
$89$ \( T + 810 \) Copy content Toggle raw display
$97$ \( T - 1106 \) Copy content Toggle raw display
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