Properties

Label 720.2.a.j
Level 720
Weight 2
Character orbit 720.a
Self dual yes
Analytic conductor 5.749
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{5} + 4q^{7} + O(q^{10}) \) \( q + q^{5} + 4q^{7} + 2q^{13} - 6q^{17} + 4q^{19} + q^{25} + 6q^{29} - 8q^{31} + 4q^{35} + 2q^{37} + 6q^{41} + 4q^{43} + 9q^{49} + 6q^{53} - 10q^{61} + 2q^{65} + 4q^{67} + 2q^{73} - 8q^{79} + 12q^{83} - 6q^{85} - 18q^{89} + 8q^{91} + 4q^{95} + 2q^{97} + O(q^{100}) \)

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 1.00000 0 4.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 720.2.a.j 1
3.b odd 2 1 240.2.a.b 1
4.b odd 2 1 90.2.a.c 1
5.b even 2 1 3600.2.a.f 1
5.c odd 4 2 3600.2.f.i 2
8.b even 2 1 2880.2.a.q 1
8.d odd 2 1 2880.2.a.a 1
12.b even 2 1 30.2.a.a 1
15.d odd 2 1 1200.2.a.k 1
15.e even 4 2 1200.2.f.e 2
20.d odd 2 1 450.2.a.d 1
20.e even 4 2 450.2.c.b 2
24.f even 2 1 960.2.a.e 1
24.h odd 2 1 960.2.a.p 1
28.d even 2 1 4410.2.a.z 1
36.f odd 6 2 810.2.e.b 2
36.h even 6 2 810.2.e.l 2
48.i odd 4 2 3840.2.k.f 2
48.k even 4 2 3840.2.k.y 2
60.h even 2 1 150.2.a.b 1
60.l odd 4 2 150.2.c.a 2
84.h odd 2 1 1470.2.a.d 1
84.j odd 6 2 1470.2.i.q 2
84.n even 6 2 1470.2.i.o 2
120.i odd 2 1 4800.2.a.d 1
120.m even 2 1 4800.2.a.cq 1
120.q odd 4 2 4800.2.f.p 2
120.w even 4 2 4800.2.f.w 2
132.d odd 2 1 3630.2.a.w 1
156.h even 2 1 5070.2.a.w 1
156.l odd 4 2 5070.2.b.k 2
204.h even 2 1 8670.2.a.g 1
420.o odd 2 1 7350.2.a.ct 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.a.a 1 12.b even 2 1
90.2.a.c 1 4.b odd 2 1
150.2.a.b 1 60.h even 2 1
150.2.c.a 2 60.l odd 4 2
240.2.a.b 1 3.b odd 2 1
450.2.a.d 1 20.d odd 2 1
450.2.c.b 2 20.e even 4 2
720.2.a.j 1 1.a even 1 1 trivial
810.2.e.b 2 36.f odd 6 2
810.2.e.l 2 36.h even 6 2
960.2.a.e 1 24.f even 2 1
960.2.a.p 1 24.h odd 2 1
1200.2.a.k 1 15.d odd 2 1
1200.2.f.e 2 15.e even 4 2
1470.2.a.d 1 84.h odd 2 1
1470.2.i.o 2 84.n even 6 2
1470.2.i.q 2 84.j odd 6 2
2880.2.a.a 1 8.d odd 2 1
2880.2.a.q 1 8.b even 2 1
3600.2.a.f 1 5.b even 2 1
3600.2.f.i 2 5.c odd 4 2
3630.2.a.w 1 132.d odd 2 1
3840.2.k.f 2 48.i odd 4 2
3840.2.k.y 2 48.k even 4 2
4410.2.a.z 1 28.d even 2 1
4800.2.a.d 1 120.i odd 2 1
4800.2.a.cq 1 120.m even 2 1
4800.2.f.p 2 120.q odd 4 2
4800.2.f.w 2 120.w even 4 2
5070.2.a.w 1 156.h even 2 1
5070.2.b.k 2 156.l odd 4 2
7350.2.a.ct 1 420.o odd 2 1
8670.2.a.g 1 204.h even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(720))\):

\( T_{7} - 4 \)
\( T_{11} \)
\( T_{13} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - T \)
$7$ \( 1 - 4 T + 7 T^{2} \)
$11$ \( 1 + 11 T^{2} \)
$13$ \( 1 - 2 T + 13 T^{2} \)
$17$ \( 1 + 6 T + 17 T^{2} \)
$19$ \( 1 - 4 T + 19 T^{2} \)
$23$ \( 1 + 23 T^{2} \)
$29$ \( 1 - 6 T + 29 T^{2} \)
$31$ \( 1 + 8 T + 31 T^{2} \)
$37$ \( 1 - 2 T + 37 T^{2} \)
$41$ \( 1 - 6 T + 41 T^{2} \)
$43$ \( 1 - 4 T + 43 T^{2} \)
$47$ \( 1 + 47 T^{2} \)
$53$ \( 1 - 6 T + 53 T^{2} \)
$59$ \( 1 + 59 T^{2} \)
$61$ \( 1 + 10 T + 61 T^{2} \)
$67$ \( 1 - 4 T + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 - 2 T + 73 T^{2} \)
$79$ \( 1 + 8 T + 79 T^{2} \)
$83$ \( 1 - 12 T + 83 T^{2} \)
$89$ \( 1 + 18 T + 89 T^{2} \)
$97$ \( 1 - 2 T + 97 T^{2} \)
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