Properties

Label 720.2.a.e
Level $720$
Weight $2$
Character orbit 720.a
Self dual yes
Analytic conductor $5.749$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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 40)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{5} + 4q^{7} + O(q^{10}) \) \( q - q^{5} + 4q^{7} + 4q^{11} - 2q^{13} - 2q^{17} - 4q^{19} + 4q^{23} + q^{25} + 2q^{29} + 8q^{31} - 4q^{35} + 6q^{37} + 6q^{41} + 8q^{43} + 4q^{47} + 9q^{49} - 6q^{53} - 4q^{55} - 4q^{59} - 2q^{61} + 2q^{65} - 8q^{67} - 6q^{73} + 16q^{77} - 16q^{83} + 2q^{85} + 6q^{89} - 8q^{91} + 4q^{95} - 14q^{97} + O(q^{100}) \)

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:

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 720.2.a.e 1
3.b odd 2 1 80.2.a.a 1
4.b odd 2 1 360.2.a.a 1
5.b even 2 1 3600.2.a.h 1
5.c odd 4 2 3600.2.f.t 2
8.b even 2 1 2880.2.a.bg 1
8.d odd 2 1 2880.2.a.t 1
12.b even 2 1 40.2.a.a 1
15.d odd 2 1 400.2.a.e 1
15.e even 4 2 400.2.c.d 2
20.d odd 2 1 1800.2.a.v 1
20.e even 4 2 1800.2.f.a 2
21.c even 2 1 3920.2.a.s 1
24.f even 2 1 320.2.a.c 1
24.h odd 2 1 320.2.a.d 1
33.d even 2 1 9680.2.a.q 1
36.f odd 6 2 3240.2.q.x 2
36.h even 6 2 3240.2.q.k 2
48.i odd 4 2 1280.2.d.a 2
48.k even 4 2 1280.2.d.j 2
60.h even 2 1 200.2.a.c 1
60.l odd 4 2 200.2.c.b 2
84.h odd 2 1 1960.2.a.g 1
84.j odd 6 2 1960.2.q.i 2
84.n even 6 2 1960.2.q.h 2
120.i odd 2 1 1600.2.a.k 1
120.m even 2 1 1600.2.a.o 1
120.q odd 4 2 1600.2.c.k 2
120.w even 4 2 1600.2.c.m 2
132.d odd 2 1 4840.2.a.f 1
156.h even 2 1 6760.2.a.i 1
420.o odd 2 1 9800.2.a.x 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.a.a 1 12.b even 2 1
80.2.a.a 1 3.b odd 2 1
200.2.a.c 1 60.h even 2 1
200.2.c.b 2 60.l odd 4 2
320.2.a.c 1 24.f even 2 1
320.2.a.d 1 24.h odd 2 1
360.2.a.a 1 4.b odd 2 1
400.2.a.e 1 15.d odd 2 1
400.2.c.d 2 15.e even 4 2
720.2.a.e 1 1.a even 1 1 trivial
1280.2.d.a 2 48.i odd 4 2
1280.2.d.j 2 48.k even 4 2
1600.2.a.k 1 120.i odd 2 1
1600.2.a.o 1 120.m even 2 1
1600.2.c.k 2 120.q odd 4 2
1600.2.c.m 2 120.w even 4 2
1800.2.a.v 1 20.d odd 2 1
1800.2.f.a 2 20.e even 4 2
1960.2.a.g 1 84.h odd 2 1
1960.2.q.h 2 84.n even 6 2
1960.2.q.i 2 84.j odd 6 2
2880.2.a.t 1 8.d odd 2 1
2880.2.a.bg 1 8.b even 2 1
3240.2.q.k 2 36.h even 6 2
3240.2.q.x 2 36.f odd 6 2
3600.2.a.h 1 5.b even 2 1
3600.2.f.t 2 5.c odd 4 2
3920.2.a.s 1 21.c even 2 1
4840.2.a.f 1 132.d odd 2 1
6760.2.a.i 1 156.h even 2 1
9680.2.a.q 1 33.d even 2 1
9800.2.a.x 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(720))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 1 + T \)
$7$ \( -4 + T \)
$11$ \( -4 + T \)
$13$ \( 2 + T \)
$17$ \( 2 + T \)
$19$ \( 4 + T \)
$23$ \( -4 + T \)
$29$ \( -2 + T \)
$31$ \( -8 + T \)
$37$ \( -6 + T \)
$41$ \( -6 + T \)
$43$ \( -8 + T \)
$47$ \( -4 + T \)
$53$ \( 6 + T \)
$59$ \( 4 + T \)
$61$ \( 2 + T \)
$67$ \( 8 + T \)
$71$ \( T \)
$73$ \( 6 + T \)
$79$ \( T \)
$83$ \( 16 + T \)
$89$ \( -6 + T \)
$97$ \( 14 + T \)
show more
show less