Properties

Label 720.2.a.c
Level $720$
Weight $2$
Character orbit 720.a
Self dual yes
Analytic conductor $5.749$
Analytic rank $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{5} + O(q^{10}) \) \( q - q^{5} - 4q^{11} - 2q^{13} - 2q^{17} - 4q^{19} + q^{25} + 2q^{29} - 10q^{37} - 10q^{41} - 4q^{43} + 8q^{47} - 7q^{49} + 10q^{53} + 4q^{55} - 4q^{59} - 2q^{61} + 2q^{65} - 12q^{67} - 8q^{71} + 10q^{73} + 12q^{83} + 2q^{85} + 6q^{89} + 4q^{95} + 2q^{97} + O(q^{100}) \)

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 0 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.c 1
3.b odd 2 1 240.2.a.d 1
4.b odd 2 1 45.2.a.a 1
5.b even 2 1 3600.2.a.u 1
5.c odd 4 2 3600.2.f.e 2
8.b even 2 1 2880.2.a.bc 1
8.d odd 2 1 2880.2.a.y 1
12.b even 2 1 15.2.a.a 1
15.d odd 2 1 1200.2.a.e 1
15.e even 4 2 1200.2.f.h 2
20.d odd 2 1 225.2.a.b 1
20.e even 4 2 225.2.b.b 2
24.f even 2 1 960.2.a.l 1
24.h odd 2 1 960.2.a.a 1
28.d even 2 1 2205.2.a.i 1
36.f odd 6 2 405.2.e.c 2
36.h even 6 2 405.2.e.f 2
44.c even 2 1 5445.2.a.c 1
48.i odd 4 2 3840.2.k.r 2
48.k even 4 2 3840.2.k.m 2
52.b odd 2 1 7605.2.a.g 1
60.h even 2 1 75.2.a.b 1
60.l odd 4 2 75.2.b.b 2
84.h odd 2 1 735.2.a.c 1
84.j odd 6 2 735.2.i.d 2
84.n even 6 2 735.2.i.e 2
120.i odd 2 1 4800.2.a.bz 1
120.m even 2 1 4800.2.a.t 1
120.q odd 4 2 4800.2.f.bf 2
120.w even 4 2 4800.2.f.c 2
132.d odd 2 1 1815.2.a.d 1
156.h even 2 1 2535.2.a.j 1
204.h even 2 1 4335.2.a.c 1
228.b odd 2 1 5415.2.a.j 1
276.h odd 2 1 7935.2.a.d 1
420.o odd 2 1 3675.2.a.j 1
660.g odd 2 1 9075.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.2.a.a 1 12.b even 2 1
45.2.a.a 1 4.b odd 2 1
75.2.a.b 1 60.h even 2 1
75.2.b.b 2 60.l odd 4 2
225.2.a.b 1 20.d odd 2 1
225.2.b.b 2 20.e even 4 2
240.2.a.d 1 3.b odd 2 1
405.2.e.c 2 36.f odd 6 2
405.2.e.f 2 36.h even 6 2
720.2.a.c 1 1.a even 1 1 trivial
735.2.a.c 1 84.h odd 2 1
735.2.i.d 2 84.j odd 6 2
735.2.i.e 2 84.n even 6 2
960.2.a.a 1 24.h odd 2 1
960.2.a.l 1 24.f even 2 1
1200.2.a.e 1 15.d odd 2 1
1200.2.f.h 2 15.e even 4 2
1815.2.a.d 1 132.d odd 2 1
2205.2.a.i 1 28.d even 2 1
2535.2.a.j 1 156.h even 2 1
2880.2.a.y 1 8.d odd 2 1
2880.2.a.bc 1 8.b even 2 1
3600.2.a.u 1 5.b even 2 1
3600.2.f.e 2 5.c odd 4 2
3675.2.a.j 1 420.o odd 2 1
3840.2.k.m 2 48.k even 4 2
3840.2.k.r 2 48.i odd 4 2
4335.2.a.c 1 204.h even 2 1
4800.2.a.t 1 120.m even 2 1
4800.2.a.bz 1 120.i odd 2 1
4800.2.f.c 2 120.w even 4 2
4800.2.f.bf 2 120.q odd 4 2
5415.2.a.j 1 228.b odd 2 1
5445.2.a.c 1 44.c even 2 1
7605.2.a.g 1 52.b odd 2 1
7935.2.a.d 1 276.h odd 2 1
9075.2.a.g 1 660.g odd 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

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