Properties

Label 720.4.a.b
Level $720$
Weight $4$
Character orbit 720.a
Self dual yes
Analytic conductor $42.481$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 30)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 5 q^{5} - 32 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 q^{5} - 32 q^{7} - 60 q^{11} - 34 q^{13} - 42 q^{17} + 76 q^{19} + 25 q^{25} - 6 q^{29} + 232 q^{31} + 160 q^{35} + 134 q^{37} - 234 q^{41} + 412 q^{43} - 360 q^{47} + 681 q^{49} - 222 q^{53} + 300 q^{55} + 660 q^{59} - 490 q^{61} + 170 q^{65} - 812 q^{67} + 120 q^{71} + 746 q^{73} + 1920 q^{77} - 152 q^{79} - 804 q^{83} + 210 q^{85} + 678 q^{89} + 1088 q^{91} - 380 q^{95} + 194 q^{97}+O(q^{100}) \) 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 −32.0000 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.b 1
3.b odd 2 1 240.4.a.c 1
4.b odd 2 1 90.4.a.d 1
12.b even 2 1 30.4.a.a 1
15.d odd 2 1 1200.4.a.bk 1
15.e even 4 2 1200.4.f.u 2
20.d odd 2 1 450.4.a.b 1
20.e even 4 2 450.4.c.k 2
24.f even 2 1 960.4.a.j 1
24.h odd 2 1 960.4.a.s 1
36.f odd 6 2 810.4.e.e 2
36.h even 6 2 810.4.e.m 2
60.h even 2 1 150.4.a.e 1
60.l odd 4 2 150.4.c.a 2
84.h odd 2 1 1470.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.4.a.a 1 12.b even 2 1
90.4.a.d 1 4.b odd 2 1
150.4.a.e 1 60.h even 2 1
150.4.c.a 2 60.l odd 4 2
240.4.a.c 1 3.b odd 2 1
450.4.a.b 1 20.d odd 2 1
450.4.c.k 2 20.e even 4 2
720.4.a.b 1 1.a even 1 1 trivial
810.4.e.e 2 36.f odd 6 2
810.4.e.m 2 36.h even 6 2
960.4.a.j 1 24.f even 2 1
960.4.a.s 1 24.h odd 2 1
1200.4.a.bk 1 15.d odd 2 1
1200.4.f.u 2 15.e even 4 2
1470.4.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(720))\):

\( T_{7} + 32 \) Copy content Toggle raw display
\( T_{11} + 60 \) 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
$7$ \( T + 32 \) Copy content Toggle raw display
$11$ \( T + 60 \) Copy content Toggle raw display
$13$ \( T + 34 \) Copy content Toggle raw display
$17$ \( T + 42 \) Copy content Toggle raw display
$19$ \( T - 76 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T + 6 \) Copy content Toggle raw display
$31$ \( T - 232 \) Copy content Toggle raw display
$37$ \( T - 134 \) Copy content Toggle raw display
$41$ \( T + 234 \) Copy content Toggle raw display
$43$ \( T - 412 \) Copy content Toggle raw display
$47$ \( T + 360 \) Copy content Toggle raw display
$53$ \( T + 222 \) Copy content Toggle raw display
$59$ \( T - 660 \) Copy content Toggle raw display
$61$ \( T + 490 \) Copy content Toggle raw display
$67$ \( T + 812 \) Copy content Toggle raw display
$71$ \( T - 120 \) Copy content Toggle raw display
$73$ \( T - 746 \) Copy content Toggle raw display
$79$ \( T + 152 \) Copy content Toggle raw display
$83$ \( T + 804 \) Copy content Toggle raw display
$89$ \( T - 678 \) Copy content Toggle raw display
$97$ \( T - 194 \) Copy content Toggle raw display
show more
show less