Properties

Label 720.2.o.b
Level $720$
Weight $2$
Character orbit 720.o
Analytic conductor $5.749$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.o (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{24} - 2 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{5} + ( -3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{7} +O(q^{10})\) \( q + ( -\zeta_{24} - 2 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{5} + ( -3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{7} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{11} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{13} + ( 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{17} + ( 4 - 8 \zeta_{24}^{4} ) q^{19} -6 \zeta_{24}^{6} q^{23} + ( 1 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{25} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{29} + ( 2 - 4 \zeta_{24}^{4} ) q^{31} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 6 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{35} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{37} + ( -5 \zeta_{24} + 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} ) q^{41} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{43} + 11 q^{49} + ( -2 - 3 \zeta_{24} - 3 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 3 \zeta_{24}^{5} ) q^{55} + ( -5 \zeta_{24} - 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} + 10 \zeta_{24}^{7} ) q^{59} -2 q^{61} + ( -3 \zeta_{24} + 4 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{65} + ( 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{71} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{73} + ( -12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{77} + ( 6 - 12 \zeta_{24}^{4} ) q^{79} + 12 \zeta_{24}^{6} q^{83} + ( -12 + 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{85} + ( 5 \zeta_{24} - 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} ) q^{89} + ( -6 + 12 \zeta_{24}^{4} ) q^{91} + ( 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 12 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{95} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 12 \zeta_{24}^{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{25} + 88q^{49} - 16q^{61} - 96q^{85} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
719.1
0.965926 0.258819i
−0.965926 0.258819i
0.965926 + 0.258819i
−0.965926 + 0.258819i
−0.258819 + 0.965926i
0.258819 + 0.965926i
−0.258819 0.965926i
0.258819 0.965926i
0 0 0 −1.73205 1.41421i 0 −4.24264 0 0 0
719.2 0 0 0 −1.73205 1.41421i 0 4.24264 0 0 0
719.3 0 0 0 −1.73205 + 1.41421i 0 −4.24264 0 0 0
719.4 0 0 0 −1.73205 + 1.41421i 0 4.24264 0 0 0
719.5 0 0 0 1.73205 1.41421i 0 −4.24264 0 0 0
719.6 0 0 0 1.73205 1.41421i 0 4.24264 0 0 0
719.7 0 0 0 1.73205 + 1.41421i 0 −4.24264 0 0 0
719.8 0 0 0 1.73205 + 1.41421i 0 4.24264 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 719.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.o.b 8
3.b odd 2 1 inner 720.2.o.b 8
4.b odd 2 1 inner 720.2.o.b 8
5.b even 2 1 inner 720.2.o.b 8
5.c odd 4 1 3600.2.h.f 4
5.c odd 4 1 3600.2.h.g 4
8.b even 2 1 2880.2.o.d 8
8.d odd 2 1 2880.2.o.d 8
12.b even 2 1 inner 720.2.o.b 8
15.d odd 2 1 inner 720.2.o.b 8
15.e even 4 1 3600.2.h.f 4
15.e even 4 1 3600.2.h.g 4
20.d odd 2 1 inner 720.2.o.b 8
20.e even 4 1 3600.2.h.f 4
20.e even 4 1 3600.2.h.g 4
24.f even 2 1 2880.2.o.d 8
24.h odd 2 1 2880.2.o.d 8
40.e odd 2 1 2880.2.o.d 8
40.f even 2 1 2880.2.o.d 8
60.h even 2 1 inner 720.2.o.b 8
60.l odd 4 1 3600.2.h.f 4
60.l odd 4 1 3600.2.h.g 4
120.i odd 2 1 2880.2.o.d 8
120.m even 2 1 2880.2.o.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.o.b 8 1.a even 1 1 trivial
720.2.o.b 8 3.b odd 2 1 inner
720.2.o.b 8 4.b odd 2 1 inner
720.2.o.b 8 5.b even 2 1 inner
720.2.o.b 8 12.b even 2 1 inner
720.2.o.b 8 15.d odd 2 1 inner
720.2.o.b 8 20.d odd 2 1 inner
720.2.o.b 8 60.h even 2 1 inner
2880.2.o.d 8 8.b even 2 1
2880.2.o.d 8 8.d odd 2 1
2880.2.o.d 8 24.f even 2 1
2880.2.o.d 8 24.h odd 2 1
2880.2.o.d 8 40.e odd 2 1
2880.2.o.d 8 40.f even 2 1
2880.2.o.d 8 120.i odd 2 1
2880.2.o.d 8 120.m even 2 1
3600.2.h.f 4 5.c odd 4 1
3600.2.h.f 4 15.e even 4 1
3600.2.h.f 4 20.e even 4 1
3600.2.h.f 4 60.l odd 4 1
3600.2.h.g 4 5.c odd 4 1
3600.2.h.g 4 15.e even 4 1
3600.2.h.g 4 20.e even 4 1
3600.2.h.g 4 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 18 \) acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 25 - 2 T^{2} + T^{4} )^{2} \)
$7$ \( ( -18 + T^{2} )^{4} \)
$11$ \( ( -6 + T^{2} )^{4} \)
$13$ \( ( 6 + T^{2} )^{4} \)
$17$ \( ( -48 + T^{2} )^{4} \)
$19$ \( ( 48 + T^{2} )^{4} \)
$23$ \( ( 36 + T^{2} )^{4} \)
$29$ \( ( 8 + T^{2} )^{4} \)
$31$ \( ( 12 + T^{2} )^{4} \)
$37$ \( ( 6 + T^{2} )^{4} \)
$41$ \( ( 50 + T^{2} )^{4} \)
$43$ \( ( -72 + T^{2} )^{4} \)
$47$ \( T^{8} \)
$53$ \( T^{8} \)
$59$ \( ( -150 + T^{2} )^{4} \)
$61$ \( ( 2 + T )^{8} \)
$67$ \( T^{8} \)
$71$ \( ( -96 + T^{2} )^{4} \)
$73$ \( ( 24 + T^{2} )^{4} \)
$79$ \( ( 108 + T^{2} )^{4} \)
$83$ \( ( 144 + T^{2} )^{4} \)
$89$ \( ( 50 + T^{2} )^{4} \)
$97$ \( ( 216 + T^{2} )^{4} \)
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