Properties

Label 720.2.q.h
Level $720$
Weight $2$
Character orbit 720.q
Analytic conductor $5.749$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{5} + ( 2 - \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{5} + ( 2 - \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( 2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{11} + ( 2 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{13} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{15} + ( 4 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{17} + 2 q^{19} + ( 3 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{21} + ( \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{23} + ( -1 + \zeta_{12}^{2} ) q^{25} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + ( -5 + 2 \zeta_{12} + 5 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{29} -2 \zeta_{12}^{2} q^{31} + ( 6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{33} + ( 2 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{35} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{37} + ( 6 + 2 \zeta_{12} - 6 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{39} + ( -4 \zeta_{12} - 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{41} + ( -8 + 2 \zeta_{12} + 8 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{43} -3 q^{45} + ( 2 - \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{47} + ( 4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{49} + ( -4 \zeta_{12} + 6 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{51} + 6 q^{53} + ( 2 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{55} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{57} + ( 6 \zeta_{12} + 4 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{59} + ( 5 + 2 \zeta_{12} - 5 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{61} + ( -3 \zeta_{12} + 6 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{63} + ( -2 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{65} + ( -\zeta_{12} + 8 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{67} + ( 3 + 2 \zeta_{12} - 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{69} + ( -6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{71} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{73} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{75} + ( 6 \zeta_{12} - 10 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{77} + ( 12 + 2 \zeta_{12} - 12 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{79} -9 \zeta_{12}^{2} q^{81} + ( -8 - 3 \zeta_{12} + 8 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{83} + ( -2 \zeta_{12} + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{85} + ( -6 + 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{87} + ( -3 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{89} + ( -2 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{91} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{93} + 2 \zeta_{12}^{2} q^{95} + ( 2 + 4 \zeta_{12} - 2 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{97} + ( -6 \zeta_{12} + 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{5} + 4q^{7} - 6q^{9} + O(q^{10}) \) \( 4q + 2q^{5} + 4q^{7} - 6q^{9} + 4q^{11} + 4q^{13} + 16q^{17} + 8q^{19} + 12q^{21} + 4q^{23} - 2q^{25} - 10q^{29} - 4q^{31} + 24q^{33} + 8q^{35} + 12q^{39} - 6q^{41} - 16q^{43} - 12q^{45} + 4q^{47} + 12q^{51} + 24q^{53} + 8q^{55} + 8q^{59} + 10q^{61} + 12q^{63} - 4q^{65} + 16q^{67} + 6q^{69} - 24q^{71} - 20q^{77} + 24q^{79} - 18q^{81} - 16q^{83} + 8q^{85} - 24q^{87} - 12q^{89} - 8q^{91} + 4q^{95} + 4q^{97} + 12q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\zeta_{12}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
241.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 −0.866025 1.50000i 0 0.500000 + 0.866025i 0 0.133975 0.232051i 0 −1.50000 + 2.59808i 0
241.2 0 0.866025 + 1.50000i 0 0.500000 + 0.866025i 0 1.86603 3.23205i 0 −1.50000 + 2.59808i 0
481.1 0 −0.866025 + 1.50000i 0 0.500000 0.866025i 0 0.133975 + 0.232051i 0 −1.50000 2.59808i 0
481.2 0 0.866025 1.50000i 0 0.500000 0.866025i 0 1.86603 + 3.23205i 0 −1.50000 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.q.h 4
3.b odd 2 1 2160.2.q.h 4
4.b odd 2 1 360.2.q.b 4
9.c even 3 1 inner 720.2.q.h 4
9.c even 3 1 6480.2.a.ba 2
9.d odd 6 1 2160.2.q.h 4
9.d odd 6 1 6480.2.a.bk 2
12.b even 2 1 1080.2.q.b 4
36.f odd 6 1 360.2.q.b 4
36.f odd 6 1 3240.2.a.k 2
36.h even 6 1 1080.2.q.b 4
36.h even 6 1 3240.2.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.q.b 4 4.b odd 2 1
360.2.q.b 4 36.f odd 6 1
720.2.q.h 4 1.a even 1 1 trivial
720.2.q.h 4 9.c even 3 1 inner
1080.2.q.b 4 12.b even 2 1
1080.2.q.b 4 36.h even 6 1
2160.2.q.h 4 3.b odd 2 1
2160.2.q.h 4 9.d odd 6 1
3240.2.a.k 2 36.f odd 6 1
3240.2.a.p 2 36.h even 6 1
6480.2.a.ba 2 9.c even 3 1
6480.2.a.bk 2 9.d odd 6 1

Hecke kernels

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

\( T_{7}^{4} - 4 T_{7}^{3} + 15 T_{7}^{2} - 4 T_{7} + 1 \)
\( T_{11}^{4} - 4 T_{11}^{3} + 24 T_{11}^{2} + 32 T_{11} + 64 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 + 3 T^{2} + T^{4} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( 1 - 4 T + 15 T^{2} - 4 T^{3} + T^{4} \)
$11$ \( 64 + 32 T + 24 T^{2} - 4 T^{3} + T^{4} \)
$13$ \( 64 + 32 T + 24 T^{2} - 4 T^{3} + T^{4} \)
$17$ \( ( 4 - 8 T + T^{2} )^{2} \)
$19$ \( ( -2 + T )^{4} \)
$23$ \( 1 - 4 T + 15 T^{2} - 4 T^{3} + T^{4} \)
$29$ \( 169 + 130 T + 87 T^{2} + 10 T^{3} + T^{4} \)
$31$ \( ( 4 + 2 T + T^{2} )^{2} \)
$37$ \( ( -108 + T^{2} )^{2} \)
$41$ \( 1521 - 234 T + 75 T^{2} + 6 T^{3} + T^{4} \)
$43$ \( 2704 + 832 T + 204 T^{2} + 16 T^{3} + T^{4} \)
$47$ \( 1 - 4 T + 15 T^{2} - 4 T^{3} + T^{4} \)
$53$ \( ( -6 + T )^{4} \)
$59$ \( 8464 + 736 T + 156 T^{2} - 8 T^{3} + T^{4} \)
$61$ \( 169 - 130 T + 87 T^{2} - 10 T^{3} + T^{4} \)
$67$ \( 3721 - 976 T + 195 T^{2} - 16 T^{3} + T^{4} \)
$71$ \( ( 24 + 12 T + T^{2} )^{2} \)
$73$ \( ( -48 + T^{2} )^{2} \)
$79$ \( 17424 - 3168 T + 444 T^{2} - 24 T^{3} + T^{4} \)
$83$ \( 1369 + 592 T + 219 T^{2} + 16 T^{3} + T^{4} \)
$89$ \( ( -39 + 6 T + T^{2} )^{2} \)
$97$ \( 1936 + 176 T + 60 T^{2} - 4 T^{3} + T^{4} \)
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