Properties

Label 720.2.q.g
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: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + \beta_{2} q^{5} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{7} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + \beta_{2} q^{5} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{7} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{9} + ( 1 + \beta_{1} - \beta_{2} ) q^{15} -2 q^{17} + ( 2 + 4 \beta_{1} - 2 \beta_{3} ) q^{19} + ( 1 - 2 \beta_{1} - 4 \beta_{2} ) q^{21} + ( -\beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{23} + ( -1 + \beta_{2} ) q^{25} + ( -5 + \beta_{3} ) q^{27} + ( -3 + 2 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{29} + ( -2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{31} + ( 1 + 2 \beta_{1} - \beta_{3} ) q^{35} -6 q^{37} + ( 2 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{41} + ( 2 + 4 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} ) q^{43} + ( 1 - 2 \beta_{3} ) q^{45} + ( -7 - \beta_{1} + 7 \beta_{2} + 2 \beta_{3} ) q^{47} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{49} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{51} + ( -2 + 8 \beta_{1} - 4 \beta_{3} ) q^{53} + ( 8 - 6 \beta_{2} - 4 \beta_{3} ) q^{57} + ( 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{59} + ( 3 - 3 \beta_{2} ) q^{61} + ( -8 - 3 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{63} + ( -5 \beta_{1} - \beta_{2} - 5 \beta_{3} ) q^{67} + ( -7 - 6 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{69} + ( -8 \beta_{1} + 4 \beta_{3} ) q^{71} + ( 4 + 8 \beta_{1} - 4 \beta_{3} ) q^{73} + ( 1 + \beta_{3} ) q^{75} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{79} + ( -4 \beta_{1} + 7 \beta_{2} + 4 \beta_{3} ) q^{81} + ( -3 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{83} -2 \beta_{2} q^{85} + ( 7 - 4 \beta_{1} - 8 \beta_{2} + 5 \beta_{3} ) q^{87} + ( -7 + 8 \beta_{1} - 4 \beta_{3} ) q^{89} + ( 2 + 4 \beta_{1} - 10 \beta_{2} + 4 \beta_{3} ) q^{93} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{95} + ( -2 + 2 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} + 2q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 4q - 2q^{3} + 2q^{5} + 2q^{7} + 2q^{9} + 2q^{15} - 8q^{17} + 8q^{19} - 4q^{21} - 10q^{23} - 2q^{25} - 20q^{27} - 6q^{29} + 12q^{31} + 4q^{35} - 24q^{37} + 10q^{41} + 4q^{43} + 4q^{45} - 14q^{47} + 4q^{51} - 8q^{53} + 20q^{57} - 12q^{59} + 6q^{61} - 26q^{63} - 2q^{67} - 22q^{69} + 16q^{73} + 4q^{75} - 4q^{79} + 14q^{81} - 6q^{83} - 4q^{85} + 12q^{87} - 28q^{89} - 12q^{93} + 4q^{95} - 4q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{2}\)

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
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 0.707107i
0 −1.72474 0.158919i 0 0.500000 + 0.866025i 0 −0.724745 + 1.25529i 0 2.94949 + 0.548188i 0
241.2 0 0.724745 1.57313i 0 0.500000 + 0.866025i 0 1.72474 2.98735i 0 −1.94949 2.28024i 0
481.1 0 −1.72474 + 0.158919i 0 0.500000 0.866025i 0 −0.724745 1.25529i 0 2.94949 0.548188i 0
481.2 0 0.724745 + 1.57313i 0 0.500000 0.866025i 0 1.72474 + 2.98735i 0 −1.94949 + 2.28024i 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.g 4
3.b odd 2 1 2160.2.q.g 4
4.b odd 2 1 360.2.q.c 4
9.c even 3 1 inner 720.2.q.g 4
9.c even 3 1 6480.2.a.bc 2
9.d odd 6 1 2160.2.q.g 4
9.d odd 6 1 6480.2.a.bl 2
12.b even 2 1 1080.2.q.c 4
36.f odd 6 1 360.2.q.c 4
36.f odd 6 1 3240.2.a.j 2
36.h even 6 1 1080.2.q.c 4
36.h even 6 1 3240.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.q.c 4 4.b odd 2 1
360.2.q.c 4 36.f odd 6 1
720.2.q.g 4 1.a even 1 1 trivial
720.2.q.g 4 9.c even 3 1 inner
1080.2.q.c 4 12.b even 2 1
1080.2.q.c 4 36.h even 6 1
2160.2.q.g 4 3.b odd 2 1
2160.2.q.g 4 9.d odd 6 1
3240.2.a.j 2 36.f odd 6 1
3240.2.a.o 2 36.h even 6 1
6480.2.a.bc 2 9.c even 3 1
6480.2.a.bl 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 \(S_{2}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{4} - 2 T_{7}^{3} + 9 T_{7}^{2} + 10 T_{7} + 25 \)
\( T_{11} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 + 6 T + T^{2} + 2 T^{3} + T^{4} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( 25 + 10 T + 9 T^{2} - 2 T^{3} + T^{4} \)
$11$ \( T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( 2 + T )^{4} \)
$19$ \( ( -20 - 4 T + T^{2} )^{2} \)
$23$ \( 361 + 190 T + 81 T^{2} + 10 T^{3} + T^{4} \)
$29$ \( 225 - 90 T + 51 T^{2} + 6 T^{3} + T^{4} \)
$31$ \( 144 - 144 T + 132 T^{2} - 12 T^{3} + T^{4} \)
$37$ \( ( 6 + T )^{4} \)
$41$ \( 1 - 10 T + 99 T^{2} - 10 T^{3} + T^{4} \)
$43$ \( 8464 + 368 T + 108 T^{2} - 4 T^{3} + T^{4} \)
$47$ \( 1849 + 602 T + 153 T^{2} + 14 T^{3} + T^{4} \)
$53$ \( ( -92 + 4 T + T^{2} )^{2} \)
$59$ \( 144 + 144 T + 132 T^{2} + 12 T^{3} + T^{4} \)
$61$ \( ( 9 - 3 T + T^{2} )^{2} \)
$67$ \( 22201 - 298 T + 153 T^{2} + 2 T^{3} + T^{4} \)
$71$ \( ( -96 + T^{2} )^{2} \)
$73$ \( ( -80 - 8 T + T^{2} )^{2} \)
$79$ \( 400 - 80 T + 36 T^{2} + 4 T^{3} + T^{4} \)
$83$ \( 9 + 18 T + 33 T^{2} + 6 T^{3} + T^{4} \)
$89$ \( ( -47 + 14 T + T^{2} )^{2} \)
$97$ \( ( 4 + 2 T + T^{2} )^{2} \)
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