Properties

Label 720.2.q.i
Level $720$
Weight $2$
Character orbit 720.q
Analytic conductor $5.749$
Analytic rank $0$
Dimension $6$
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.954288.1
Defining polynomial: \(x^{6} - x^{5} - 2 x^{4} + 3 x^{3} - 6 x^{2} - 9 x + 27\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{4} q^{3} + \beta_{3} q^{5} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{7} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{5} ) q^{9} +O(q^{10})\) \( q -\beta_{4} q^{3} + \beta_{3} q^{5} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{7} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{5} ) q^{9} + ( -\beta_{1} + \beta_{2} ) q^{11} + ( \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{13} -\beta_{1} q^{15} + ( -\beta_{1} - \beta_{4} + \beta_{5} ) q^{17} + ( -2 + \beta_{1} + \beta_{4} - \beta_{5} ) q^{19} + ( 1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{21} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - \beta_{5} ) q^{23} + ( -1 - \beta_{3} ) q^{25} + ( 2 - 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{27} + ( 3 + 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{29} + ( -4 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{31} + ( 2 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{33} + ( -2 - \beta_{2} + \beta_{4} ) q^{35} + ( 2 + \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{37} + ( 4 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{39} + ( -2 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{41} + ( 2 + 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{43} + ( -1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{45} + ( 6 - 4 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{47} + ( -4 \beta_{1} - \beta_{2} - \beta_{4} - 3 \beta_{5} ) q^{49} + ( -2 - \beta_{2} - 4 \beta_{3} + \beta_{4} - \beta_{5} ) q^{51} + ( 2 \beta_{2} - 2 \beta_{4} ) q^{53} + ( \beta_{1} + \beta_{4} - \beta_{5} ) q^{55} + ( 2 + \beta_{2} + 4 \beta_{3} + \beta_{4} + \beta_{5} ) q^{57} + ( 2 \beta_{1} + 2 \beta_{5} ) q^{59} + ( 1 - 3 \beta_{1} + 3 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{61} + ( -6 - \beta_{1} - 3 \beta_{2} + \beta_{4} ) q^{63} + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{65} + ( \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{67} + ( 9 + \beta_{1} + 3 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{69} + ( 6 - 3 \beta_{1} + 2 \beta_{2} - 5 \beta_{4} + 3 \beta_{5} ) q^{71} + ( -4 - 4 \beta_{2} + 4 \beta_{4} ) q^{73} + ( \beta_{1} + \beta_{4} ) q^{75} + ( 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{77} + ( 2 - 4 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 4 \beta_{5} ) q^{79} + ( -4 - 2 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{81} + ( -6 + 3 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{83} + ( \beta_{2} + \beta_{4} - \beta_{5} ) q^{85} + ( 2 - \beta_{1} - 2 \beta_{2} - 8 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} ) q^{87} -3 q^{89} + ( -4 + \beta_{1} + \beta_{4} - \beta_{5} ) q^{91} + ( -8 + 5 \beta_{1} - 7 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} ) q^{93} + ( -\beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{95} + ( 4 + 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{97} + ( -4 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{3} - 3q^{5} + 5q^{7} - 7q^{9} + O(q^{10}) \) \( 6q - q^{3} - 3q^{5} + 5q^{7} - 7q^{9} - 2q^{11} - 4q^{13} - q^{15} - 4q^{17} - 8q^{19} + 3q^{23} - 3q^{25} + 2q^{27} + 7q^{29} + 8q^{31} + 20q^{33} - 10q^{35} + 12q^{37} + 14q^{39} + 13q^{41} + 10q^{43} + 2q^{45} + 13q^{47} + 2q^{49} + 4q^{51} - 4q^{53} + 4q^{55} - 2q^{57} - 2q^{59} - q^{61} - 33q^{63} - 4q^{65} + 11q^{67} + 39q^{69} + 20q^{71} - 16q^{73} + 2q^{75} + 2q^{79} - 19q^{81} - 15q^{83} + 2q^{85} + 26q^{87} - 18q^{89} - 20q^{91} - 42q^{93} + 4q^{95} + 18q^{97} - 22q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 2 x^{4} + 3 x^{3} - 6 x^{2} - 9 x + 27\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 5 \nu^{4} + \nu^{3} + 9 \nu^{2} - 6 \nu - 45 \)\()/27\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{5} - \nu^{4} - 2 \nu^{3} + 12 \nu + 9 \)\()/27\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} + 2 \nu^{4} - 2 \nu^{3} - 6 \nu - 18 \)\()/9\)
\(\beta_{5}\)\(=\)\((\)\( 4 \nu^{5} + 2 \nu^{4} - 5 \nu^{3} + 18 \nu^{2} - 24 \nu - 72 \)\()/27\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2\)
\(\nu^{3}\)\(=\)\(-\beta_{5} - 2 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 2\)
\(\nu^{4}\)\(=\)\(-2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 4 \beta_{2} + 5\)
\(\nu^{5}\)\(=\)\(2 \beta_{5} + \beta_{4} - 10 \beta_{3} - 4 \beta_{2} + 6 \beta_{1} + 4\)

Character values

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

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(\beta_{3}\)

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
0.403374 1.68443i
−1.62241 + 0.606458i
1.71903 + 0.211943i
0.403374 + 1.68443i
−1.62241 0.606458i
1.71903 0.211943i
0 −1.25707 1.19154i 0 −0.500000 0.866025i 0 −0.257068 + 0.445256i 0 0.160442 + 2.99571i 0
241.2 0 −0.285997 + 1.70828i 0 −0.500000 0.866025i 0 0.714003 1.23669i 0 −2.83641 0.977122i 0
241.3 0 1.04307 1.38276i 0 −0.500000 0.866025i 0 2.04307 3.53869i 0 −0.824030 2.88461i 0
481.1 0 −1.25707 + 1.19154i 0 −0.500000 + 0.866025i 0 −0.257068 0.445256i 0 0.160442 2.99571i 0
481.2 0 −0.285997 1.70828i 0 −0.500000 + 0.866025i 0 0.714003 + 1.23669i 0 −2.83641 + 0.977122i 0
481.3 0 1.04307 + 1.38276i 0 −0.500000 + 0.866025i 0 2.04307 + 3.53869i 0 −0.824030 + 2.88461i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 481.3
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.i 6
3.b odd 2 1 2160.2.q.k 6
4.b odd 2 1 45.2.e.b 6
9.c even 3 1 inner 720.2.q.i 6
9.c even 3 1 6480.2.a.bv 3
9.d odd 6 1 2160.2.q.k 6
9.d odd 6 1 6480.2.a.bs 3
12.b even 2 1 135.2.e.b 6
20.d odd 2 1 225.2.e.b 6
20.e even 4 2 225.2.k.b 12
36.f odd 6 1 45.2.e.b 6
36.f odd 6 1 405.2.a.j 3
36.h even 6 1 135.2.e.b 6
36.h even 6 1 405.2.a.i 3
60.h even 2 1 675.2.e.b 6
60.l odd 4 2 675.2.k.b 12
180.n even 6 1 675.2.e.b 6
180.n even 6 1 2025.2.a.o 3
180.p odd 6 1 225.2.e.b 6
180.p odd 6 1 2025.2.a.n 3
180.v odd 12 2 675.2.k.b 12
180.v odd 12 2 2025.2.b.m 6
180.x even 12 2 225.2.k.b 12
180.x even 12 2 2025.2.b.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.e.b 6 4.b odd 2 1
45.2.e.b 6 36.f odd 6 1
135.2.e.b 6 12.b even 2 1
135.2.e.b 6 36.h even 6 1
225.2.e.b 6 20.d odd 2 1
225.2.e.b 6 180.p odd 6 1
225.2.k.b 12 20.e even 4 2
225.2.k.b 12 180.x even 12 2
405.2.a.i 3 36.h even 6 1
405.2.a.j 3 36.f odd 6 1
675.2.e.b 6 60.h even 2 1
675.2.e.b 6 180.n even 6 1
675.2.k.b 12 60.l odd 4 2
675.2.k.b 12 180.v odd 12 2
720.2.q.i 6 1.a even 1 1 trivial
720.2.q.i 6 9.c even 3 1 inner
2025.2.a.n 3 180.p odd 6 1
2025.2.a.o 3 180.n even 6 1
2025.2.b.l 6 180.x even 12 2
2025.2.b.m 6 180.v odd 12 2
2160.2.q.k 6 3.b odd 2 1
2160.2.q.k 6 9.d odd 6 1
6480.2.a.bs 3 9.d odd 6 1
6480.2.a.bv 3 9.c even 3 1

Hecke kernels

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

\( T_{7}^{6} - 5 T_{7}^{5} + 22 T_{7}^{4} - 21 T_{7}^{3} + 24 T_{7}^{2} + 9 T_{7} + 9 \)
\( T_{11}^{6} + 2 T_{11}^{5} + 12 T_{11}^{4} + 8 T_{11}^{3} + 88 T_{11}^{2} + 96 T_{11} + 144 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( 27 + 9 T + 12 T^{2} + 3 T^{3} + 4 T^{4} + T^{5} + T^{6} \)
$5$ \( ( 1 + T + T^{2} )^{3} \)
$7$ \( 9 + 9 T + 24 T^{2} - 21 T^{3} + 22 T^{4} - 5 T^{5} + T^{6} \)
$11$ \( 144 + 96 T + 88 T^{2} + 8 T^{3} + 12 T^{4} + 2 T^{5} + T^{6} \)
$13$ \( 16 + 16 T + 32 T^{2} - 8 T^{3} + 20 T^{4} + 4 T^{5} + T^{6} \)
$17$ \( ( -12 - 8 T + 2 T^{2} + T^{3} )^{2} \)
$19$ \( ( -4 - 4 T + 4 T^{2} + T^{3} )^{2} \)
$23$ \( 13689 - 3861 T + 1440 T^{2} - 135 T^{3} + 42 T^{4} - 3 T^{5} + T^{6} \)
$29$ \( 2601 - 1479 T + 1198 T^{2} + 101 T^{3} + 78 T^{4} - 7 T^{5} + T^{6} \)
$31$ \( 219024 - 28080 T + 7344 T^{2} - 456 T^{3} + 124 T^{4} - 8 T^{5} + T^{6} \)
$37$ \( ( 4 - 12 T - 6 T^{2} + T^{3} )^{2} \)
$41$ \( 9 - 57 T + 322 T^{2} - 241 T^{3} + 150 T^{4} - 13 T^{5} + T^{6} \)
$43$ \( 16 - 16 T + 56 T^{2} + 32 T^{3} + 104 T^{4} - 10 T^{5} + T^{6} \)
$47$ \( 136161 - 4059 T + 4918 T^{2} - 595 T^{3} + 180 T^{4} - 13 T^{5} + T^{6} \)
$53$ \( ( -24 - 20 T + 2 T^{2} + T^{3} )^{2} \)
$59$ \( 576 + 480 T + 448 T^{2} + 8 T^{3} + 24 T^{4} + 2 T^{5} + T^{6} \)
$61$ \( 5041 - 2627 T + 1298 T^{2} - 179 T^{3} + 38 T^{4} + T^{5} + T^{6} \)
$67$ \( 257049 - 19773 T + 7098 T^{2} - 585 T^{3} + 160 T^{4} - 11 T^{5} + T^{6} \)
$71$ \( ( 708 - 92 T - 10 T^{2} + T^{3} )^{2} \)
$73$ \( ( -128 - 64 T + 8 T^{2} + T^{3} )^{2} \)
$79$ \( 576 + 2016 T + 7008 T^{2} + 216 T^{3} + 88 T^{4} - 2 T^{5} + T^{6} \)
$83$ \( 6561 - 2187 T + 1944 T^{2} + 567 T^{3} + 198 T^{4} + 15 T^{5} + T^{6} \)
$89$ \( ( 3 + T )^{6} \)
$97$ \( 1700416 - 46944 T + 24768 T^{2} - 1960 T^{3} + 360 T^{4} - 18 T^{5} + T^{6} \)
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