Properties

Label 720.2.w.e
Level $720$
Weight $2$
Character orbit 720.w
Analytic conductor $5.749$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.40960000.1
Defining polynomial: \(x^{8} + 7 x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{7} q^{5} + ( 1 + \beta_{2} + \beta_{5} ) q^{7} +O(q^{10})\) \( q -\beta_{7} q^{5} + ( 1 + \beta_{2} + \beta_{5} ) q^{7} + ( \beta_{1} + \beta_{6} + \beta_{7} ) q^{11} + ( -2 + 2 \beta_{2} + \beta_{3} ) q^{13} + ( -\beta_{4} + \beta_{6} - 2 \beta_{7} ) q^{17} + ( -2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{19} + ( -2 \beta_{4} - 2 \beta_{6} ) q^{23} -5 \beta_{2} q^{25} + ( \beta_{1} + 2 \beta_{4} - \beta_{7} ) q^{29} + ( -6 - \beta_{3} + \beta_{5} ) q^{31} + ( \beta_{1} + 5 \beta_{4} - \beta_{7} ) q^{35} + \beta_{5} q^{37} -5 \beta_{6} q^{41} + ( 2 - 2 \beta_{2} + 2 \beta_{3} ) q^{43} + ( -\beta_{4} + \beta_{6} - 2 \beta_{7} ) q^{47} + ( 5 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{49} + ( 5 + 5 \beta_{2} + \beta_{5} ) q^{55} + ( \beta_{1} - 3 \beta_{4} - \beta_{7} ) q^{59} + ( 6 - \beta_{3} + \beta_{5} ) q^{61} + ( 2 \beta_{1} + 5 \beta_{6} + 2 \beta_{7} ) q^{65} + ( 6 + 6 \beta_{2} - 2 \beta_{5} ) q^{67} + ( -4 \beta_{1} - 4 \beta_{7} ) q^{71} + ( -5 + 5 \beta_{2} - 2 \beta_{3} ) q^{73} + ( -6 \beta_{4} + 6 \beta_{6} + 4 \beta_{7} ) q^{77} + ( -10 \beta_{2} - \beta_{3} - \beta_{5} ) q^{79} + ( 2 \beta_{1} - 3 \beta_{4} - 3 \beta_{6} ) q^{83} + ( -10 \beta_{2} + \beta_{3} + \beta_{5} ) q^{85} + ( -2 \beta_{1} + 5 \beta_{4} + 2 \beta_{7} ) q^{89} + ( -14 + 3 \beta_{3} - 3 \beta_{5} ) q^{91} + ( -2 \beta_{1} + 5 \beta_{4} + 5 \beta_{6} ) q^{95} + ( 1 + \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{7} + O(q^{10}) \) \( 8q + 8q^{7} - 16q^{13} - 48q^{31} + 16q^{43} + 40q^{55} + 48q^{61} + 48q^{67} - 40q^{73} - 112q^{91} + 8q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 7 x^{4} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} + 11 \nu \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 8 \nu^{2} \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{6} + 2 \nu^{4} - 18 \nu^{2} + 7 \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{7} - \nu^{5} + 13 \nu^{3} - 5 \nu \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{6} - 2 \nu^{4} - 18 \nu^{2} - 7 \)\()/3\)
\(\beta_{6}\)\(=\)\((\)\( -2 \nu^{7} - \nu^{5} - 13 \nu^{3} - 5 \nu \)\()/3\)
\(\beta_{7}\)\(=\)\((\)\( 4 \nu^{7} + 29 \nu^{3} \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} + \beta_{4} + 2 \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + \beta_{3} + 6 \beta_{2}\)\()/4\)
\(\nu^{3}\)\(=\)\(\beta_{7} + \beta_{6} - \beta_{4}\)
\(\nu^{4}\)\(=\)\((\)\(-3 \beta_{5} + 3 \beta_{3} - 14\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-11 \beta_{6} - 11 \beta_{4} - 10 \beta_{1}\)\()/4\)
\(\nu^{6}\)\(=\)\(-2 \beta_{5} - 2 \beta_{3} - 9 \beta_{2}\)
\(\nu^{7}\)\(=\)\((\)\(-26 \beta_{7} - 29 \beta_{6} + 29 \beta_{4}\)\()/4\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−0.437016 0.437016i
−1.14412 1.14412i
0.437016 + 0.437016i
1.14412 + 1.14412i
−0.437016 + 0.437016i
−1.14412 + 1.14412i
0.437016 0.437016i
1.14412 1.14412i
0 0 0 −1.58114 + 1.58114i 0 −1.23607 1.23607i 0 0 0
17.2 0 0 0 −1.58114 + 1.58114i 0 3.23607 + 3.23607i 0 0 0
17.3 0 0 0 1.58114 1.58114i 0 −1.23607 1.23607i 0 0 0
17.4 0 0 0 1.58114 1.58114i 0 3.23607 + 3.23607i 0 0 0
593.1 0 0 0 −1.58114 1.58114i 0 −1.23607 + 1.23607i 0 0 0
593.2 0 0 0 −1.58114 1.58114i 0 3.23607 3.23607i 0 0 0
593.3 0 0 0 1.58114 + 1.58114i 0 −1.23607 + 1.23607i 0 0 0
593.4 0 0 0 1.58114 + 1.58114i 0 3.23607 3.23607i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 593.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.w.e 8
3.b odd 2 1 inner 720.2.w.e 8
4.b odd 2 1 360.2.s.b 8
5.b even 2 1 3600.2.w.j 8
5.c odd 4 1 inner 720.2.w.e 8
5.c odd 4 1 3600.2.w.j 8
8.b even 2 1 2880.2.w.o 8
8.d odd 2 1 2880.2.w.m 8
12.b even 2 1 360.2.s.b 8
15.d odd 2 1 3600.2.w.j 8
15.e even 4 1 inner 720.2.w.e 8
15.e even 4 1 3600.2.w.j 8
20.d odd 2 1 1800.2.s.e 8
20.e even 4 1 360.2.s.b 8
20.e even 4 1 1800.2.s.e 8
24.f even 2 1 2880.2.w.m 8
24.h odd 2 1 2880.2.w.o 8
40.i odd 4 1 2880.2.w.o 8
40.k even 4 1 2880.2.w.m 8
60.h even 2 1 1800.2.s.e 8
60.l odd 4 1 360.2.s.b 8
60.l odd 4 1 1800.2.s.e 8
120.q odd 4 1 2880.2.w.m 8
120.w even 4 1 2880.2.w.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.s.b 8 4.b odd 2 1
360.2.s.b 8 12.b even 2 1
360.2.s.b 8 20.e even 4 1
360.2.s.b 8 60.l odd 4 1
720.2.w.e 8 1.a even 1 1 trivial
720.2.w.e 8 3.b odd 2 1 inner
720.2.w.e 8 5.c odd 4 1 inner
720.2.w.e 8 15.e even 4 1 inner
1800.2.s.e 8 20.d odd 2 1
1800.2.s.e 8 20.e even 4 1
1800.2.s.e 8 60.h even 2 1
1800.2.s.e 8 60.l odd 4 1
2880.2.w.m 8 8.d odd 2 1
2880.2.w.m 8 24.f even 2 1
2880.2.w.m 8 40.k even 4 1
2880.2.w.m 8 120.q odd 4 1
2880.2.w.o 8 8.b even 2 1
2880.2.w.o 8 24.h odd 2 1
2880.2.w.o 8 40.i odd 4 1
2880.2.w.o 8 120.w even 4 1
3600.2.w.j 8 5.b even 2 1
3600.2.w.j 8 5.c odd 4 1
3600.2.w.j 8 15.d odd 2 1
3600.2.w.j 8 15.e even 4 1

Hecke kernels

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

\( T_{7}^{4} - 4 T_{7}^{3} + 8 T_{7}^{2} + 32 T_{7} + 64 \)
\( T_{13}^{4} + 8 T_{13}^{3} + 32 T_{13}^{2} - 16 T_{13} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 25 + T^{4} )^{2} \)
$7$ \( ( 64 + 32 T + 8 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$11$ \( ( 64 + 24 T^{2} + T^{4} )^{2} \)
$13$ \( ( 4 - 16 T + 32 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$17$ \( 65536 + 1792 T^{4} + T^{8} \)
$19$ \( ( 256 + 48 T^{2} + T^{4} )^{2} \)
$23$ \( ( 256 + T^{4} )^{2} \)
$29$ \( ( 4 - 36 T^{2} + T^{4} )^{2} \)
$31$ \( ( 16 + 12 T + T^{2} )^{4} \)
$37$ \( ( 100 + T^{4} )^{2} \)
$41$ \( ( 50 + T^{2} )^{4} \)
$43$ \( ( 1024 + 256 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$47$ \( 65536 + 1792 T^{4} + T^{8} \)
$53$ \( T^{8} \)
$59$ \( ( 64 - 56 T^{2} + T^{4} )^{2} \)
$61$ \( ( 16 - 12 T + T^{2} )^{4} \)
$67$ \( ( 1024 - 768 T + 288 T^{2} - 24 T^{3} + T^{4} )^{2} \)
$71$ \( ( 160 + T^{2} )^{4} \)
$73$ \( ( 100 + 200 T + 200 T^{2} + 20 T^{3} + T^{4} )^{2} \)
$79$ \( ( 6400 + 240 T^{2} + T^{4} )^{2} \)
$83$ \( 65536 + 12032 T^{4} + T^{8} \)
$89$ \( ( 100 - 180 T^{2} + T^{4} )^{2} \)
$97$ \( ( 2 - 2 T + T^{2} )^{4} \)
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