Properties

Label 882.3.s.i
Level $882$
Weight $3$
Character orbit 882.s
Analytic conductor $24.033$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 882.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.0327593166\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12745506816.5
Defining polynomial: \(x^{8} - 8 x^{6} + 55 x^{4} - 72 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -\beta_{2} - \beta_{6} ) q^{2} + 2 \beta_{1} q^{4} + ( -\beta_{2} - \beta_{6} + \beta_{7} ) q^{5} -2 \beta_{6} q^{8} +O(q^{10})\) \( q + ( -\beta_{2} - \beta_{6} ) q^{2} + 2 \beta_{1} q^{4} + ( -\beta_{2} - \beta_{6} + \beta_{7} ) q^{5} -2 \beta_{6} q^{8} + ( 2 \beta_{1} - \beta_{5} ) q^{10} + ( -2 \beta_{2} + 2 \beta_{4} ) q^{11} + ( 8 - \beta_{3} ) q^{13} + ( -4 + 4 \beta_{1} ) q^{16} + ( -13 \beta_{2} + \beta_{4} ) q^{17} + ( 20 - 20 \beta_{1} ) q^{19} + ( -2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{20} + ( 4 + 2 \beta_{3} ) q^{22} + ( 2 \beta_{2} + 2 \beta_{6} - 2 \beta_{7} ) q^{23} + ( 33 \beta_{1} - 2 \beta_{5} ) q^{25} + ( -8 \beta_{2} - 8 \beta_{6} - 2 \beta_{7} ) q^{26} + ( 2 \beta_{4} - 19 \beta_{6} - 2 \beta_{7} ) q^{29} + ( 4 \beta_{1} + 2 \beta_{5} ) q^{31} + 4 \beta_{2} q^{32} + ( 26 + \beta_{3} ) q^{34} + ( -38 + 38 \beta_{1} ) q^{37} -20 \beta_{2} q^{38} + ( -4 + 4 \beta_{1} - 2 \beta_{3} - 2 \beta_{5} ) q^{40} + ( 3 \beta_{4} - 27 \beta_{6} - 3 \beta_{7} ) q^{41} + ( 20 - 6 \beta_{3} ) q^{43} + ( -4 \beta_{2} - 4 \beta_{6} + 4 \beta_{7} ) q^{44} + ( -4 \beta_{1} + 2 \beta_{5} ) q^{46} + ( 12 \beta_{2} + 12 \beta_{6} ) q^{47} + ( -4 \beta_{4} - 33 \beta_{6} + 4 \beta_{7} ) q^{50} + ( 16 \beta_{1} + 2 \beta_{5} ) q^{52} + ( -3 \beta_{2} - 12 \beta_{4} ) q^{53} + ( 116 + 4 \beta_{3} ) q^{55} + ( -38 + 38 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} ) q^{58} + ( -20 \beta_{2} - 4 \beta_{4} ) q^{59} + ( 58 - 58 \beta_{1} + 4 \beta_{3} + 4 \beta_{5} ) q^{61} + ( 4 \beta_{4} - 4 \beta_{6} - 4 \beta_{7} ) q^{62} -8 q^{64} + ( 48 \beta_{2} + 48 \beta_{6} + 6 \beta_{7} ) q^{65} + ( 48 \beta_{1} + 8 \beta_{5} ) q^{67} + ( -26 \beta_{2} - 26 \beta_{6} + 2 \beta_{7} ) q^{68} + ( -2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{71} + ( -24 \beta_{1} - 5 \beta_{5} ) q^{73} + 38 \beta_{2} q^{74} + 40 q^{76} + ( -76 + 76 \beta_{1} - 4 \beta_{3} - 4 \beta_{5} ) q^{79} + ( 4 \beta_{2} - 4 \beta_{4} ) q^{80} + ( -54 + 54 \beta_{1} + 3 \beta_{3} + 3 \beta_{5} ) q^{82} + ( -4 \beta_{4} - 64 \beta_{6} + 4 \beta_{7} ) q^{83} + ( 82 + 14 \beta_{3} ) q^{85} + ( -20 \beta_{2} - 20 \beta_{6} - 12 \beta_{7} ) q^{86} + ( 8 \beta_{1} - 4 \beta_{5} ) q^{88} + ( 51 \beta_{2} + 51 \beta_{6} - 9 \beta_{7} ) q^{89} + ( 4 \beta_{4} + 4 \beta_{6} - 4 \beta_{7} ) q^{92} -24 \beta_{1} q^{94} + ( -20 \beta_{2} + 20 \beta_{4} ) q^{95} + ( 72 - 11 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{4} + O(q^{10}) \) \( 8q + 8q^{4} + 8q^{10} + 64q^{13} - 16q^{16} + 80q^{19} + 32q^{22} + 132q^{25} + 16q^{31} + 208q^{34} - 152q^{37} - 16q^{40} + 160q^{43} - 16q^{46} + 64q^{52} + 928q^{55} - 152q^{58} + 232q^{61} - 64q^{64} + 192q^{67} - 96q^{73} + 320q^{76} - 304q^{79} - 216q^{82} + 656q^{85} + 32q^{88} - 96q^{94} + 576q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 8 x^{6} + 55 x^{4} - 72 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -8 \nu^{6} + 55 \nu^{4} - 440 \nu^{2} + 576 \)\()/495\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + 203 \nu \)\()/165\)
\(\beta_{3}\)\(=\)\((\)\( -4 \nu^{6} - 592 \)\()/55\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{7} + 1066 \nu \)\()/165\)
\(\beta_{5}\)\(=\)\((\)\( -92 \nu^{6} + 880 \nu^{4} - 5060 \nu^{2} + 6624 \)\()/495\)
\(\beta_{6}\)\(=\)\((\)\( -8 \nu^{7} + 55 \nu^{5} - 341 \nu^{3} + 81 \nu \)\()/297\)
\(\beta_{7}\)\(=\)\((\)\( -158 \nu^{7} + 1210 \nu^{5} - 8690 \nu^{3} + 11376 \nu \)\()/1485\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} - 2 \beta_{2}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + \beta_{3} - 16 \beta_{1} + 16\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{7} + 22 \beta_{6} + 5 \beta_{4}\)\()/4\)
\(\nu^{4}\)\(=\)\(2 \beta_{5} - 23 \beta_{1}\)
\(\nu^{5}\)\(=\)\((\)\(-31 \beta_{7} + 158 \beta_{6} + 158 \beta_{2}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(-55 \beta_{3} - 592\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-203 \beta_{4} + 1066 \beta_{2}\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-1 + \beta_{1}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
557.1
−2.23256 1.28897i
1.00781 + 0.581861i
−1.00781 0.581861i
2.23256 + 1.28897i
−2.23256 + 1.28897i
1.00781 0.581861i
−1.00781 + 0.581861i
2.23256 1.28897i
−1.22474 + 0.707107i 0 1.00000 1.73205i −7.70549 + 4.44876i 0 0 2.82843i 0 6.29150 10.8972i
557.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 5.25600 3.03455i 0 0 2.82843i 0 −4.29150 + 7.43310i
557.3 1.22474 0.707107i 0 1.00000 1.73205i −5.25600 + 3.03455i 0 0 2.82843i 0 −4.29150 + 7.43310i
557.4 1.22474 0.707107i 0 1.00000 1.73205i 7.70549 4.44876i 0 0 2.82843i 0 6.29150 10.8972i
863.1 −1.22474 0.707107i 0 1.00000 + 1.73205i −7.70549 4.44876i 0 0 2.82843i 0 6.29150 + 10.8972i
863.2 −1.22474 0.707107i 0 1.00000 + 1.73205i 5.25600 + 3.03455i 0 0 2.82843i 0 −4.29150 7.43310i
863.3 1.22474 + 0.707107i 0 1.00000 + 1.73205i −5.25600 3.03455i 0 0 2.82843i 0 −4.29150 7.43310i
863.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 7.70549 + 4.44876i 0 0 2.82843i 0 6.29150 + 10.8972i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 863.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.3.s.i 8
3.b odd 2 1 inner 882.3.s.i 8
7.b odd 2 1 882.3.s.e 8
7.c even 3 1 882.3.b.f 4
7.c even 3 1 inner 882.3.s.i 8
7.d odd 6 1 126.3.b.a 4
7.d odd 6 1 882.3.s.e 8
21.c even 2 1 882.3.s.e 8
21.g even 6 1 126.3.b.a 4
21.g even 6 1 882.3.s.e 8
21.h odd 6 1 882.3.b.f 4
21.h odd 6 1 inner 882.3.s.i 8
28.f even 6 1 1008.3.d.a 4
35.i odd 6 1 3150.3.e.e 4
35.k even 12 2 3150.3.c.b 8
56.j odd 6 1 4032.3.d.i 4
56.m even 6 1 4032.3.d.j 4
63.i even 6 1 1134.3.q.c 8
63.k odd 6 1 1134.3.q.c 8
63.s even 6 1 1134.3.q.c 8
63.t odd 6 1 1134.3.q.c 8
84.j odd 6 1 1008.3.d.a 4
105.p even 6 1 3150.3.e.e 4
105.w odd 12 2 3150.3.c.b 8
168.ba even 6 1 4032.3.d.i 4
168.be odd 6 1 4032.3.d.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.b.a 4 7.d odd 6 1
126.3.b.a 4 21.g even 6 1
882.3.b.f 4 7.c even 3 1
882.3.b.f 4 21.h odd 6 1
882.3.s.e 8 7.b odd 2 1
882.3.s.e 8 7.d odd 6 1
882.3.s.e 8 21.c even 2 1
882.3.s.e 8 21.g even 6 1
882.3.s.i 8 1.a even 1 1 trivial
882.3.s.i 8 3.b odd 2 1 inner
882.3.s.i 8 7.c even 3 1 inner
882.3.s.i 8 21.h odd 6 1 inner
1008.3.d.a 4 28.f even 6 1
1008.3.d.a 4 84.j odd 6 1
1134.3.q.c 8 63.i even 6 1
1134.3.q.c 8 63.k odd 6 1
1134.3.q.c 8 63.s even 6 1
1134.3.q.c 8 63.t odd 6 1
3150.3.c.b 8 35.k even 12 2
3150.3.c.b 8 105.w odd 12 2
3150.3.e.e 4 35.i odd 6 1
3150.3.e.e 4 105.p even 6 1
4032.3.d.i 4 56.j odd 6 1
4032.3.d.i 4 168.ba even 6 1
4032.3.d.j 4 56.m even 6 1
4032.3.d.j 4 168.be odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{8} - 116 T_{5}^{6} + 10540 T_{5}^{4} - 338256 T_{5}^{2} + 8503056 \)
\( T_{11}^{8} - 464 T_{11}^{6} + 168640 T_{11}^{4} - 21648384 T_{11}^{2} + 2176782336 \)
\( T_{13}^{2} - 16 T_{13} - 48 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 - 2 T^{2} + T^{4} )^{2} \)
$3$ \( T^{8} \)
$5$ \( 8503056 - 338256 T^{2} + 10540 T^{4} - 116 T^{6} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( 2176782336 - 21648384 T^{2} + 168640 T^{4} - 464 T^{6} + T^{8} \)
$13$ \( ( -48 - 16 T + T^{2} )^{4} \)
$17$ \( 6324066576 - 62664912 T^{2} + 541420 T^{4} - 788 T^{6} + T^{8} \)
$19$ \( ( 400 - 20 T + T^{2} )^{4} \)
$23$ \( 2176782336 - 21648384 T^{2} + 168640 T^{4} - 464 T^{6} + T^{8} \)
$29$ \( ( 248004 + 1892 T^{2} + T^{4} )^{2} \)
$31$ \( ( 186624 + 3456 T + 496 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$37$ \( ( 1444 + 38 T + T^{2} )^{4} \)
$41$ \( ( 910116 + 3924 T^{2} + T^{4} )^{2} \)
$43$ \( ( -3632 - 40 T + T^{2} )^{4} \)
$47$ \( ( 82944 - 288 T^{2} + T^{4} )^{2} \)
$53$ \( 4191023663229456 - 1046426907024 T^{2} + 196536780 T^{4} - 16164 T^{6} + T^{8} \)
$59$ \( 84934656 - 31260672 T^{2} + 11496448 T^{4} - 3392 T^{6} + T^{8} \)
$61$ \( ( 2471184 - 182352 T + 11884 T^{2} - 116 T^{3} + T^{4} )^{2} \)
$67$ \( ( 23658496 + 466944 T + 14080 T^{2} - 96 T^{3} + T^{4} )^{2} \)
$71$ \( ( 46656 + 464 T^{2} + T^{4} )^{2} \)
$73$ \( ( 4946176 - 106752 T + 4528 T^{2} + 48 T^{3} + T^{4} )^{2} \)
$79$ \( ( 15872256 + 605568 T + 19120 T^{2} + 152 T^{3} + T^{4} )^{2} \)
$83$ \( ( 53231616 + 18176 T^{2} + T^{4} )^{2} \)
$89$ \( 196741925136 - 8638696656 T^{2} + 378871020 T^{4} - 19476 T^{6} + T^{8} \)
$97$ \( ( -8368 - 144 T + T^{2} )^{4} \)
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