Properties

Label 882.4.g.be
Level $882$
Weight $4$
Character orbit 882.g
Analytic conductor $52.040$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 882.g (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(52.0396846251\)
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: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 294)
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.

\(f(q)\) \(=\) \( q -2 \beta_{2} q^{2} + ( -4 - 4 \beta_{2} ) q^{4} + ( -\beta_{1} + 6 \beta_{2} - \beta_{3} ) q^{5} -8 q^{8} +O(q^{10})\) \( q -2 \beta_{2} q^{2} + ( -4 - 4 \beta_{2} ) q^{4} + ( -\beta_{1} + 6 \beta_{2} - \beta_{3} ) q^{5} -8 q^{8} + ( 12 - 2 \beta_{1} + 12 \beta_{2} ) q^{10} + ( 2 - 6 \beta_{1} + 2 \beta_{2} ) q^{11} + ( -24 + 15 \beta_{3} ) q^{13} + 16 \beta_{2} q^{16} + ( -66 - 11 \beta_{1} - 66 \beta_{2} ) q^{17} + ( 46 \beta_{1} - 60 \beta_{2} + 46 \beta_{3} ) q^{19} + ( 24 + 4 \beta_{3} ) q^{20} + ( 4 + 12 \beta_{3} ) q^{22} + ( -102 \beta_{1} + 38 \beta_{2} - 102 \beta_{3} ) q^{23} + ( 87 + 12 \beta_{1} + 87 \beta_{2} ) q^{25} + ( 30 \beta_{1} + 48 \beta_{2} + 30 \beta_{3} ) q^{26} + ( 56 - 150 \beta_{3} ) q^{29} + ( 216 + 54 \beta_{1} + 216 \beta_{2} ) q^{31} + ( 32 + 32 \beta_{2} ) q^{32} + ( -132 + 22 \beta_{3} ) q^{34} + ( 180 \beta_{1} - 140 \beta_{2} + 180 \beta_{3} ) q^{37} + ( -120 + 92 \beta_{1} - 120 \beta_{2} ) q^{38} + ( 8 \beta_{1} - 48 \beta_{2} + 8 \beta_{3} ) q^{40} + ( 18 + 127 \beta_{3} ) q^{41} + ( -64 + 288 \beta_{3} ) q^{43} + ( 24 \beta_{1} - 8 \beta_{2} + 24 \beta_{3} ) q^{44} + ( 76 - 204 \beta_{1} + 76 \beta_{2} ) q^{46} + ( 338 \beta_{1} - 132 \beta_{2} + 338 \beta_{3} ) q^{47} + ( 174 - 24 \beta_{3} ) q^{50} + ( 96 + 60 \beta_{1} + 96 \beta_{2} ) q^{52} + ( 134 + 192 \beta_{1} + 134 \beta_{2} ) q^{53} + ( -24 - 38 \beta_{3} ) q^{55} + ( -300 \beta_{1} - 112 \beta_{2} - 300 \beta_{3} ) q^{58} + ( -168 + 298 \beta_{1} - 168 \beta_{2} ) q^{59} + ( 353 \beta_{1} + 252 \beta_{2} + 353 \beta_{3} ) q^{61} + ( 432 - 108 \beta_{3} ) q^{62} + 64 q^{64} + ( -66 \beta_{1} - 114 \beta_{2} - 66 \beta_{3} ) q^{65} + ( 192 - 144 \beta_{1} + 192 \beta_{2} ) q^{67} + ( 44 \beta_{1} + 264 \beta_{2} + 44 \beta_{3} ) q^{68} + ( 198 - 342 \beta_{3} ) q^{71} + ( -156 + 595 \beta_{1} - 156 \beta_{2} ) q^{73} + ( -280 + 360 \beta_{1} - 280 \beta_{2} ) q^{74} + ( -240 - 184 \beta_{3} ) q^{76} + ( 300 \beta_{1} - 424 \beta_{2} + 300 \beta_{3} ) q^{79} + ( -96 + 16 \beta_{1} - 96 \beta_{2} ) q^{80} + ( 254 \beta_{1} - 36 \beta_{2} + 254 \beta_{3} ) q^{82} + ( -324 + 80 \beta_{3} ) q^{83} + 374 q^{85} + ( 576 \beta_{1} + 128 \beta_{2} + 576 \beta_{3} ) q^{86} + ( -16 + 48 \beta_{1} - 16 \beta_{2} ) q^{88} + ( 175 \beta_{1} - 306 \beta_{2} + 175 \beta_{3} ) q^{89} + ( 152 + 408 \beta_{3} ) q^{92} + ( -264 + 676 \beta_{1} - 264 \beta_{2} ) q^{94} + ( 452 - 336 \beta_{1} + 452 \beta_{2} ) q^{95} + ( 1092 - 133 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - 8q^{4} - 12q^{5} - 32q^{8} + O(q^{10}) \) \( 4q + 4q^{2} - 8q^{4} - 12q^{5} - 32q^{8} + 24q^{10} + 4q^{11} - 96q^{13} - 32q^{16} - 132q^{17} + 120q^{19} + 96q^{20} + 16q^{22} - 76q^{23} + 174q^{25} - 96q^{26} + 224q^{29} + 432q^{31} + 64q^{32} - 528q^{34} + 280q^{37} - 240q^{38} + 96q^{40} + 72q^{41} - 256q^{43} + 16q^{44} + 152q^{46} + 264q^{47} + 696q^{50} + 192q^{52} + 268q^{53} - 96q^{55} + 224q^{58} - 336q^{59} - 504q^{61} + 1728q^{62} + 256q^{64} + 228q^{65} + 384q^{67} - 528q^{68} + 792q^{71} - 312q^{73} - 560q^{74} - 960q^{76} + 848q^{79} - 192q^{80} + 72q^{82} - 1296q^{83} + 1496q^{85} - 256q^{86} - 32q^{88} + 612q^{89} + 608q^{92} - 528q^{94} + 904q^{95} + 4368q^{97} + O(q^{100}) \)

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 \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
1.00000 + 1.73205i 0 −2.00000 + 3.46410i −3.70711 6.42090i 0 0 −8.00000 0 7.41421 12.8418i
361.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −2.29289 3.97141i 0 0 −8.00000 0 4.58579 7.94282i
667.1 1.00000 1.73205i 0 −2.00000 3.46410i −3.70711 + 6.42090i 0 0 −8.00000 0 7.41421 + 12.8418i
667.2 1.00000 1.73205i 0 −2.00000 3.46410i −2.29289 + 3.97141i 0 0 −8.00000 0 4.58579 + 7.94282i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.4.g.be 4
3.b odd 2 1 294.4.e.m 4
7.b odd 2 1 882.4.g.bk 4
7.c even 3 1 882.4.a.bb 2
7.c even 3 1 inner 882.4.g.be 4
7.d odd 6 1 882.4.a.t 2
7.d odd 6 1 882.4.g.bk 4
21.c even 2 1 294.4.e.k 4
21.g even 6 1 294.4.a.o yes 2
21.g even 6 1 294.4.e.k 4
21.h odd 6 1 294.4.a.l 2
21.h odd 6 1 294.4.e.m 4
84.j odd 6 1 2352.4.a.bu 2
84.n even 6 1 2352.4.a.bw 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.4.a.l 2 21.h odd 6 1
294.4.a.o yes 2 21.g even 6 1
294.4.e.k 4 21.c even 2 1
294.4.e.k 4 21.g even 6 1
294.4.e.m 4 3.b odd 2 1
294.4.e.m 4 21.h odd 6 1
882.4.a.t 2 7.d odd 6 1
882.4.a.bb 2 7.c even 3 1
882.4.g.be 4 1.a even 1 1 trivial
882.4.g.be 4 7.c even 3 1 inner
882.4.g.bk 4 7.b odd 2 1
882.4.g.bk 4 7.d odd 6 1
2352.4.a.bu 2 84.j odd 6 1
2352.4.a.bw 2 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{4} + 12 T_{5}^{3} + 110 T_{5}^{2} + 408 T_{5} + 1156 \)
\( T_{11}^{4} - 4 T_{11}^{3} + 84 T_{11}^{2} + 272 T_{11} + 4624 \)
\( T_{13}^{2} + 48 T_{13} + 126 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 - 2 T + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 1156 + 408 T + 110 T^{2} + 12 T^{3} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 4624 + 272 T + 84 T^{2} - 4 T^{3} + T^{4} \)
$13$ \( ( 126 + 48 T + T^{2} )^{2} \)
$17$ \( 16924996 + 543048 T + 13310 T^{2} + 132 T^{3} + T^{4} \)
$19$ \( 399424 + 75840 T + 15032 T^{2} - 120 T^{3} + T^{4} \)
$23$ \( 374964496 - 1471664 T + 25140 T^{2} + 76 T^{3} + T^{4} \)
$29$ \( ( -41864 - 112 T + T^{2} )^{2} \)
$31$ \( 1666598976 - 17635968 T + 145800 T^{2} - 432 T^{3} + T^{4} \)
$37$ \( 2043040000 + 12656000 T + 123600 T^{2} - 280 T^{3} + T^{4} \)
$41$ \( ( -31934 - 36 T + T^{2} )^{2} \)
$43$ \( ( -161792 + 128 T + T^{2} )^{2} \)
$47$ \( 44548012096 + 55720896 T + 280760 T^{2} - 264 T^{3} + T^{4} \)
$53$ \( 3110515984 + 14946896 T + 127596 T^{2} - 268 T^{3} + T^{4} \)
$59$ \( 22315579456 - 50193024 T + 262280 T^{2} + 336 T^{3} + T^{4} \)
$61$ \( 34489689796 - 93599856 T + 439730 T^{2} + 504 T^{3} + T^{4} \)
$67$ \( 21233664 + 1769472 T + 152064 T^{2} - 384 T^{3} + T^{4} \)
$71$ \( ( -194724 - 396 T + T^{2} )^{2} \)
$73$ \( 467464833796 - 213318768 T + 781058 T^{2} + 312 T^{3} + T^{4} \)
$79$ \( 50176 + 189952 T + 719328 T^{2} - 848 T^{3} + T^{4} \)
$83$ \( ( 92176 + 648 T + T^{2} )^{2} \)
$89$ \( 1048852996 - 19820232 T + 342158 T^{2} - 612 T^{3} + T^{4} \)
$97$ \( ( 1157086 - 2184 T + T^{2} )^{2} \)
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