Properties

Label 882.4.g.bf
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: \(\Q(\sqrt{-3}, \sqrt{1345})\)
Defining polynomial: \( x^{4} - x^{3} + 337x^{2} + 336x + 112896 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
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.

\(f(q)\) \(=\) \( q + 2 \beta_{2} q^{2} + (4 \beta_{2} - 4) q^{4} + ( - 2 \beta_{2} - \beta_1) q^{5} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_{2} q^{2} + (4 \beta_{2} - 4) q^{4} + ( - 2 \beta_{2} - \beta_1) q^{5} - 8 q^{8} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 6) q^{10} + (\beta_{3} - 34 \beta_{2} + \beta_1 + 33) q^{11} + ( - \beta_{3} - 20) q^{13} - 16 \beta_{2} q^{16} + ( - 4 \beta_{3} - 44 \beta_{2} - 4 \beta_1 + 48) q^{17} + (23 \beta_{2} - 3 \beta_1) q^{19} + ( - 4 \beta_{3} + 12) q^{20} + (2 \beta_{3} + 66) q^{22} + (76 \beta_{2} - 4 \beta_1) q^{23} + (5 \beta_{3} + 215 \beta_{2} + 5 \beta_1 - 220) q^{25} + ( - 42 \beta_{2} + 2 \beta_1) q^{26} + ( - 11 \beta_{3} - 33) q^{29} + ( - 2 \beta_{3} + 261 \beta_{2} - 2 \beta_1 - 259) q^{31} + ( - 32 \beta_{2} + 32) q^{32} + ( - 8 \beta_{3} + 96) q^{34} + (\beta_{2} - 9 \beta_1) q^{37} + ( - 6 \beta_{3} + 46 \beta_{2} - 6 \beta_1 - 40) q^{38} + (16 \beta_{2} + 8 \beta_1) q^{40} + (6 \beta_{3} - 216) q^{41} + ( - 15 \beta_{3} - 46) q^{43} + (136 \beta_{2} - 4 \beta_1) q^{44} + ( - 8 \beta_{3} + 152 \beta_{2} - 8 \beta_1 - 144) q^{46} + ( - 282 \beta_{2} - 12 \beta_1) q^{47} + (10 \beta_{3} - 440) q^{50} + (4 \beta_{3} - 84 \beta_{2} + 4 \beta_1 + 80) q^{52} + (3 \beta_{3} + 120 \beta_{2} + 3 \beta_1 - 123) q^{53} + (31 \beta_{3} + 237) q^{55} + ( - 88 \beta_{2} + 22 \beta_1) q^{58} + (25 \beta_{3} - 16 \beta_{2} + 25 \beta_1 - 9) q^{59} + (114 \beta_{2} - 4 \beta_1) q^{61} + ( - 4 \beta_{3} - 518) q^{62} + 64 q^{64} + ( - 294 \beta_{2} + 18 \beta_1) q^{65} + ( - 11 \beta_{3} + 349 \beta_{2} - 11 \beta_1 - 338) q^{67} + (176 \beta_{2} + 16 \beta_1) q^{68} + (20 \beta_{3} - 246) q^{71} + (35 \beta_{3} + 443 \beta_{2} + 35 \beta_1 - 478) q^{73} + ( - 18 \beta_{3} + 2 \beta_{2} - 18 \beta_1 + 16) q^{74} + ( - 12 \beta_{3} - 80) q^{76} + (267 \beta_{2} - 8 \beta_1) q^{79} + (16 \beta_{3} + 32 \beta_{2} + 16 \beta_1 - 48) q^{80} + ( - 420 \beta_{2} - 12 \beta_1) q^{82} + (25 \beta_{3} - 123) q^{83} + (56 \beta_{3} - 1488) q^{85} + ( - 122 \beta_{2} + 30 \beta_1) q^{86} + ( - 8 \beta_{3} + 272 \beta_{2} - 8 \beta_1 - 264) q^{88} + ( - 408 \beta_{2} + 42 \beta_1) q^{89} + ( - 16 \beta_{3} - 288) q^{92} + ( - 24 \beta_{3} - 564 \beta_{2} - 24 \beta_1 + 588) q^{94} + ( - 14 \beta_{3} + 962 \beta_{2} - 14 \beta_1 - 948) q^{95} + ( - 35 \beta_{3} - 959) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 8 q^{4} - 5 q^{5} - 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 8 q^{4} - 5 q^{5} - 32 q^{8} + 10 q^{10} + 67 q^{11} - 82 q^{13} - 32 q^{16} + 92 q^{17} + 43 q^{19} + 40 q^{20} + 268 q^{22} + 148 q^{23} - 435 q^{25} - 82 q^{26} - 154 q^{29} - 520 q^{31} + 64 q^{32} + 368 q^{34} - 7 q^{37} - 86 q^{38} + 40 q^{40} - 852 q^{41} - 214 q^{43} + 268 q^{44} - 296 q^{46} - 576 q^{47} - 1740 q^{50} + 164 q^{52} - 243 q^{53} + 1010 q^{55} - 154 q^{58} + 7 q^{59} + 224 q^{61} - 2080 q^{62} + 256 q^{64} - 570 q^{65} - 687 q^{67} + 368 q^{68} - 944 q^{71} - 921 q^{73} + 14 q^{74} - 344 q^{76} + 526 q^{79} - 80 q^{80} - 852 q^{82} - 442 q^{83} - 5840 q^{85} - 214 q^{86} - 536 q^{88} - 774 q^{89} - 1184 q^{92} + 1152 q^{94} - 1910 q^{95} - 3906 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 337x^{2} + 336x + 112896 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 337\nu^{2} - 337\nu + 112896 ) / 113232 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 673 ) / 337 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 336\beta_{2} + \beta _1 - 337 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 337\beta_{3} - 673 \) Copy content Toggle raw display

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
9.41856 + 16.3134i
−8.91856 15.4474i
9.41856 16.3134i
−8.91856 + 15.4474i
1.00000 + 1.73205i 0 −2.00000 + 3.46410i −10.4186 18.0455i 0 0 −8.00000 0 20.8371 36.0910i
361.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 7.91856 + 13.7153i 0 0 −8.00000 0 −15.8371 + 27.4307i
667.1 1.00000 1.73205i 0 −2.00000 3.46410i −10.4186 + 18.0455i 0 0 −8.00000 0 20.8371 + 36.0910i
667.2 1.00000 1.73205i 0 −2.00000 3.46410i 7.91856 13.7153i 0 0 −8.00000 0 −15.8371 27.4307i
\(n\): 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.bf 4
3.b odd 2 1 294.4.e.l 4
7.b odd 2 1 126.4.g.g 4
7.c even 3 1 882.4.a.z 2
7.c even 3 1 inner 882.4.g.bf 4
7.d odd 6 1 126.4.g.g 4
7.d odd 6 1 882.4.a.v 2
21.c even 2 1 42.4.e.c 4
21.g even 6 1 42.4.e.c 4
21.g even 6 1 294.4.a.n 2
21.h odd 6 1 294.4.a.m 2
21.h odd 6 1 294.4.e.l 4
84.h odd 2 1 336.4.q.j 4
84.j odd 6 1 336.4.q.j 4
84.j odd 6 1 2352.4.a.bq 2
84.n even 6 1 2352.4.a.ca 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.c 4 21.c even 2 1
42.4.e.c 4 21.g even 6 1
126.4.g.g 4 7.b odd 2 1
126.4.g.g 4 7.d odd 6 1
294.4.a.m 2 21.h odd 6 1
294.4.a.n 2 21.g even 6 1
294.4.e.l 4 3.b odd 2 1
294.4.e.l 4 21.h odd 6 1
336.4.q.j 4 84.h odd 2 1
336.4.q.j 4 84.j odd 6 1
882.4.a.v 2 7.d odd 6 1
882.4.a.z 2 7.c even 3 1
882.4.g.bf 4 1.a even 1 1 trivial
882.4.g.bf 4 7.c even 3 1 inner
2352.4.a.bq 2 84.j odd 6 1
2352.4.a.ca 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 \(S_{4}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{4} + 5T_{5}^{3} + 355T_{5}^{2} - 1650T_{5} + 108900 \) Copy content Toggle raw display
\( T_{11}^{4} - 67T_{11}^{3} + 3703T_{11}^{2} - 52662T_{11} + 617796 \) Copy content Toggle raw display
\( T_{13}^{2} + 41T_{13} + 84 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 5 T^{3} + 355 T^{2} + \cdots + 108900 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 67 T^{3} + 3703 T^{2} + \cdots + 617796 \) Copy content Toggle raw display
$13$ \( (T^{2} + 41 T + 84)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 92 T^{3} + 11728 T^{2} + \cdots + 10653696 \) Copy content Toggle raw display
$19$ \( T^{4} - 43 T^{3} + 4413 T^{2} + \cdots + 6574096 \) Copy content Toggle raw display
$23$ \( T^{4} - 148 T^{3} + 21808 T^{2} + \cdots + 9216 \) Copy content Toggle raw display
$29$ \( (T^{2} + 77 T - 39204)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 520 T^{3} + \cdots + 4389725025 \) Copy content Toggle raw display
$37$ \( T^{4} + 7 T^{3} + 27273 T^{2} + \cdots + 741146176 \) Copy content Toggle raw display
$41$ \( (T^{2} + 426 T + 33264)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 107 T - 72794)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 576 T^{3} + \cdots + 1191906576 \) Copy content Toggle raw display
$53$ \( T^{4} + 243 T^{3} + \cdots + 137733696 \) Copy content Toggle raw display
$59$ \( T^{4} - 7 T^{3} + \cdots + 44160500736 \) Copy content Toggle raw display
$61$ \( T^{4} - 224 T^{3} + \cdots + 51322896 \) Copy content Toggle raw display
$67$ \( T^{4} + 687 T^{3} + \cdots + 5976217636 \) Copy content Toggle raw display
$71$ \( (T^{2} + 472 T - 78804)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 921 T^{3} + \cdots + 39938423716 \) Copy content Toggle raw display
$79$ \( T^{4} - 526 T^{3} + \cdots + 2270427201 \) Copy content Toggle raw display
$83$ \( (T^{2} + 221 T - 197946)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 774 T^{3} + \cdots + 196582277376 \) Copy content Toggle raw display
$97$ \( (T^{2} + 1953 T + 541646)^{2} \) Copy content Toggle raw display
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