Properties

Label 882.4.g.c
Level $882$
Weight $4$
Character orbit 882.g
Analytic conductor $52.040$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} - 8 \zeta_{6} q^{5} + 8 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} - 8 \zeta_{6} q^{5} + 8 q^{8} + (16 \zeta_{6} - 16) q^{10} + ( - 40 \zeta_{6} + 40) q^{11} - 4 q^{13} - 16 \zeta_{6} q^{16} + ( - 84 \zeta_{6} + 84) q^{17} + 148 \zeta_{6} q^{19} + 32 q^{20} - 80 q^{22} + 84 \zeta_{6} q^{23} + ( - 61 \zeta_{6} + 61) q^{25} + 8 \zeta_{6} q^{26} - 58 q^{29} + (136 \zeta_{6} - 136) q^{31} + (32 \zeta_{6} - 32) q^{32} - 168 q^{34} + 222 \zeta_{6} q^{37} + ( - 296 \zeta_{6} + 296) q^{38} - 64 \zeta_{6} q^{40} + 420 q^{41} - 164 q^{43} + 160 \zeta_{6} q^{44} + ( - 168 \zeta_{6} + 168) q^{46} - 488 \zeta_{6} q^{47} - 122 q^{50} + ( - 16 \zeta_{6} + 16) q^{52} + ( - 478 \zeta_{6} + 478) q^{53} - 320 q^{55} + 116 \zeta_{6} q^{58} + (548 \zeta_{6} - 548) q^{59} + 692 \zeta_{6} q^{61} + 272 q^{62} + 64 q^{64} + 32 \zeta_{6} q^{65} + ( - 908 \zeta_{6} + 908) q^{67} + 336 \zeta_{6} q^{68} + 524 q^{71} + ( - 440 \zeta_{6} + 440) q^{73} + ( - 444 \zeta_{6} + 444) q^{74} - 592 q^{76} - 1216 \zeta_{6} q^{79} + (128 \zeta_{6} - 128) q^{80} - 840 \zeta_{6} q^{82} - 684 q^{83} - 672 q^{85} + 328 \zeta_{6} q^{86} + ( - 320 \zeta_{6} + 320) q^{88} - 604 \zeta_{6} q^{89} - 336 q^{92} + (976 \zeta_{6} - 976) q^{94} + ( - 1184 \zeta_{6} + 1184) q^{95} + 832 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{4} - 8 q^{5} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 4 q^{4} - 8 q^{5} + 16 q^{8} - 16 q^{10} + 40 q^{11} - 8 q^{13} - 16 q^{16} + 84 q^{17} + 148 q^{19} + 64 q^{20} - 160 q^{22} + 84 q^{23} + 61 q^{25} + 8 q^{26} - 116 q^{29} - 136 q^{31} - 32 q^{32} - 336 q^{34} + 222 q^{37} + 296 q^{38} - 64 q^{40} + 840 q^{41} - 328 q^{43} + 160 q^{44} + 168 q^{46} - 488 q^{47} - 244 q^{50} + 16 q^{52} + 478 q^{53} - 640 q^{55} + 116 q^{58} - 548 q^{59} + 692 q^{61} + 544 q^{62} + 128 q^{64} + 32 q^{65} + 908 q^{67} + 336 q^{68} + 1048 q^{71} + 440 q^{73} + 444 q^{74} - 1184 q^{76} - 1216 q^{79} - 128 q^{80} - 840 q^{82} - 1368 q^{83} - 1344 q^{85} + 328 q^{86} + 320 q^{88} - 604 q^{89} - 672 q^{92} - 976 q^{94} + 1184 q^{95} + 1664 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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
0.500000 + 0.866025i
0.500000 0.866025i
−1.00000 1.73205i 0 −2.00000 + 3.46410i −4.00000 6.92820i 0 0 8.00000 0 −8.00000 + 13.8564i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i −4.00000 + 6.92820i 0 0 8.00000 0 −8.00000 13.8564i
\(n\): 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.c 2
3.b odd 2 1 294.4.e.j 2
7.b odd 2 1 882.4.g.j 2
7.c even 3 1 882.4.a.q 1
7.c even 3 1 inner 882.4.g.c 2
7.d odd 6 1 882.4.a.j 1
7.d odd 6 1 882.4.g.j 2
21.c even 2 1 294.4.e.f 2
21.g even 6 1 294.4.a.f yes 1
21.g even 6 1 294.4.e.f 2
21.h odd 6 1 294.4.a.b 1
21.h odd 6 1 294.4.e.j 2
84.j odd 6 1 2352.4.a.m 1
84.n even 6 1 2352.4.a.z 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.4.a.b 1 21.h odd 6 1
294.4.a.f yes 1 21.g even 6 1
294.4.e.f 2 21.c even 2 1
294.4.e.f 2 21.g even 6 1
294.4.e.j 2 3.b odd 2 1
294.4.e.j 2 21.h odd 6 1
882.4.a.j 1 7.d odd 6 1
882.4.a.q 1 7.c even 3 1
882.4.g.c 2 1.a even 1 1 trivial
882.4.g.c 2 7.c even 3 1 inner
882.4.g.j 2 7.b odd 2 1
882.4.g.j 2 7.d odd 6 1
2352.4.a.m 1 84.j odd 6 1
2352.4.a.z 1 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 \(S_{4}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{2} + 8T_{5} + 64 \) Copy content Toggle raw display
\( T_{11}^{2} - 40T_{11} + 1600 \) Copy content Toggle raw display
\( T_{13} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 40T + 1600 \) Copy content Toggle raw display
$13$ \( (T + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 84T + 7056 \) Copy content Toggle raw display
$19$ \( T^{2} - 148T + 21904 \) Copy content Toggle raw display
$23$ \( T^{2} - 84T + 7056 \) Copy content Toggle raw display
$29$ \( (T + 58)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 136T + 18496 \) Copy content Toggle raw display
$37$ \( T^{2} - 222T + 49284 \) Copy content Toggle raw display
$41$ \( (T - 420)^{2} \) Copy content Toggle raw display
$43$ \( (T + 164)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 488T + 238144 \) Copy content Toggle raw display
$53$ \( T^{2} - 478T + 228484 \) Copy content Toggle raw display
$59$ \( T^{2} + 548T + 300304 \) Copy content Toggle raw display
$61$ \( T^{2} - 692T + 478864 \) Copy content Toggle raw display
$67$ \( T^{2} - 908T + 824464 \) Copy content Toggle raw display
$71$ \( (T - 524)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 440T + 193600 \) Copy content Toggle raw display
$79$ \( T^{2} + 1216 T + 1478656 \) Copy content Toggle raw display
$83$ \( (T + 684)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 604T + 364816 \) Copy content Toggle raw display
$97$ \( (T - 832)^{2} \) Copy content Toggle raw display
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