Properties

Label 882.4.g.l
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: \(\Z[a_1, \ldots, a_{5}]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} + 15 \zeta_{6} q^{5} + 8 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} + 15 \zeta_{6} q^{5} + 8 q^{8} + ( - 30 \zeta_{6} + 30) q^{10} + (9 \zeta_{6} - 9) q^{11} + 88 q^{13} - 16 \zeta_{6} q^{16} + ( - 84 \zeta_{6} + 84) q^{17} + 104 \zeta_{6} q^{19} - 60 q^{20} + 18 q^{22} - 84 \zeta_{6} q^{23} + (100 \zeta_{6} - 100) q^{25} - 176 \zeta_{6} q^{26} - 51 q^{29} + ( - 185 \zeta_{6} + 185) q^{31} + (32 \zeta_{6} - 32) q^{32} - 168 q^{34} - 44 \zeta_{6} q^{37} + ( - 208 \zeta_{6} + 208) q^{38} + 120 \zeta_{6} q^{40} - 168 q^{41} + 326 q^{43} - 36 \zeta_{6} q^{44} + (168 \zeta_{6} - 168) q^{46} + 138 \zeta_{6} q^{47} + 200 q^{50} + (352 \zeta_{6} - 352) q^{52} + ( - 639 \zeta_{6} + 639) q^{53} - 135 q^{55} + 102 \zeta_{6} q^{58} + (159 \zeta_{6} - 159) q^{59} + 722 \zeta_{6} q^{61} - 370 q^{62} + 64 q^{64} + 1320 \zeta_{6} q^{65} + ( - 166 \zeta_{6} + 166) q^{67} + 336 \zeta_{6} q^{68} - 1086 q^{71} + ( - 218 \zeta_{6} + 218) q^{73} + (88 \zeta_{6} - 88) q^{74} - 416 q^{76} + 583 \zeta_{6} q^{79} + ( - 240 \zeta_{6} + 240) q^{80} + 336 \zeta_{6} q^{82} - 597 q^{83} + 1260 q^{85} - 652 \zeta_{6} q^{86} + (72 \zeta_{6} - 72) q^{88} + 1038 \zeta_{6} q^{89} + 336 q^{92} + ( - 276 \zeta_{6} + 276) q^{94} + (1560 \zeta_{6} - 1560) q^{95} + 169 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{4} + 15 q^{5} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 4 q^{4} + 15 q^{5} + 16 q^{8} + 30 q^{10} - 9 q^{11} + 176 q^{13} - 16 q^{16} + 84 q^{17} + 104 q^{19} - 120 q^{20} + 36 q^{22} - 84 q^{23} - 100 q^{25} - 176 q^{26} - 102 q^{29} + 185 q^{31} - 32 q^{32} - 336 q^{34} - 44 q^{37} + 208 q^{38} + 120 q^{40} - 336 q^{41} + 652 q^{43} - 36 q^{44} - 168 q^{46} + 138 q^{47} + 400 q^{50} - 352 q^{52} + 639 q^{53} - 270 q^{55} + 102 q^{58} - 159 q^{59} + 722 q^{61} - 740 q^{62} + 128 q^{64} + 1320 q^{65} + 166 q^{67} + 336 q^{68} - 2172 q^{71} + 218 q^{73} - 88 q^{74} - 832 q^{76} + 583 q^{79} + 240 q^{80} + 336 q^{82} - 1194 q^{83} + 2520 q^{85} - 652 q^{86} - 72 q^{88} + 1038 q^{89} + 672 q^{92} + 276 q^{94} - 1560 q^{95} + 338 q^{97}+O(q^{100}) \) 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 7.50000 + 12.9904i 0 0 8.00000 0 15.0000 25.9808i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i 7.50000 12.9904i 0 0 8.00000 0 15.0000 + 25.9808i
\(n\): 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.l 2
3.b odd 2 1 294.4.e.e 2
7.b odd 2 1 126.4.g.a 2
7.c even 3 1 882.4.a.h 1
7.c even 3 1 inner 882.4.g.l 2
7.d odd 6 1 126.4.g.a 2
7.d odd 6 1 882.4.a.r 1
21.c even 2 1 42.4.e.b 2
21.g even 6 1 42.4.e.b 2
21.g even 6 1 294.4.a.a 1
21.h odd 6 1 294.4.a.g 1
21.h odd 6 1 294.4.e.e 2
84.h odd 2 1 336.4.q.d 2
84.j odd 6 1 336.4.q.d 2
84.j odd 6 1 2352.4.a.u 1
84.n even 6 1 2352.4.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.b 2 21.c even 2 1
42.4.e.b 2 21.g even 6 1
126.4.g.a 2 7.b odd 2 1
126.4.g.a 2 7.d odd 6 1
294.4.a.a 1 21.g even 6 1
294.4.a.g 1 21.h odd 6 1
294.4.e.e 2 3.b odd 2 1
294.4.e.e 2 21.h odd 6 1
336.4.q.d 2 84.h odd 2 1
336.4.q.d 2 84.j odd 6 1
882.4.a.h 1 7.c even 3 1
882.4.a.r 1 7.d odd 6 1
882.4.g.l 2 1.a even 1 1 trivial
882.4.g.l 2 7.c even 3 1 inner
2352.4.a.q 1 84.n even 6 1
2352.4.a.u 1 84.j odd 6 1

Hecke kernels

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

\( T_{5}^{2} - 15T_{5} + 225 \) Copy content Toggle raw display
\( T_{11}^{2} + 9T_{11} + 81 \) Copy content Toggle raw display
\( T_{13} - 88 \) 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
$5$ \( T^{2} - 15T + 225 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$13$ \( (T - 88)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 84T + 7056 \) Copy content Toggle raw display
$19$ \( T^{2} - 104T + 10816 \) Copy content Toggle raw display
$23$ \( T^{2} + 84T + 7056 \) Copy content Toggle raw display
$29$ \( (T + 51)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 185T + 34225 \) Copy content Toggle raw display
$37$ \( T^{2} + 44T + 1936 \) Copy content Toggle raw display
$41$ \( (T + 168)^{2} \) Copy content Toggle raw display
$43$ \( (T - 326)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 138T + 19044 \) Copy content Toggle raw display
$53$ \( T^{2} - 639T + 408321 \) Copy content Toggle raw display
$59$ \( T^{2} + 159T + 25281 \) Copy content Toggle raw display
$61$ \( T^{2} - 722T + 521284 \) Copy content Toggle raw display
$67$ \( T^{2} - 166T + 27556 \) Copy content Toggle raw display
$71$ \( (T + 1086)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 218T + 47524 \) Copy content Toggle raw display
$79$ \( T^{2} - 583T + 339889 \) Copy content Toggle raw display
$83$ \( (T + 597)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 1038 T + 1077444 \) Copy content Toggle raw display
$97$ \( (T - 169)^{2} \) Copy content Toggle raw display
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