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: \(x^{2} - x + 1\)
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.

\(f(q)\) \(=\) \( q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 15 \zeta_{6} q^{5} + 8 q^{8} +O(q^{10})\) \( q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 15 \zeta_{6} q^{5} + 8 q^{8} + ( 30 - 30 \zeta_{6} ) q^{10} + ( -9 + 9 \zeta_{6} ) q^{11} + 88 q^{13} -16 \zeta_{6} q^{16} + ( 84 - 84 \zeta_{6} ) q^{17} + 104 \zeta_{6} q^{19} -60 q^{20} + 18 q^{22} -84 \zeta_{6} q^{23} + ( -100 + 100 \zeta_{6} ) q^{25} -176 \zeta_{6} q^{26} -51 q^{29} + ( 185 - 185 \zeta_{6} ) q^{31} + ( -32 + 32 \zeta_{6} ) q^{32} -168 q^{34} -44 \zeta_{6} q^{37} + ( 208 - 208 \zeta_{6} ) q^{38} + 120 \zeta_{6} q^{40} -168 q^{41} + 326 q^{43} -36 \zeta_{6} q^{44} + ( -168 + 168 \zeta_{6} ) q^{46} + 138 \zeta_{6} q^{47} + 200 q^{50} + ( -352 + 352 \zeta_{6} ) q^{52} + ( 639 - 639 \zeta_{6} ) q^{53} -135 q^{55} + 102 \zeta_{6} q^{58} + ( -159 + 159 \zeta_{6} ) q^{59} + 722 \zeta_{6} q^{61} -370 q^{62} + 64 q^{64} + 1320 \zeta_{6} q^{65} + ( 166 - 166 \zeta_{6} ) q^{67} + 336 \zeta_{6} q^{68} -1086 q^{71} + ( 218 - 218 \zeta_{6} ) q^{73} + ( -88 + 88 \zeta_{6} ) q^{74} -416 q^{76} + 583 \zeta_{6} q^{79} + ( 240 - 240 \zeta_{6} ) q^{80} + 336 \zeta_{6} q^{82} -597 q^{83} + 1260 q^{85} -652 \zeta_{6} q^{86} + ( -72 + 72 \zeta_{6} ) q^{88} + 1038 \zeta_{6} q^{89} + 336 q^{92} + ( 276 - 276 \zeta_{6} ) q^{94} + ( -1560 + 1560 \zeta_{6} ) q^{95} + 169 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 4q^{4} + 15q^{5} + 16q^{8} + O(q^{10}) \) \( 2q - 2q^{2} - 4q^{4} + 15q^{5} + 16q^{8} + 30q^{10} - 9q^{11} + 176q^{13} - 16q^{16} + 84q^{17} + 104q^{19} - 120q^{20} + 36q^{22} - 84q^{23} - 100q^{25} - 176q^{26} - 102q^{29} + 185q^{31} - 32q^{32} - 336q^{34} - 44q^{37} + 208q^{38} + 120q^{40} - 336q^{41} + 652q^{43} - 36q^{44} - 168q^{46} + 138q^{47} + 400q^{50} - 352q^{52} + 639q^{53} - 270q^{55} + 102q^{58} - 159q^{59} + 722q^{61} - 740q^{62} + 128q^{64} + 1320q^{65} + 166q^{67} + 336q^{68} - 2172q^{71} + 218q^{73} - 88q^{74} - 832q^{76} + 583q^{79} + 240q^{80} + 336q^{82} - 1194q^{83} + 2520q^{85} - 652q^{86} - 72q^{88} + 1038q^{89} + 672q^{92} + 276q^{94} - 1560q^{95} + 338q^{97} + O(q^{100}) \)

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 \(S_{4}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{2} - 15 T_{5} + 225 \)
\( T_{11}^{2} + 9 T_{11} + 81 \)
\( T_{13} - 88 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 2 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 225 - 15 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 81 + 9 T + T^{2} \)
$13$ \( ( -88 + T )^{2} \)
$17$ \( 7056 - 84 T + T^{2} \)
$19$ \( 10816 - 104 T + T^{2} \)
$23$ \( 7056 + 84 T + T^{2} \)
$29$ \( ( 51 + T )^{2} \)
$31$ \( 34225 - 185 T + T^{2} \)
$37$ \( 1936 + 44 T + T^{2} \)
$41$ \( ( 168 + T )^{2} \)
$43$ \( ( -326 + T )^{2} \)
$47$ \( 19044 - 138 T + T^{2} \)
$53$ \( 408321 - 639 T + T^{2} \)
$59$ \( 25281 + 159 T + T^{2} \)
$61$ \( 521284 - 722 T + T^{2} \)
$67$ \( 27556 - 166 T + T^{2} \)
$71$ \( ( 1086 + T )^{2} \)
$73$ \( 47524 - 218 T + T^{2} \)
$79$ \( 339889 - 583 T + T^{2} \)
$83$ \( ( 597 + T )^{2} \)
$89$ \( 1077444 - 1038 T + T^{2} \)
$97$ \( ( -169 + T )^{2} \)
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