Properties

Label 882.4.g.h
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: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
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.

\(f(q)\) \(=\) \( q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 6 \zeta_{6} q^{5} + 8 q^{8} +O(q^{10})\) \( q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 6 \zeta_{6} q^{5} + 8 q^{8} + ( 12 - 12 \zeta_{6} ) q^{10} + ( -30 + 30 \zeta_{6} ) q^{11} -2 q^{13} -16 \zeta_{6} q^{16} + ( 66 - 66 \zeta_{6} ) q^{17} -52 \zeta_{6} q^{19} -24 q^{20} + 60 q^{22} -114 \zeta_{6} q^{23} + ( 89 - 89 \zeta_{6} ) q^{25} + 4 \zeta_{6} q^{26} + 72 q^{29} + ( -196 + 196 \zeta_{6} ) q^{31} + ( -32 + 32 \zeta_{6} ) q^{32} -132 q^{34} + 286 \zeta_{6} q^{37} + ( -104 + 104 \zeta_{6} ) q^{38} + 48 \zeta_{6} q^{40} + 378 q^{41} + 164 q^{43} -120 \zeta_{6} q^{44} + ( -228 + 228 \zeta_{6} ) q^{46} -228 \zeta_{6} q^{47} -178 q^{50} + ( 8 - 8 \zeta_{6} ) q^{52} + ( 348 - 348 \zeta_{6} ) q^{53} -180 q^{55} -144 \zeta_{6} q^{58} + ( -348 + 348 \zeta_{6} ) q^{59} -106 \zeta_{6} q^{61} + 392 q^{62} + 64 q^{64} -12 \zeta_{6} q^{65} + ( -596 + 596 \zeta_{6} ) q^{67} + 264 \zeta_{6} q^{68} + 630 q^{71} + ( -1042 + 1042 \zeta_{6} ) q^{73} + ( 572 - 572 \zeta_{6} ) q^{74} + 208 q^{76} + 88 \zeta_{6} q^{79} + ( 96 - 96 \zeta_{6} ) q^{80} -756 \zeta_{6} q^{82} + 1440 q^{83} + 396 q^{85} -328 \zeta_{6} q^{86} + ( -240 + 240 \zeta_{6} ) q^{88} + 1374 \zeta_{6} q^{89} + 456 q^{92} + ( -456 + 456 \zeta_{6} ) q^{94} + ( 312 - 312 \zeta_{6} ) q^{95} + 34 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 4q^{4} + 6q^{5} + 16q^{8} + O(q^{10}) \) \( 2q - 2q^{2} - 4q^{4} + 6q^{5} + 16q^{8} + 12q^{10} - 30q^{11} - 4q^{13} - 16q^{16} + 66q^{17} - 52q^{19} - 48q^{20} + 120q^{22} - 114q^{23} + 89q^{25} + 4q^{26} + 144q^{29} - 196q^{31} - 32q^{32} - 264q^{34} + 286q^{37} - 104q^{38} + 48q^{40} + 756q^{41} + 328q^{43} - 120q^{44} - 228q^{46} - 228q^{47} - 356q^{50} + 8q^{52} + 348q^{53} - 360q^{55} - 144q^{58} - 348q^{59} - 106q^{61} + 784q^{62} + 128q^{64} - 12q^{65} - 596q^{67} + 264q^{68} + 1260q^{71} - 1042q^{73} + 572q^{74} + 416q^{76} + 88q^{79} + 96q^{80} - 756q^{82} + 2880q^{83} + 792q^{85} - 328q^{86} - 240q^{88} + 1374q^{89} + 912q^{92} - 456q^{94} + 312q^{95} + 68q^{97} + O(q^{100}) \)

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 3.00000 + 5.19615i 0 0 8.00000 0 6.00000 10.3923i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i 3.00000 5.19615i 0 0 8.00000 0 6.00000 + 10.3923i
\(n\): 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.h 2
3.b odd 2 1 882.4.g.q 2
7.b odd 2 1 882.4.g.e 2
7.c even 3 1 882.4.a.m 1
7.c even 3 1 inner 882.4.g.h 2
7.d odd 6 1 126.4.a.g yes 1
7.d odd 6 1 882.4.g.e 2
21.c even 2 1 882.4.g.t 2
21.g even 6 1 126.4.a.b 1
21.g even 6 1 882.4.g.t 2
21.h odd 6 1 882.4.a.e 1
21.h odd 6 1 882.4.g.q 2
28.f even 6 1 1008.4.a.n 1
84.j odd 6 1 1008.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.a.b 1 21.g even 6 1
126.4.a.g yes 1 7.d odd 6 1
882.4.a.e 1 21.h odd 6 1
882.4.a.m 1 7.c even 3 1
882.4.g.e 2 7.b odd 2 1
882.4.g.e 2 7.d odd 6 1
882.4.g.h 2 1.a even 1 1 trivial
882.4.g.h 2 7.c even 3 1 inner
882.4.g.q 2 3.b odd 2 1
882.4.g.q 2 21.h odd 6 1
882.4.g.t 2 21.c even 2 1
882.4.g.t 2 21.g even 6 1
1008.4.a.g 1 84.j odd 6 1
1008.4.a.n 1 28.f even 6 1

Hecke kernels

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

\( T_{5}^{2} - 6 T_{5} + 36 \)
\( T_{11}^{2} + 30 T_{11} + 900 \)
\( T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 2 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 36 - 6 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 900 + 30 T + T^{2} \)
$13$ \( ( 2 + T )^{2} \)
$17$ \( 4356 - 66 T + T^{2} \)
$19$ \( 2704 + 52 T + T^{2} \)
$23$ \( 12996 + 114 T + T^{2} \)
$29$ \( ( -72 + T )^{2} \)
$31$ \( 38416 + 196 T + T^{2} \)
$37$ \( 81796 - 286 T + T^{2} \)
$41$ \( ( -378 + T )^{2} \)
$43$ \( ( -164 + T )^{2} \)
$47$ \( 51984 + 228 T + T^{2} \)
$53$ \( 121104 - 348 T + T^{2} \)
$59$ \( 121104 + 348 T + T^{2} \)
$61$ \( 11236 + 106 T + T^{2} \)
$67$ \( 355216 + 596 T + T^{2} \)
$71$ \( ( -630 + T )^{2} \)
$73$ \( 1085764 + 1042 T + T^{2} \)
$79$ \( 7744 - 88 T + T^{2} \)
$83$ \( ( -1440 + T )^{2} \)
$89$ \( 1887876 - 1374 T + T^{2} \)
$97$ \( ( -34 + T )^{2} \)
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