Properties

Label 882.4.g.t
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.t 2
3.b odd 2 1 882.4.g.e 2
7.b odd 2 1 882.4.g.q 2
7.c even 3 1 126.4.a.b 1
7.c even 3 1 inner 882.4.g.t 2
7.d odd 6 1 882.4.a.e 1
7.d odd 6 1 882.4.g.q 2
21.c even 2 1 882.4.g.h 2
21.g even 6 1 882.4.a.m 1
21.g even 6 1 882.4.g.h 2
21.h odd 6 1 126.4.a.g yes 1
21.h odd 6 1 882.4.g.e 2
28.g odd 6 1 1008.4.a.g 1
84.n even 6 1 1008.4.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.a.b 1 7.c even 3 1
126.4.a.g yes 1 21.h odd 6 1
882.4.a.e 1 7.d odd 6 1
882.4.a.m 1 21.g even 6 1
882.4.g.e 2 3.b odd 2 1
882.4.g.e 2 21.h odd 6 1
882.4.g.h 2 21.c even 2 1
882.4.g.h 2 21.g even 6 1
882.4.g.q 2 7.b odd 2 1
882.4.g.q 2 7.d odd 6 1
882.4.g.t 2 1.a even 1 1 trivial
882.4.g.t 2 7.c even 3 1 inner
1008.4.a.g 1 28.g odd 6 1
1008.4.a.n 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 \(S_{4}^{\mathrm{new}}(882, [\chi])\):

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

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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