Properties

Label 882.4.g.o
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 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.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} -18 \zeta_{6} q^{5} -8 q^{8} +O(q^{10})\) \( q + 2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} -18 \zeta_{6} q^{5} -8 q^{8} + ( 36 - 36 \zeta_{6} ) q^{10} + ( -72 + 72 \zeta_{6} ) q^{11} + 34 q^{13} -16 \zeta_{6} q^{16} + ( -6 + 6 \zeta_{6} ) q^{17} + 92 \zeta_{6} q^{19} + 72 q^{20} -144 q^{22} -180 \zeta_{6} q^{23} + ( -199 + 199 \zeta_{6} ) q^{25} + 68 \zeta_{6} q^{26} + 114 q^{29} + ( 56 - 56 \zeta_{6} ) q^{31} + ( 32 - 32 \zeta_{6} ) q^{32} -12 q^{34} + 34 \zeta_{6} q^{37} + ( -184 + 184 \zeta_{6} ) q^{38} + 144 \zeta_{6} q^{40} + 6 q^{41} + 164 q^{43} -288 \zeta_{6} q^{44} + ( 360 - 360 \zeta_{6} ) q^{46} -168 \zeta_{6} q^{47} -398 q^{50} + ( -136 + 136 \zeta_{6} ) q^{52} + ( 654 - 654 \zeta_{6} ) q^{53} + 1296 q^{55} + 228 \zeta_{6} q^{58} + ( 492 - 492 \zeta_{6} ) q^{59} -250 \zeta_{6} q^{61} + 112 q^{62} + 64 q^{64} -612 \zeta_{6} q^{65} + ( 124 - 124 \zeta_{6} ) q^{67} -24 \zeta_{6} q^{68} -36 q^{71} + ( 1010 - 1010 \zeta_{6} ) q^{73} + ( -68 + 68 \zeta_{6} ) q^{74} -368 q^{76} -56 \zeta_{6} q^{79} + ( -288 + 288 \zeta_{6} ) q^{80} + 12 \zeta_{6} q^{82} + 228 q^{83} + 108 q^{85} + 328 \zeta_{6} q^{86} + ( 576 - 576 \zeta_{6} ) q^{88} -390 \zeta_{6} q^{89} + 720 q^{92} + ( 336 - 336 \zeta_{6} ) q^{94} + ( 1656 - 1656 \zeta_{6} ) q^{95} + 70 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 4q^{4} - 18q^{5} - 16q^{8} + O(q^{10}) \) \( 2q + 2q^{2} - 4q^{4} - 18q^{5} - 16q^{8} + 36q^{10} - 72q^{11} + 68q^{13} - 16q^{16} - 6q^{17} + 92q^{19} + 144q^{20} - 288q^{22} - 180q^{23} - 199q^{25} + 68q^{26} + 228q^{29} + 56q^{31} + 32q^{32} - 24q^{34} + 34q^{37} - 184q^{38} + 144q^{40} + 12q^{41} + 328q^{43} - 288q^{44} + 360q^{46} - 168q^{47} - 796q^{50} - 136q^{52} + 654q^{53} + 2592q^{55} + 228q^{58} + 492q^{59} - 250q^{61} + 224q^{62} + 128q^{64} - 612q^{65} + 124q^{67} - 24q^{68} - 72q^{71} + 1010q^{73} - 68q^{74} - 736q^{76} - 56q^{79} - 288q^{80} + 12q^{82} + 456q^{83} + 216q^{85} + 328q^{86} + 576q^{88} - 390q^{89} + 1440q^{92} + 336q^{94} + 1656q^{95} + 140q^{97} + O(q^{100}) \)

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 −9.00000 15.5885i 0 0 −8.00000 0 18.0000 31.1769i
667.1 1.00000 1.73205i 0 −2.00000 3.46410i −9.00000 + 15.5885i 0 0 −8.00000 0 18.0000 + 31.1769i
\(n\): 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.o 2
3.b odd 2 1 294.4.e.b 2
7.b odd 2 1 882.4.g.w 2
7.c even 3 1 882.4.a.g 1
7.c even 3 1 inner 882.4.g.o 2
7.d odd 6 1 126.4.a.a 1
7.d odd 6 1 882.4.g.w 2
21.c even 2 1 294.4.e.c 2
21.g even 6 1 42.4.a.a 1
21.g even 6 1 294.4.e.c 2
21.h odd 6 1 294.4.a.i 1
21.h odd 6 1 294.4.e.b 2
28.f even 6 1 1008.4.a.b 1
84.j odd 6 1 336.4.a.l 1
84.n even 6 1 2352.4.a.a 1
105.p even 6 1 1050.4.a.g 1
105.w odd 12 2 1050.4.g.a 2
168.ba even 6 1 1344.4.a.o 1
168.be odd 6 1 1344.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.a.a 1 21.g even 6 1
126.4.a.a 1 7.d odd 6 1
294.4.a.i 1 21.h odd 6 1
294.4.e.b 2 3.b odd 2 1
294.4.e.b 2 21.h odd 6 1
294.4.e.c 2 21.c even 2 1
294.4.e.c 2 21.g even 6 1
336.4.a.l 1 84.j odd 6 1
882.4.a.g 1 7.c even 3 1
882.4.g.o 2 1.a even 1 1 trivial
882.4.g.o 2 7.c even 3 1 inner
882.4.g.w 2 7.b odd 2 1
882.4.g.w 2 7.d odd 6 1
1008.4.a.b 1 28.f even 6 1
1050.4.a.g 1 105.p even 6 1
1050.4.g.a 2 105.w odd 12 2
1344.4.a.a 1 168.be odd 6 1
1344.4.a.o 1 168.ba even 6 1
2352.4.a.a 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 \(S_{4}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{2} + 18 T_{5} + 324 \)
\( T_{11}^{2} + 72 T_{11} + 5184 \)
\( T_{13} - 34 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 - 2 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 324 + 18 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 5184 + 72 T + T^{2} \)
$13$ \( ( -34 + T )^{2} \)
$17$ \( 36 + 6 T + T^{2} \)
$19$ \( 8464 - 92 T + T^{2} \)
$23$ \( 32400 + 180 T + T^{2} \)
$29$ \( ( -114 + T )^{2} \)
$31$ \( 3136 - 56 T + T^{2} \)
$37$ \( 1156 - 34 T + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( ( -164 + T )^{2} \)
$47$ \( 28224 + 168 T + T^{2} \)
$53$ \( 427716 - 654 T + T^{2} \)
$59$ \( 242064 - 492 T + T^{2} \)
$61$ \( 62500 + 250 T + T^{2} \)
$67$ \( 15376 - 124 T + T^{2} \)
$71$ \( ( 36 + T )^{2} \)
$73$ \( 1020100 - 1010 T + T^{2} \)
$79$ \( 3136 + 56 T + T^{2} \)
$83$ \( ( -228 + T )^{2} \)
$89$ \( 152100 + 390 T + T^{2} \)
$97$ \( ( -70 + T )^{2} \)
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