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 $2$

Related objects

Downloads

Learn more

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: \(\Z[a_1, \ldots, a_{25}]\)
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.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} + 6 \zeta_{6} q^{5} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} + 6 \zeta_{6} q^{5} - 8 q^{8} + (12 \zeta_{6} - 12) q^{10} + ( - 30 \zeta_{6} + 30) q^{11} + 2 q^{13} - 16 \zeta_{6} q^{16} + ( - 66 \zeta_{6} + 66) q^{17} + 52 \zeta_{6} q^{19} - 24 q^{20} + 60 q^{22} + 114 \zeta_{6} q^{23} + ( - 89 \zeta_{6} + 89) q^{25} + 4 \zeta_{6} q^{26} - 72 q^{29} + ( - 196 \zeta_{6} + 196) q^{31} + ( - 32 \zeta_{6} + 32) q^{32} + 132 q^{34} + 286 \zeta_{6} q^{37} + (104 \zeta_{6} - 104) q^{38} - 48 \zeta_{6} q^{40} + 378 q^{41} + 164 q^{43} + 120 \zeta_{6} q^{44} + (228 \zeta_{6} - 228) q^{46} - 228 \zeta_{6} q^{47} + 178 q^{50} + (8 \zeta_{6} - 8) q^{52} + (348 \zeta_{6} - 348) q^{53} + 180 q^{55} - 144 \zeta_{6} q^{58} + (348 \zeta_{6} - 348) q^{59} + 106 \zeta_{6} q^{61} + 392 q^{62} + 64 q^{64} + 12 \zeta_{6} q^{65} + (596 \zeta_{6} - 596) q^{67} + 264 \zeta_{6} q^{68} - 630 q^{71} + ( - 1042 \zeta_{6} + 1042) q^{73} + (572 \zeta_{6} - 572) q^{74} - 208 q^{76} + 88 \zeta_{6} q^{79} + ( - 96 \zeta_{6} + 96) q^{80} + 756 \zeta_{6} q^{82} + 1440 q^{83} + 396 q^{85} + 328 \zeta_{6} q^{86} + (240 \zeta_{6} - 240) q^{88} + 1374 \zeta_{6} q^{89} - 456 q^{92} + ( - 456 \zeta_{6} + 456) q^{94} + (312 \zeta_{6} - 312) q^{95} - 34 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 4 q^{4} + 6 q^{5} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 4 q^{4} + 6 q^{5} - 16 q^{8} - 12 q^{10} + 30 q^{11} + 4 q^{13} - 16 q^{16} + 66 q^{17} + 52 q^{19} - 48 q^{20} + 120 q^{22} + 114 q^{23} + 89 q^{25} + 4 q^{26} - 144 q^{29} + 196 q^{31} + 32 q^{32} + 264 q^{34} + 286 q^{37} - 104 q^{38} - 48 q^{40} + 756 q^{41} + 328 q^{43} + 120 q^{44} - 228 q^{46} - 228 q^{47} + 356 q^{50} - 8 q^{52} - 348 q^{53} + 360 q^{55} - 144 q^{58} - 348 q^{59} + 106 q^{61} + 784 q^{62} + 128 q^{64} + 12 q^{65} - 596 q^{67} + 264 q^{68} - 1260 q^{71} + 1042 q^{73} - 572 q^{74} - 416 q^{76} + 88 q^{79} + 96 q^{80} + 756 q^{82} + 2880 q^{83} + 792 q^{85} + 328 q^{86} - 240 q^{88} + 1374 q^{89} - 912 q^{92} + 456 q^{94} - 312 q^{95} - 68 q^{97}+O(q^{100}) \) 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 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 \(S_{4}^{\mathrm{new}}(882, [\chi])\):

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