Properties

Label 882.4.g.x
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: \( x^{2} - x + 1 \) 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.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} + 22 \zeta_{6} q^{5} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} + 22 \zeta_{6} q^{5} - 8 q^{8} + (44 \zeta_{6} - 44) q^{10} + (26 \zeta_{6} - 26) q^{11} + 54 q^{13} - 16 \zeta_{6} q^{16} + ( - 74 \zeta_{6} + 74) q^{17} + 116 \zeta_{6} q^{19} - 88 q^{20} - 52 q^{22} + 58 \zeta_{6} q^{23} + (359 \zeta_{6} - 359) q^{25} + 108 \zeta_{6} q^{26} + 208 q^{29} + (252 \zeta_{6} - 252) q^{31} + ( - 32 \zeta_{6} + 32) q^{32} + 148 q^{34} - 50 \zeta_{6} q^{37} + (232 \zeta_{6} - 232) q^{38} - 176 \zeta_{6} q^{40} - 126 q^{41} + 164 q^{43} - 104 \zeta_{6} q^{44} + (116 \zeta_{6} - 116) q^{46} - 444 \zeta_{6} q^{47} - 718 q^{50} + (216 \zeta_{6} - 216) q^{52} + (12 \zeta_{6} - 12) q^{53} - 572 q^{55} + 416 \zeta_{6} q^{58} + ( - 124 \zeta_{6} + 124) q^{59} - 162 \zeta_{6} q^{61} - 504 q^{62} + 64 q^{64} + 1188 \zeta_{6} q^{65} + ( - 860 \zeta_{6} + 860) q^{67} + 296 \zeta_{6} q^{68} - 238 q^{71} + (146 \zeta_{6} - 146) q^{73} + ( - 100 \zeta_{6} + 100) q^{74} - 464 q^{76} + 984 \zeta_{6} q^{79} + ( - 352 \zeta_{6} + 352) q^{80} - 252 \zeta_{6} q^{82} - 656 q^{83} + 1628 q^{85} + 328 \zeta_{6} q^{86} + ( - 208 \zeta_{6} + 208) q^{88} - 954 \zeta_{6} q^{89} - 232 q^{92} + ( - 888 \zeta_{6} + 888) q^{94} + (2552 \zeta_{6} - 2552) q^{95} - 526 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 4 q^{4} + 22 q^{5} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 4 q^{4} + 22 q^{5} - 16 q^{8} - 44 q^{10} - 26 q^{11} + 108 q^{13} - 16 q^{16} + 74 q^{17} + 116 q^{19} - 176 q^{20} - 104 q^{22} + 58 q^{23} - 359 q^{25} + 108 q^{26} + 416 q^{29} - 252 q^{31} + 32 q^{32} + 296 q^{34} - 50 q^{37} - 232 q^{38} - 176 q^{40} - 252 q^{41} + 328 q^{43} - 104 q^{44} - 116 q^{46} - 444 q^{47} - 1436 q^{50} - 216 q^{52} - 12 q^{53} - 1144 q^{55} + 416 q^{58} + 124 q^{59} - 162 q^{61} - 1008 q^{62} + 128 q^{64} + 1188 q^{65} + 860 q^{67} + 296 q^{68} - 476 q^{71} - 146 q^{73} + 100 q^{74} - 928 q^{76} + 984 q^{79} + 352 q^{80} - 252 q^{82} - 1312 q^{83} + 3256 q^{85} + 328 q^{86} + 208 q^{88} - 954 q^{89} - 464 q^{92} + 888 q^{94} - 2552 q^{95} - 1052 q^{97}+O(q^{100}) \) 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 11.0000 + 19.0526i 0 0 −8.00000 0 −22.0000 + 38.1051i
667.1 1.00000 1.73205i 0 −2.00000 3.46410i 11.0000 19.0526i 0 0 −8.00000 0 −22.0000 38.1051i
\(n\): 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.x 2
3.b odd 2 1 882.4.g.a 2
7.b odd 2 1 882.4.g.n 2
7.c even 3 1 882.4.a.a 1
7.c even 3 1 inner 882.4.g.x 2
7.d odd 6 1 126.4.a.e 1
7.d odd 6 1 882.4.g.n 2
21.c even 2 1 882.4.g.m 2
21.g even 6 1 126.4.a.f yes 1
21.g even 6 1 882.4.g.m 2
21.h odd 6 1 882.4.a.s 1
21.h odd 6 1 882.4.g.a 2
28.f even 6 1 1008.4.a.w 1
84.j odd 6 1 1008.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.a.e 1 7.d odd 6 1
126.4.a.f yes 1 21.g even 6 1
882.4.a.a 1 7.c even 3 1
882.4.a.s 1 21.h odd 6 1
882.4.g.a 2 3.b odd 2 1
882.4.g.a 2 21.h odd 6 1
882.4.g.m 2 21.c even 2 1
882.4.g.m 2 21.g even 6 1
882.4.g.n 2 7.b odd 2 1
882.4.g.n 2 7.d odd 6 1
882.4.g.x 2 1.a even 1 1 trivial
882.4.g.x 2 7.c even 3 1 inner
1008.4.a.a 1 84.j odd 6 1
1008.4.a.w 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 \(S_{4}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{2} - 22T_{5} + 484 \) Copy content Toggle raw display
\( T_{11}^{2} + 26T_{11} + 676 \) Copy content Toggle raw display
\( T_{13} - 54 \) 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
$5$ \( T^{2} - 22T + 484 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 26T + 676 \) Copy content Toggle raw display
$13$ \( (T - 54)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 74T + 5476 \) Copy content Toggle raw display
$19$ \( T^{2} - 116T + 13456 \) Copy content Toggle raw display
$23$ \( T^{2} - 58T + 3364 \) Copy content Toggle raw display
$29$ \( (T - 208)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 252T + 63504 \) Copy content Toggle raw display
$37$ \( T^{2} + 50T + 2500 \) Copy content Toggle raw display
$41$ \( (T + 126)^{2} \) Copy content Toggle raw display
$43$ \( (T - 164)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 444T + 197136 \) Copy content Toggle raw display
$53$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$59$ \( T^{2} - 124T + 15376 \) Copy content Toggle raw display
$61$ \( T^{2} + 162T + 26244 \) Copy content Toggle raw display
$67$ \( T^{2} - 860T + 739600 \) Copy content Toggle raw display
$71$ \( (T + 238)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 146T + 21316 \) Copy content Toggle raw display
$79$ \( T^{2} - 984T + 968256 \) Copy content Toggle raw display
$83$ \( (T + 656)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 954T + 910116 \) Copy content Toggle raw display
$97$ \( (T + 526)^{2} \) Copy content Toggle raw display
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