Properties

Label 882.4.g.u
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
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.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} + 9 \zeta_{6} q^{5} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} + 9 \zeta_{6} q^{5} - 8 q^{8} + (18 \zeta_{6} - 18) q^{10} + (57 \zeta_{6} - 57) q^{11} + 70 q^{13} - 16 \zeta_{6} q^{16} + (51 \zeta_{6} - 51) q^{17} + 5 \zeta_{6} q^{19} - 36 q^{20} - 114 q^{22} + 69 \zeta_{6} q^{23} + ( - 44 \zeta_{6} + 44) q^{25} + 140 \zeta_{6} q^{26} - 114 q^{29} + ( - 23 \zeta_{6} + 23) q^{31} + ( - 32 \zeta_{6} + 32) q^{32} - 102 q^{34} + 253 \zeta_{6} q^{37} + (10 \zeta_{6} - 10) q^{38} - 72 \zeta_{6} q^{40} - 42 q^{41} - 124 q^{43} - 228 \zeta_{6} q^{44} + (138 \zeta_{6} - 138) q^{46} - 201 \zeta_{6} q^{47} + 88 q^{50} + (280 \zeta_{6} - 280) q^{52} + (393 \zeta_{6} - 393) q^{53} - 513 q^{55} - 228 \zeta_{6} q^{58} + (219 \zeta_{6} - 219) q^{59} - 709 \zeta_{6} q^{61} + 46 q^{62} + 64 q^{64} + 630 \zeta_{6} q^{65} + (419 \zeta_{6} - 419) q^{67} - 204 \zeta_{6} q^{68} + 96 q^{71} + (313 \zeta_{6} - 313) q^{73} + (506 \zeta_{6} - 506) q^{74} - 20 q^{76} - 461 \zeta_{6} q^{79} + ( - 144 \zeta_{6} + 144) q^{80} - 84 \zeta_{6} q^{82} - 588 q^{83} - 459 q^{85} - 248 \zeta_{6} q^{86} + ( - 456 \zeta_{6} + 456) q^{88} + 1017 \zeta_{6} q^{89} - 276 q^{92} + ( - 402 \zeta_{6} + 402) q^{94} + (45 \zeta_{6} - 45) q^{95} + 1834 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 4 q^{4} + 9 q^{5} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 4 q^{4} + 9 q^{5} - 16 q^{8} - 18 q^{10} - 57 q^{11} + 140 q^{13} - 16 q^{16} - 51 q^{17} + 5 q^{19} - 72 q^{20} - 228 q^{22} + 69 q^{23} + 44 q^{25} + 140 q^{26} - 228 q^{29} + 23 q^{31} + 32 q^{32} - 204 q^{34} + 253 q^{37} - 10 q^{38} - 72 q^{40} - 84 q^{41} - 248 q^{43} - 228 q^{44} - 138 q^{46} - 201 q^{47} + 176 q^{50} - 280 q^{52} - 393 q^{53} - 1026 q^{55} - 228 q^{58} - 219 q^{59} - 709 q^{61} + 92 q^{62} + 128 q^{64} + 630 q^{65} - 419 q^{67} - 204 q^{68} + 192 q^{71} - 313 q^{73} - 506 q^{74} - 40 q^{76} - 461 q^{79} + 144 q^{80} - 84 q^{82} - 1176 q^{83} - 918 q^{85} - 248 q^{86} + 456 q^{88} + 1017 q^{89} - 552 q^{92} + 402 q^{94} - 45 q^{95} + 3668 q^{97}+O(q^{100}) \) 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 4.50000 + 7.79423i 0 0 −8.00000 0 −9.00000 + 15.5885i
667.1 1.00000 1.73205i 0 −2.00000 3.46410i 4.50000 7.79423i 0 0 −8.00000 0 −9.00000 15.5885i
\(n\): 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.u 2
3.b odd 2 1 98.4.c.a 2
7.b odd 2 1 126.4.g.d 2
7.c even 3 1 882.4.a.c 1
7.c even 3 1 inner 882.4.g.u 2
7.d odd 6 1 126.4.g.d 2
7.d odd 6 1 882.4.a.f 1
21.c even 2 1 14.4.c.a 2
21.g even 6 1 14.4.c.a 2
21.g even 6 1 98.4.a.d 1
21.h odd 6 1 98.4.a.f 1
21.h odd 6 1 98.4.c.a 2
84.h odd 2 1 112.4.i.a 2
84.j odd 6 1 112.4.i.a 2
84.j odd 6 1 784.4.a.p 1
84.n even 6 1 784.4.a.c 1
105.g even 2 1 350.4.e.e 2
105.k odd 4 2 350.4.j.b 4
105.o odd 6 1 2450.4.a.d 1
105.p even 6 1 350.4.e.e 2
105.p even 6 1 2450.4.a.q 1
105.w odd 12 2 350.4.j.b 4
168.e odd 2 1 448.4.i.e 2
168.i even 2 1 448.4.i.b 2
168.ba even 6 1 448.4.i.b 2
168.be odd 6 1 448.4.i.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.a 2 21.c even 2 1
14.4.c.a 2 21.g even 6 1
98.4.a.d 1 21.g even 6 1
98.4.a.f 1 21.h odd 6 1
98.4.c.a 2 3.b odd 2 1
98.4.c.a 2 21.h odd 6 1
112.4.i.a 2 84.h odd 2 1
112.4.i.a 2 84.j odd 6 1
126.4.g.d 2 7.b odd 2 1
126.4.g.d 2 7.d odd 6 1
350.4.e.e 2 105.g even 2 1
350.4.e.e 2 105.p even 6 1
350.4.j.b 4 105.k odd 4 2
350.4.j.b 4 105.w odd 12 2
448.4.i.b 2 168.i even 2 1
448.4.i.b 2 168.ba even 6 1
448.4.i.e 2 168.e odd 2 1
448.4.i.e 2 168.be odd 6 1
784.4.a.c 1 84.n even 6 1
784.4.a.p 1 84.j odd 6 1
882.4.a.c 1 7.c even 3 1
882.4.a.f 1 7.d odd 6 1
882.4.g.u 2 1.a even 1 1 trivial
882.4.g.u 2 7.c even 3 1 inner
2450.4.a.d 1 105.o odd 6 1
2450.4.a.q 1 105.p even 6 1

Hecke kernels

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

\( T_{5}^{2} - 9T_{5} + 81 \) Copy content Toggle raw display
\( T_{11}^{2} + 57T_{11} + 3249 \) Copy content Toggle raw display
\( T_{13} - 70 \) 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
$5$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 57T + 3249 \) Copy content Toggle raw display
$13$ \( (T - 70)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 51T + 2601 \) Copy content Toggle raw display
$19$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$23$ \( T^{2} - 69T + 4761 \) Copy content Toggle raw display
$29$ \( (T + 114)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 23T + 529 \) Copy content Toggle raw display
$37$ \( T^{2} - 253T + 64009 \) Copy content Toggle raw display
$41$ \( (T + 42)^{2} \) Copy content Toggle raw display
$43$ \( (T + 124)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 201T + 40401 \) Copy content Toggle raw display
$53$ \( T^{2} + 393T + 154449 \) Copy content Toggle raw display
$59$ \( T^{2} + 219T + 47961 \) Copy content Toggle raw display
$61$ \( T^{2} + 709T + 502681 \) Copy content Toggle raw display
$67$ \( T^{2} + 419T + 175561 \) Copy content Toggle raw display
$71$ \( (T - 96)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 313T + 97969 \) Copy content Toggle raw display
$79$ \( T^{2} + 461T + 212521 \) Copy content Toggle raw display
$83$ \( (T + 588)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 1017 T + 1034289 \) Copy content Toggle raw display
$97$ \( (T - 1834)^{2} \) Copy content Toggle raw display
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