Properties

Label 882.4.g.f
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 6)
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 \( q - 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} - 6 \zeta_{6} q^{5} + 8 q^{8} + (12 \zeta_{6} - 12) q^{10} + ( - 12 \zeta_{6} + 12) q^{11} - 38 q^{13} - 16 \zeta_{6} q^{16} + ( - 126 \zeta_{6} + 126) q^{17} + 20 \zeta_{6} q^{19} + 24 q^{20} - 24 q^{22} + 168 \zeta_{6} q^{23} + ( - 89 \zeta_{6} + 89) q^{25} + 76 \zeta_{6} q^{26} - 30 q^{29} + (88 \zeta_{6} - 88) q^{31} + (32 \zeta_{6} - 32) q^{32} - 252 q^{34} - 254 \zeta_{6} q^{37} + ( - 40 \zeta_{6} + 40) q^{38} - 48 \zeta_{6} q^{40} + 42 q^{41} - 52 q^{43} + 48 \zeta_{6} q^{44} + ( - 336 \zeta_{6} + 336) q^{46} + 96 \zeta_{6} q^{47} - 178 q^{50} + ( - 152 \zeta_{6} + 152) q^{52} + ( - 198 \zeta_{6} + 198) q^{53} - 72 q^{55} + 60 \zeta_{6} q^{58} + ( - 660 \zeta_{6} + 660) q^{59} - 538 \zeta_{6} q^{61} + 176 q^{62} + 64 q^{64} + 228 \zeta_{6} q^{65} + (884 \zeta_{6} - 884) q^{67} + 504 \zeta_{6} q^{68} - 792 q^{71} + ( - 218 \zeta_{6} + 218) q^{73} + (508 \zeta_{6} - 508) q^{74} - 80 q^{76} + 520 \zeta_{6} q^{79} + (96 \zeta_{6} - 96) q^{80} - 84 \zeta_{6} q^{82} - 492 q^{83} - 756 q^{85} + 104 \zeta_{6} q^{86} + ( - 96 \zeta_{6} + 96) q^{88} - 810 \zeta_{6} q^{89} - 672 q^{92} + ( - 192 \zeta_{6} + 192) q^{94} + ( - 120 \zeta_{6} + 120) q^{95} - 1154 q^{97} +O(q^{100}) \) 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 \( 2 q - 2 q^{2} - 4 q^{4} - 6 q^{5} + 16 q^{8} - 12 q^{10} + 12 q^{11} - 76 q^{13} - 16 q^{16} + 126 q^{17} + 20 q^{19} + 48 q^{20} - 48 q^{22} + 168 q^{23} + 89 q^{25} + 76 q^{26} - 60 q^{29} - 88 q^{31} - 32 q^{32} - 504 q^{34} - 254 q^{37} + 40 q^{38} - 48 q^{40} + 84 q^{41} - 104 q^{43} + 48 q^{44} + 336 q^{46} + 96 q^{47} - 356 q^{50} + 152 q^{52} + 198 q^{53} - 144 q^{55} + 60 q^{58} + 660 q^{59} - 538 q^{61} + 352 q^{62} + 128 q^{64} + 228 q^{65} - 884 q^{67} + 504 q^{68} - 1584 q^{71} + 218 q^{73} - 508 q^{74} - 160 q^{76} + 520 q^{79} - 96 q^{80} - 84 q^{82} - 984 q^{83} - 1512 q^{85} + 104 q^{86} + 96 q^{88} - 810 q^{89} - 1344 q^{92} + 192 q^{94} + 120 q^{95} - 2308 q^{97}+O(q^{100}) \) 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.f 2
3.b odd 2 1 294.4.e.g 2
7.b odd 2 1 882.4.g.i 2
7.c even 3 1 882.4.a.n 1
7.c even 3 1 inner 882.4.g.f 2
7.d odd 6 1 18.4.a.a 1
7.d odd 6 1 882.4.g.i 2
21.c even 2 1 294.4.e.h 2
21.g even 6 1 6.4.a.a 1
21.g even 6 1 294.4.e.h 2
21.h odd 6 1 294.4.a.e 1
21.h odd 6 1 294.4.e.g 2
28.f even 6 1 144.4.a.c 1
35.i odd 6 1 450.4.a.h 1
35.k even 12 2 450.4.c.e 2
56.j odd 6 1 576.4.a.q 1
56.m even 6 1 576.4.a.r 1
63.i even 6 1 162.4.c.f 2
63.k odd 6 1 162.4.c.c 2
63.s even 6 1 162.4.c.f 2
63.t odd 6 1 162.4.c.c 2
77.i even 6 1 2178.4.a.e 1
84.j odd 6 1 48.4.a.c 1
84.n even 6 1 2352.4.a.e 1
105.p even 6 1 150.4.a.i 1
105.w odd 12 2 150.4.c.d 2
168.ba even 6 1 192.4.a.i 1
168.be odd 6 1 192.4.a.c 1
231.k odd 6 1 726.4.a.f 1
273.ba even 6 1 1014.4.a.g 1
273.cb odd 12 2 1014.4.b.d 2
336.bo even 12 2 768.4.d.n 2
336.br odd 12 2 768.4.d.c 2
357.s even 6 1 1734.4.a.d 1
399.s odd 6 1 2166.4.a.i 1
420.be odd 6 1 1200.4.a.b 1
420.br even 12 2 1200.4.f.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 21.g even 6 1
18.4.a.a 1 7.d odd 6 1
48.4.a.c 1 84.j odd 6 1
144.4.a.c 1 28.f even 6 1
150.4.a.i 1 105.p even 6 1
150.4.c.d 2 105.w odd 12 2
162.4.c.c 2 63.k odd 6 1
162.4.c.c 2 63.t odd 6 1
162.4.c.f 2 63.i even 6 1
162.4.c.f 2 63.s even 6 1
192.4.a.c 1 168.be odd 6 1
192.4.a.i 1 168.ba even 6 1
294.4.a.e 1 21.h odd 6 1
294.4.e.g 2 3.b odd 2 1
294.4.e.g 2 21.h odd 6 1
294.4.e.h 2 21.c even 2 1
294.4.e.h 2 21.g even 6 1
450.4.a.h 1 35.i odd 6 1
450.4.c.e 2 35.k even 12 2
576.4.a.q 1 56.j odd 6 1
576.4.a.r 1 56.m even 6 1
726.4.a.f 1 231.k odd 6 1
768.4.d.c 2 336.br odd 12 2
768.4.d.n 2 336.bo even 12 2
882.4.a.n 1 7.c even 3 1
882.4.g.f 2 1.a even 1 1 trivial
882.4.g.f 2 7.c even 3 1 inner
882.4.g.i 2 7.b odd 2 1
882.4.g.i 2 7.d odd 6 1
1014.4.a.g 1 273.ba even 6 1
1014.4.b.d 2 273.cb odd 12 2
1200.4.a.b 1 420.be odd 6 1
1200.4.f.j 2 420.br even 12 2
1734.4.a.d 1 357.s even 6 1
2166.4.a.i 1 399.s odd 6 1
2178.4.a.e 1 77.i even 6 1
2352.4.a.e 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
\( T_{11}^{2} - 12T_{11} + 144 \) Copy content Toggle raw display
\( T_{13} + 38 \) 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
$11$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$13$ \( (T + 38)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 126T + 15876 \) Copy content Toggle raw display
$19$ \( T^{2} - 20T + 400 \) Copy content Toggle raw display
$23$ \( T^{2} - 168T + 28224 \) Copy content Toggle raw display
$29$ \( (T + 30)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 88T + 7744 \) Copy content Toggle raw display
$37$ \( T^{2} + 254T + 64516 \) Copy content Toggle raw display
$41$ \( (T - 42)^{2} \) Copy content Toggle raw display
$43$ \( (T + 52)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 96T + 9216 \) Copy content Toggle raw display
$53$ \( T^{2} - 198T + 39204 \) Copy content Toggle raw display
$59$ \( T^{2} - 660T + 435600 \) Copy content Toggle raw display
$61$ \( T^{2} + 538T + 289444 \) Copy content Toggle raw display
$67$ \( T^{2} + 884T + 781456 \) Copy content Toggle raw display
$71$ \( (T + 792)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 218T + 47524 \) Copy content Toggle raw display
$79$ \( T^{2} - 520T + 270400 \) Copy content Toggle raw display
$83$ \( (T + 492)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 810T + 656100 \) Copy content Toggle raw display
$97$ \( (T + 1154)^{2} \) Copy content Toggle raw display
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