Properties

Label 882.4.g.g
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: \(\Z[a_1, \ldots, a_{19}]\)
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.

\(f(q)\) \(=\) \( q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 6 \zeta_{6} q^{5} + 8 q^{8} +O(q^{10})\) \( q -2 \zeta_{6} q^{2} + ( -4 + 4 \zeta_{6} ) q^{4} + 6 \zeta_{6} q^{5} + 8 q^{8} + ( 12 - 12 \zeta_{6} ) q^{10} + ( -30 + 30 \zeta_{6} ) q^{11} -53 q^{13} -16 \zeta_{6} q^{16} + ( 84 - 84 \zeta_{6} ) q^{17} -97 \zeta_{6} q^{19} -24 q^{20} + 60 q^{22} + 84 \zeta_{6} q^{23} + ( 89 - 89 \zeta_{6} ) q^{25} + 106 \zeta_{6} q^{26} + 180 q^{29} + ( 179 - 179 \zeta_{6} ) q^{31} + ( -32 + 32 \zeta_{6} ) q^{32} -168 q^{34} + 145 \zeta_{6} q^{37} + ( -194 + 194 \zeta_{6} ) q^{38} + 48 \zeta_{6} q^{40} + 126 q^{41} -325 q^{43} -120 \zeta_{6} q^{44} + ( 168 - 168 \zeta_{6} ) q^{46} + 366 \zeta_{6} q^{47} -178 q^{50} + ( 212 - 212 \zeta_{6} ) q^{52} + ( -768 + 768 \zeta_{6} ) q^{53} -180 q^{55} -360 \zeta_{6} q^{58} + ( 264 - 264 \zeta_{6} ) q^{59} + 818 \zeta_{6} q^{61} -358 q^{62} + 64 q^{64} -318 \zeta_{6} q^{65} + ( 523 - 523 \zeta_{6} ) q^{67} + 336 \zeta_{6} q^{68} + 342 q^{71} + ( -43 + 43 \zeta_{6} ) q^{73} + ( 290 - 290 \zeta_{6} ) q^{74} + 388 q^{76} + 1171 \zeta_{6} q^{79} + ( 96 - 96 \zeta_{6} ) q^{80} -252 \zeta_{6} q^{82} -810 q^{83} + 504 q^{85} + 650 \zeta_{6} q^{86} + ( -240 + 240 \zeta_{6} ) q^{88} + 600 \zeta_{6} q^{89} -336 q^{92} + ( 732 - 732 \zeta_{6} ) q^{94} + ( 582 - 582 \zeta_{6} ) q^{95} -386 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 4q^{4} + 6q^{5} + 16q^{8} + O(q^{10}) \) \( 2q - 2q^{2} - 4q^{4} + 6q^{5} + 16q^{8} + 12q^{10} - 30q^{11} - 106q^{13} - 16q^{16} + 84q^{17} - 97q^{19} - 48q^{20} + 120q^{22} + 84q^{23} + 89q^{25} + 106q^{26} + 360q^{29} + 179q^{31} - 32q^{32} - 336q^{34} + 145q^{37} - 194q^{38} + 48q^{40} + 252q^{41} - 650q^{43} - 120q^{44} + 168q^{46} + 366q^{47} - 356q^{50} + 212q^{52} - 768q^{53} - 360q^{55} - 360q^{58} + 264q^{59} + 818q^{61} - 716q^{62} + 128q^{64} - 318q^{65} + 523q^{67} + 336q^{68} + 684q^{71} - 43q^{73} + 290q^{74} + 776q^{76} + 1171q^{79} + 96q^{80} - 252q^{82} - 1620q^{83} + 1008q^{85} + 650q^{86} - 240q^{88} + 600q^{89} - 672q^{92} + 732q^{94} + 582q^{95} - 772q^{97} + O(q^{100}) \)

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 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.g 2
3.b odd 2 1 294.4.e.i 2
7.b odd 2 1 126.4.g.b 2
7.c even 3 1 882.4.a.l 1
7.c even 3 1 inner 882.4.g.g 2
7.d odd 6 1 126.4.g.b 2
7.d odd 6 1 882.4.a.o 1
21.c even 2 1 42.4.e.a 2
21.g even 6 1 42.4.e.a 2
21.g even 6 1 294.4.a.d 1
21.h odd 6 1 294.4.a.c 1
21.h odd 6 1 294.4.e.i 2
84.h odd 2 1 336.4.q.f 2
84.j odd 6 1 336.4.q.f 2
84.j odd 6 1 2352.4.a.f 1
84.n even 6 1 2352.4.a.bf 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.a 2 21.c even 2 1
42.4.e.a 2 21.g even 6 1
126.4.g.b 2 7.b odd 2 1
126.4.g.b 2 7.d odd 6 1
294.4.a.c 1 21.h odd 6 1
294.4.a.d 1 21.g even 6 1
294.4.e.i 2 3.b odd 2 1
294.4.e.i 2 21.h odd 6 1
336.4.q.f 2 84.h odd 2 1
336.4.q.f 2 84.j odd 6 1
882.4.a.l 1 7.c even 3 1
882.4.a.o 1 7.d odd 6 1
882.4.g.g 2 1.a even 1 1 trivial
882.4.g.g 2 7.c even 3 1 inner
2352.4.a.f 1 84.j odd 6 1
2352.4.a.bf 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 \(S_{4}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{2} - 6 T_{5} + 36 \)
\( T_{11}^{2} + 30 T_{11} + 900 \)
\( T_{13} + 53 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + 2 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 36 - 6 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 900 + 30 T + T^{2} \)
$13$ \( ( 53 + T )^{2} \)
$17$ \( 7056 - 84 T + T^{2} \)
$19$ \( 9409 + 97 T + T^{2} \)
$23$ \( 7056 - 84 T + T^{2} \)
$29$ \( ( -180 + T )^{2} \)
$31$ \( 32041 - 179 T + T^{2} \)
$37$ \( 21025 - 145 T + T^{2} \)
$41$ \( ( -126 + T )^{2} \)
$43$ \( ( 325 + T )^{2} \)
$47$ \( 133956 - 366 T + T^{2} \)
$53$ \( 589824 + 768 T + T^{2} \)
$59$ \( 69696 - 264 T + T^{2} \)
$61$ \( 669124 - 818 T + T^{2} \)
$67$ \( 273529 - 523 T + T^{2} \)
$71$ \( ( -342 + T )^{2} \)
$73$ \( 1849 + 43 T + T^{2} \)
$79$ \( 1371241 - 1171 T + T^{2} \)
$83$ \( ( 810 + T )^{2} \)
$89$ \( 360000 - 600 T + T^{2} \)
$97$ \( ( 386 + T )^{2} \)
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