Properties

Label 882.2.f.i
Level 882
Weight 2
Character orbit 882.f
Analytic conductor 7.043
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.04280545828\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{2} + ( 2 - \zeta_{6} ) q^{3} -\zeta_{6} q^{4} -3 \zeta_{6} q^{5} + ( 1 - 2 \zeta_{6} ) q^{6} - q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{2} + ( 2 - \zeta_{6} ) q^{3} -\zeta_{6} q^{4} -3 \zeta_{6} q^{5} + ( 1 - 2 \zeta_{6} ) q^{6} - q^{8} + ( 3 - 3 \zeta_{6} ) q^{9} -3 q^{10} + ( 3 - 3 \zeta_{6} ) q^{11} + ( -1 - \zeta_{6} ) q^{12} + \zeta_{6} q^{13} + ( -3 - 3 \zeta_{6} ) q^{15} + ( -1 + \zeta_{6} ) q^{16} + 3 q^{17} -3 \zeta_{6} q^{18} -7 q^{19} + ( -3 + 3 \zeta_{6} ) q^{20} -3 \zeta_{6} q^{22} + 9 \zeta_{6} q^{23} + ( -2 + \zeta_{6} ) q^{24} + ( -4 + 4 \zeta_{6} ) q^{25} + q^{26} + ( 3 - 6 \zeta_{6} ) q^{27} + ( -3 + 3 \zeta_{6} ) q^{29} + ( -6 + 3 \zeta_{6} ) q^{30} -8 \zeta_{6} q^{31} + \zeta_{6} q^{32} + ( 3 - 6 \zeta_{6} ) q^{33} + ( 3 - 3 \zeta_{6} ) q^{34} -3 q^{36} - q^{37} + ( -7 + 7 \zeta_{6} ) q^{38} + ( 1 + \zeta_{6} ) q^{39} + 3 \zeta_{6} q^{40} -3 \zeta_{6} q^{41} + ( 1 - \zeta_{6} ) q^{43} -3 q^{44} -9 q^{45} + 9 q^{46} + ( -1 + 2 \zeta_{6} ) q^{48} + 4 \zeta_{6} q^{50} + ( 6 - 3 \zeta_{6} ) q^{51} + ( 1 - \zeta_{6} ) q^{52} + 3 q^{53} + ( -3 - 3 \zeta_{6} ) q^{54} -9 q^{55} + ( -14 + 7 \zeta_{6} ) q^{57} + 3 \zeta_{6} q^{58} + ( -3 + 6 \zeta_{6} ) q^{60} + ( -2 + 2 \zeta_{6} ) q^{61} -8 q^{62} + q^{64} + ( 3 - 3 \zeta_{6} ) q^{65} + ( -3 - 3 \zeta_{6} ) q^{66} + 4 \zeta_{6} q^{67} -3 \zeta_{6} q^{68} + ( 9 + 9 \zeta_{6} ) q^{69} + 12 q^{71} + ( -3 + 3 \zeta_{6} ) q^{72} + 11 q^{73} + ( -1 + \zeta_{6} ) q^{74} + ( -4 + 8 \zeta_{6} ) q^{75} + 7 \zeta_{6} q^{76} + ( 2 - \zeta_{6} ) q^{78} + ( 16 - 16 \zeta_{6} ) q^{79} + 3 q^{80} -9 \zeta_{6} q^{81} -3 q^{82} + ( 9 - 9 \zeta_{6} ) q^{83} -9 \zeta_{6} q^{85} -\zeta_{6} q^{86} + ( -3 + 6 \zeta_{6} ) q^{87} + ( -3 + 3 \zeta_{6} ) q^{88} + 3 q^{89} + ( -9 + 9 \zeta_{6} ) q^{90} + ( 9 - 9 \zeta_{6} ) q^{92} + ( -8 - 8 \zeta_{6} ) q^{93} + 21 \zeta_{6} q^{95} + ( 1 + \zeta_{6} ) q^{96} + ( 1 - \zeta_{6} ) q^{97} -9 \zeta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 3q^{3} - q^{4} - 3q^{5} - 2q^{8} + 3q^{9} + O(q^{10}) \) \( 2q + q^{2} + 3q^{3} - q^{4} - 3q^{5} - 2q^{8} + 3q^{9} - 6q^{10} + 3q^{11} - 3q^{12} + q^{13} - 9q^{15} - q^{16} + 6q^{17} - 3q^{18} - 14q^{19} - 3q^{20} - 3q^{22} + 9q^{23} - 3q^{24} - 4q^{25} + 2q^{26} - 3q^{29} - 9q^{30} - 8q^{31} + q^{32} + 3q^{34} - 6q^{36} - 2q^{37} - 7q^{38} + 3q^{39} + 3q^{40} - 3q^{41} + q^{43} - 6q^{44} - 18q^{45} + 18q^{46} + 4q^{50} + 9q^{51} + q^{52} + 6q^{53} - 9q^{54} - 18q^{55} - 21q^{57} + 3q^{58} - 2q^{61} - 16q^{62} + 2q^{64} + 3q^{65} - 9q^{66} + 4q^{67} - 3q^{68} + 27q^{69} + 24q^{71} - 3q^{72} + 22q^{73} - q^{74} + 7q^{76} + 3q^{78} + 16q^{79} + 6q^{80} - 9q^{81} - 6q^{82} + 9q^{83} - 9q^{85} - q^{86} - 3q^{88} + 6q^{89} - 9q^{90} + 9q^{92} - 24q^{93} + 21q^{95} + 3q^{96} + q^{97} - 9q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
295.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i 1.50000 0.866025i −0.500000 0.866025i −1.50000 2.59808i 1.73205i 0 −1.00000 1.50000 2.59808i −3.00000
589.1 0.500000 + 0.866025i 1.50000 + 0.866025i −0.500000 + 0.866025i −1.50000 + 2.59808i 1.73205i 0 −1.00000 1.50000 + 2.59808i −3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.2.f.i 2
3.b odd 2 1 2646.2.f.d 2
7.b odd 2 1 882.2.f.g 2
7.c even 3 1 126.2.e.a 2
7.c even 3 1 126.2.h.b yes 2
7.d odd 6 1 882.2.e.c 2
7.d odd 6 1 882.2.h.i 2
9.c even 3 1 inner 882.2.f.i 2
9.c even 3 1 7938.2.a.m 1
9.d odd 6 1 2646.2.f.d 2
9.d odd 6 1 7938.2.a.t 1
21.c even 2 1 2646.2.f.a 2
21.g even 6 1 2646.2.e.g 2
21.g even 6 1 2646.2.h.d 2
21.h odd 6 1 378.2.e.b 2
21.h odd 6 1 378.2.h.a 2
28.g odd 6 1 1008.2.q.a 2
28.g odd 6 1 1008.2.t.f 2
63.g even 3 1 126.2.e.a 2
63.g even 3 1 1134.2.g.e 2
63.h even 3 1 126.2.h.b yes 2
63.h even 3 1 1134.2.g.e 2
63.i even 6 1 2646.2.h.d 2
63.j odd 6 1 378.2.h.a 2
63.j odd 6 1 1134.2.g.c 2
63.k odd 6 1 882.2.e.c 2
63.l odd 6 1 882.2.f.g 2
63.l odd 6 1 7938.2.a.b 1
63.n odd 6 1 378.2.e.b 2
63.n odd 6 1 1134.2.g.c 2
63.o even 6 1 2646.2.f.a 2
63.o even 6 1 7938.2.a.be 1
63.s even 6 1 2646.2.e.g 2
63.t odd 6 1 882.2.h.i 2
84.n even 6 1 3024.2.q.f 2
84.n even 6 1 3024.2.t.a 2
252.o even 6 1 3024.2.q.f 2
252.u odd 6 1 1008.2.t.f 2
252.bb even 6 1 3024.2.t.a 2
252.bl odd 6 1 1008.2.q.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.a 2 7.c even 3 1
126.2.e.a 2 63.g even 3 1
126.2.h.b yes 2 7.c even 3 1
126.2.h.b yes 2 63.h even 3 1
378.2.e.b 2 21.h odd 6 1
378.2.e.b 2 63.n odd 6 1
378.2.h.a 2 21.h odd 6 1
378.2.h.a 2 63.j odd 6 1
882.2.e.c 2 7.d odd 6 1
882.2.e.c 2 63.k odd 6 1
882.2.f.g 2 7.b odd 2 1
882.2.f.g 2 63.l odd 6 1
882.2.f.i 2 1.a even 1 1 trivial
882.2.f.i 2 9.c even 3 1 inner
882.2.h.i 2 7.d odd 6 1
882.2.h.i 2 63.t odd 6 1
1008.2.q.a 2 28.g odd 6 1
1008.2.q.a 2 252.bl odd 6 1
1008.2.t.f 2 28.g odd 6 1
1008.2.t.f 2 252.u odd 6 1
1134.2.g.c 2 63.j odd 6 1
1134.2.g.c 2 63.n odd 6 1
1134.2.g.e 2 63.g even 3 1
1134.2.g.e 2 63.h even 3 1
2646.2.e.g 2 21.g even 6 1
2646.2.e.g 2 63.s even 6 1
2646.2.f.a 2 21.c even 2 1
2646.2.f.a 2 63.o even 6 1
2646.2.f.d 2 3.b odd 2 1
2646.2.f.d 2 9.d odd 6 1
2646.2.h.d 2 21.g even 6 1
2646.2.h.d 2 63.i even 6 1
3024.2.q.f 2 84.n even 6 1
3024.2.q.f 2 252.o even 6 1
3024.2.t.a 2 84.n even 6 1
3024.2.t.a 2 252.bb even 6 1
7938.2.a.b 1 63.l odd 6 1
7938.2.a.m 1 9.c even 3 1
7938.2.a.t 1 9.d odd 6 1
7938.2.a.be 1 63.o even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{2} + 3 T_{5} + 9 \)
\( T_{11}^{2} - 3 T_{11} + 9 \)
\( T_{13}^{2} - T_{13} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 - 3 T + 3 T^{2} \)
$5$ \( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \)
$7$ 1
$11$ \( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \)
$13$ \( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} \)
$17$ \( ( 1 - 3 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 + 7 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4} \)
$29$ \( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} \)
$31$ \( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \)
$37$ \( ( 1 + T + 37 T^{2} )^{2} \)
$41$ \( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} \)
$43$ \( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 - 3 T + 53 T^{2} )^{2} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 11 T + 73 T^{2} )^{2} \)
$79$ \( 1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \)
$89$ \( ( 1 - 3 T + 89 T^{2} )^{2} \)
$97$ \( 1 - T - 96 T^{2} - 97 T^{3} + 9409 T^{4} \)
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