Properties

Label 882.2.e.b
Level 882
Weight 2
Character orbit 882.e
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.e (of order \(3\), degree \(2\), not 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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q - q^{2} + ( -1 - \zeta_{6} ) q^{3} + q^{4} + ( 3 - 3 \zeta_{6} ) q^{5} + ( 1 + \zeta_{6} ) q^{6} - q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q - q^{2} + ( -1 - \zeta_{6} ) q^{3} + q^{4} + ( 3 - 3 \zeta_{6} ) q^{5} + ( 1 + \zeta_{6} ) q^{6} - q^{8} + 3 \zeta_{6} q^{9} + ( -3 + 3 \zeta_{6} ) q^{10} + 6 \zeta_{6} q^{11} + ( -1 - \zeta_{6} ) q^{12} -2 \zeta_{6} q^{13} + ( -6 + 3 \zeta_{6} ) q^{15} + q^{16} + ( -6 + 6 \zeta_{6} ) q^{17} -3 \zeta_{6} q^{18} + 7 \zeta_{6} q^{19} + ( 3 - 3 \zeta_{6} ) q^{20} -6 \zeta_{6} q^{22} + ( -3 + 3 \zeta_{6} ) q^{23} + ( 1 + \zeta_{6} ) q^{24} -4 \zeta_{6} q^{25} + 2 \zeta_{6} q^{26} + ( 3 - 6 \zeta_{6} ) q^{27} + ( -6 + 6 \zeta_{6} ) q^{29} + ( 6 - 3 \zeta_{6} ) q^{30} + 2 q^{31} - q^{32} + ( 6 - 12 \zeta_{6} ) q^{33} + ( 6 - 6 \zeta_{6} ) q^{34} + 3 \zeta_{6} q^{36} -2 \zeta_{6} q^{37} -7 \zeta_{6} q^{38} + ( -2 + 4 \zeta_{6} ) q^{39} + ( -3 + 3 \zeta_{6} ) q^{40} + ( -2 + 2 \zeta_{6} ) q^{43} + 6 \zeta_{6} q^{44} + 9 q^{45} + ( 3 - 3 \zeta_{6} ) q^{46} + ( -1 - \zeta_{6} ) q^{48} + 4 \zeta_{6} q^{50} + ( 12 - 6 \zeta_{6} ) q^{51} -2 \zeta_{6} q^{52} + ( -6 + 6 \zeta_{6} ) q^{53} + ( -3 + 6 \zeta_{6} ) q^{54} + 18 q^{55} + ( 7 - 14 \zeta_{6} ) q^{57} + ( 6 - 6 \zeta_{6} ) q^{58} + ( -6 + 3 \zeta_{6} ) q^{60} + 5 q^{61} -2 q^{62} + q^{64} -6 q^{65} + ( -6 + 12 \zeta_{6} ) q^{66} + 8 q^{67} + ( -6 + 6 \zeta_{6} ) q^{68} + ( 6 - 3 \zeta_{6} ) q^{69} + 3 q^{71} -3 \zeta_{6} q^{72} + ( -2 + 2 \zeta_{6} ) q^{73} + 2 \zeta_{6} q^{74} + ( -4 + 8 \zeta_{6} ) q^{75} + 7 \zeta_{6} q^{76} + ( 2 - 4 \zeta_{6} ) q^{78} + 5 q^{79} + ( 3 - 3 \zeta_{6} ) q^{80} + ( -9 + 9 \zeta_{6} ) q^{81} + ( -12 + 12 \zeta_{6} ) q^{83} + 18 \zeta_{6} q^{85} + ( 2 - 2 \zeta_{6} ) q^{86} + ( 12 - 6 \zeta_{6} ) q^{87} -6 \zeta_{6} q^{88} -9 q^{90} + ( -3 + 3 \zeta_{6} ) q^{92} + ( -2 - 2 \zeta_{6} ) q^{93} + 21 q^{95} + ( 1 + \zeta_{6} ) q^{96} + ( -2 + 2 \zeta_{6} ) q^{97} + ( -18 + 18 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 3q^{3} + 2q^{4} + 3q^{5} + 3q^{6} - 2q^{8} + 3q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 3q^{3} + 2q^{4} + 3q^{5} + 3q^{6} - 2q^{8} + 3q^{9} - 3q^{10} + 6q^{11} - 3q^{12} - 2q^{13} - 9q^{15} + 2q^{16} - 6q^{17} - 3q^{18} + 7q^{19} + 3q^{20} - 6q^{22} - 3q^{23} + 3q^{24} - 4q^{25} + 2q^{26} - 6q^{29} + 9q^{30} + 4q^{31} - 2q^{32} + 6q^{34} + 3q^{36} - 2q^{37} - 7q^{38} - 3q^{40} - 2q^{43} + 6q^{44} + 18q^{45} + 3q^{46} - 3q^{48} + 4q^{50} + 18q^{51} - 2q^{52} - 6q^{53} + 36q^{55} + 6q^{58} - 9q^{60} + 10q^{61} - 4q^{62} + 2q^{64} - 12q^{65} + 16q^{67} - 6q^{68} + 9q^{69} + 6q^{71} - 3q^{72} - 2q^{73} + 2q^{74} + 7q^{76} + 10q^{79} + 3q^{80} - 9q^{81} - 12q^{83} + 18q^{85} + 2q^{86} + 18q^{87} - 6q^{88} - 18q^{90} - 3q^{92} - 6q^{93} + 42q^{95} + 3q^{96} - 2q^{97} - 18q^{99} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
373.1
0.500000 0.866025i
0.500000 + 0.866025i
−1.00000 −1.50000 + 0.866025i 1.00000 1.50000 + 2.59808i 1.50000 0.866025i 0 −1.00000 1.50000 2.59808i −1.50000 2.59808i
655.1 −1.00000 −1.50000 0.866025i 1.00000 1.50000 2.59808i 1.50000 + 0.866025i 0 −1.00000 1.50000 + 2.59808i −1.50000 + 2.59808i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h 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.e.b 2
3.b odd 2 1 2646.2.e.f 2
7.b odd 2 1 882.2.e.d 2
7.c even 3 1 126.2.f.a 2
7.c even 3 1 882.2.h.j 2
7.d odd 6 1 882.2.f.h 2
7.d odd 6 1 882.2.h.f 2
9.c even 3 1 882.2.h.j 2
9.d odd 6 1 2646.2.h.e 2
21.c even 2 1 2646.2.e.j 2
21.g even 6 1 2646.2.f.c 2
21.g even 6 1 2646.2.h.a 2
21.h odd 6 1 378.2.f.a 2
21.h odd 6 1 2646.2.h.e 2
28.g odd 6 1 1008.2.r.d 2
63.g even 3 1 126.2.f.a 2
63.h even 3 1 inner 882.2.e.b 2
63.h even 3 1 1134.2.a.a 1
63.i even 6 1 2646.2.e.j 2
63.i even 6 1 7938.2.a.u 1
63.j odd 6 1 1134.2.a.h 1
63.j odd 6 1 2646.2.e.f 2
63.k odd 6 1 882.2.f.h 2
63.l odd 6 1 882.2.h.f 2
63.n odd 6 1 378.2.f.a 2
63.o even 6 1 2646.2.h.a 2
63.s even 6 1 2646.2.f.c 2
63.t odd 6 1 882.2.e.d 2
63.t odd 6 1 7938.2.a.l 1
84.n even 6 1 3024.2.r.a 2
252.o even 6 1 3024.2.r.a 2
252.u odd 6 1 9072.2.a.c 1
252.bb even 6 1 9072.2.a.w 1
252.bl odd 6 1 1008.2.r.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.a 2 7.c even 3 1
126.2.f.a 2 63.g even 3 1
378.2.f.a 2 21.h odd 6 1
378.2.f.a 2 63.n odd 6 1
882.2.e.b 2 1.a even 1 1 trivial
882.2.e.b 2 63.h even 3 1 inner
882.2.e.d 2 7.b odd 2 1
882.2.e.d 2 63.t odd 6 1
882.2.f.h 2 7.d odd 6 1
882.2.f.h 2 63.k odd 6 1
882.2.h.f 2 7.d odd 6 1
882.2.h.f 2 63.l odd 6 1
882.2.h.j 2 7.c even 3 1
882.2.h.j 2 9.c even 3 1
1008.2.r.d 2 28.g odd 6 1
1008.2.r.d 2 252.bl odd 6 1
1134.2.a.a 1 63.h even 3 1
1134.2.a.h 1 63.j odd 6 1
2646.2.e.f 2 3.b odd 2 1
2646.2.e.f 2 63.j odd 6 1
2646.2.e.j 2 21.c even 2 1
2646.2.e.j 2 63.i even 6 1
2646.2.f.c 2 21.g even 6 1
2646.2.f.c 2 63.s even 6 1
2646.2.h.a 2 21.g even 6 1
2646.2.h.a 2 63.o even 6 1
2646.2.h.e 2 9.d odd 6 1
2646.2.h.e 2 21.h odd 6 1
3024.2.r.a 2 84.n even 6 1
3024.2.r.a 2 252.o even 6 1
7938.2.a.l 1 63.t odd 6 1
7938.2.a.u 1 63.i even 6 1
9072.2.a.c 1 252.u odd 6 1
9072.2.a.w 1 252.bb even 6 1

Hecke kernels

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

\( T_{5}^{2} - 3 T_{5} + 9 \)
\( T_{11}^{2} - 6 T_{11} + 36 \)
\( T_{13}^{2} + 2 T_{13} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( 1 + 3 T + 3 T^{2} \)
$5$ \( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} \)
$7$ 1
$11$ \( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \)
$17$ \( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \)
$23$ \( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \)
$29$ \( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} \)
$31$ \( ( 1 - 2 T + 31 T^{2} )^{2} \)
$37$ \( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 41 T^{2} + 1681 T^{4} \)
$43$ \( 1 + 2 T - 39 T^{2} + 86 T^{3} + 1849 T^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 5 T + 61 T^{2} )^{2} \)
$67$ \( ( 1 - 8 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 - 3 T + 71 T^{2} )^{2} \)
$73$ \( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 - 5 T + 79 T^{2} )^{2} \)
$83$ \( 1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 89 T^{2} + 7921 T^{4} \)
$97$ \( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \)
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