Properties

Label 882.2.h.j
Level $882$
Weight $2$
Character orbit 882.h
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.h (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: \(\Z[a_1, a_2]\)
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.

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i 1.50000 + 0.866025i −0.500000 0.866025i −3.00000 1.50000 0.866025i 0 −1.00000 1.50000 + 2.59808i −1.50000 + 2.59808i
79.1 0.500000 + 0.866025i 1.50000 0.866025i −0.500000 + 0.866025i −3.00000 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.g 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.h.j 2
3.b odd 2 1 2646.2.h.e 2
7.b odd 2 1 882.2.h.f 2
7.c even 3 1 126.2.f.a 2
7.c even 3 1 882.2.e.b 2
7.d odd 6 1 882.2.e.d 2
7.d odd 6 1 882.2.f.h 2
9.c even 3 1 882.2.e.b 2
9.d odd 6 1 2646.2.e.f 2
21.c even 2 1 2646.2.h.a 2
21.g even 6 1 2646.2.e.j 2
21.g even 6 1 2646.2.f.c 2
21.h odd 6 1 378.2.f.a 2
21.h odd 6 1 2646.2.e.f 2
28.g odd 6 1 1008.2.r.d 2
63.g even 3 1 inner 882.2.h.j 2
63.g even 3 1 1134.2.a.a 1
63.h even 3 1 126.2.f.a 2
63.i even 6 1 2646.2.f.c 2
63.j odd 6 1 378.2.f.a 2
63.k odd 6 1 882.2.h.f 2
63.k odd 6 1 7938.2.a.l 1
63.l odd 6 1 882.2.e.d 2
63.n odd 6 1 1134.2.a.h 1
63.n odd 6 1 2646.2.h.e 2
63.o even 6 1 2646.2.e.j 2
63.s even 6 1 2646.2.h.a 2
63.s even 6 1 7938.2.a.u 1
63.t odd 6 1 882.2.f.h 2
84.n even 6 1 3024.2.r.a 2
252.o even 6 1 9072.2.a.w 1
252.u odd 6 1 1008.2.r.d 2
252.bb even 6 1 3024.2.r.a 2
252.bl odd 6 1 9072.2.a.c 1
    
        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.h even 3 1
378.2.f.a 2 21.h odd 6 1
378.2.f.a 2 63.j odd 6 1
882.2.e.b 2 7.c even 3 1
882.2.e.b 2 9.c even 3 1
882.2.e.d 2 7.d odd 6 1
882.2.e.d 2 63.l odd 6 1
882.2.f.h 2 7.d odd 6 1
882.2.f.h 2 63.t odd 6 1
882.2.h.f 2 7.b odd 2 1
882.2.h.f 2 63.k odd 6 1
882.2.h.j 2 1.a even 1 1 trivial
882.2.h.j 2 63.g even 3 1 inner
1008.2.r.d 2 28.g odd 6 1
1008.2.r.d 2 252.u odd 6 1
1134.2.a.a 1 63.g even 3 1
1134.2.a.h 1 63.n odd 6 1
2646.2.e.f 2 9.d odd 6 1
2646.2.e.f 2 21.h odd 6 1
2646.2.e.j 2 21.g even 6 1
2646.2.e.j 2 63.o even 6 1
2646.2.f.c 2 21.g even 6 1
2646.2.f.c 2 63.i even 6 1
2646.2.h.a 2 21.c even 2 1
2646.2.h.a 2 63.s even 6 1
2646.2.h.e 2 3.b odd 2 1
2646.2.h.e 2 63.n odd 6 1
3024.2.r.a 2 84.n even 6 1
3024.2.r.a 2 252.bb even 6 1
7938.2.a.l 1 63.k odd 6 1
7938.2.a.u 1 63.s even 6 1
9072.2.a.c 1 252.bl odd 6 1
9072.2.a.w 1 252.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 \(S_{2}^{\mathrm{new}}(882, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 - 3 T + 3 T^{2} \)
$5$ \( ( 1 + 3 T + 5 T^{2} )^{2} \)
$7$ 1
$11$ \( ( 1 + 6 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \)
$17$ \( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \)
$23$ \( ( 1 - 3 T + 23 T^{2} )^{2} \)
$29$ \( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} \)
$31$ \( 1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4} \)
$37$ \( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 41 T^{2} + 1681 T^{4} \)
$43$ \( 1 + 2 T - 39 T^{2} + 86 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 59 T^{2} + 3481 T^{4} \)
$61$ \( 1 + 5 T - 36 T^{2} + 305 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 8 T - 3 T^{2} + 536 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 3 T + 71 T^{2} )^{2} \)
$73$ \( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 5 T - 54 T^{2} + 395 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 89 T^{2} + 7921 T^{4} \)
$97$ \( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} \)
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