Properties

Label 882.2.e.m
Level $882$
Weight $2$
Character orbit 882.e
Analytic conductor $7.043$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{3} + q^{4} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{6} + q^{8} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{9} +O(q^{10})\) \( q + q^{2} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{3} + q^{4} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{6} + q^{8} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{9} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{10} + ( -2 + 2 \beta_{2} ) q^{11} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{12} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{13} + ( -3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{15} + q^{16} -2 \beta_{2} q^{17} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{18} + ( -5 + \beta_{1} + 5 \beta_{2} - 2 \beta_{3} ) q^{19} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{20} + ( -2 + 2 \beta_{2} ) q^{22} + \beta_{2} q^{23} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{24} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{25} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{26} + ( -5 - \beta_{3} ) q^{27} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{29} + ( -3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{30} + 6 q^{31} + q^{32} + ( 2 - 2 \beta_{3} ) q^{33} -2 \beta_{2} q^{34} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{36} + ( -2 - 4 \beta_{1} + 2 \beta_{2} + 8 \beta_{3} ) q^{37} + ( -5 + \beta_{1} + 5 \beta_{2} - 2 \beta_{3} ) q^{38} + ( 4 + 4 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{39} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{40} + ( -4 \beta_{1} + 8 \beta_{3} ) q^{41} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{43} + ( -2 + 2 \beta_{2} ) q^{44} + ( -5 + 2 \beta_{1} + 8 \beta_{2} - 3 \beta_{3} ) q^{45} + \beta_{2} q^{46} + ( 8 \beta_{1} - 4 \beta_{3} ) q^{47} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{48} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{50} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{51} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{52} + ( 2 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{53} + ( -5 - \beta_{3} ) q^{54} + ( 2 - 4 \beta_{1} + 2 \beta_{3} ) q^{55} + ( 3 - 2 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{57} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{58} -2 q^{59} + ( -3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{60} + ( 9 - 2 \beta_{1} + \beta_{3} ) q^{61} + 6 q^{62} + q^{64} + ( -12 + 4 \beta_{1} - 2 \beta_{3} ) q^{65} + ( 2 - 2 \beta_{3} ) q^{66} + ( -8 - 4 \beta_{1} + 2 \beta_{3} ) q^{67} -2 \beta_{2} q^{68} + ( 1 - \beta_{1} - \beta_{2} ) q^{69} + ( 5 - 4 \beta_{1} + 2 \beta_{3} ) q^{71} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{72} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{73} + ( -2 - 4 \beta_{1} + 2 \beta_{2} + 8 \beta_{3} ) q^{74} + ( -2 - 4 \beta_{1} + 8 \beta_{2} ) q^{75} + ( -5 + \beta_{1} + 5 \beta_{2} - 2 \beta_{3} ) q^{76} + ( 4 + 4 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{78} + ( 3 - 4 \beta_{1} + 2 \beta_{3} ) q^{79} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{80} + ( 4 \beta_{1} + 7 \beta_{2} - 4 \beta_{3} ) q^{81} + ( -4 \beta_{1} + 8 \beta_{3} ) q^{82} -2 \beta_{2} q^{83} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{85} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{86} + ( -2 - 6 \beta_{2} - 4 \beta_{3} ) q^{87} + ( -2 + 2 \beta_{2} ) q^{88} + ( 12 + 2 \beta_{1} - 12 \beta_{2} - 4 \beta_{3} ) q^{89} + ( -5 + 2 \beta_{1} + 8 \beta_{2} - 3 \beta_{3} ) q^{90} + \beta_{2} q^{92} + ( -6 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{93} + ( 8 \beta_{1} - 4 \beta_{3} ) q^{94} + ( 11 - 12 \beta_{1} + 6 \beta_{3} ) q^{95} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{96} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{97} + ( -4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - 2q^{3} + 4q^{4} - 2q^{5} - 2q^{6} + 4q^{8} + 2q^{9} + O(q^{10}) \) \( 4q + 4q^{2} - 2q^{3} + 4q^{4} - 2q^{5} - 2q^{6} + 4q^{8} + 2q^{9} - 2q^{10} - 4q^{11} - 2q^{12} - 14q^{15} + 4q^{16} - 4q^{17} + 2q^{18} - 10q^{19} - 2q^{20} - 4q^{22} + 2q^{23} - 2q^{24} - 4q^{25} - 20q^{27} + 4q^{29} - 14q^{30} + 24q^{31} + 4q^{32} + 8q^{33} - 4q^{34} + 2q^{36} - 4q^{37} - 10q^{38} - 2q^{40} - 4q^{43} - 4q^{44} - 4q^{45} + 2q^{46} - 2q^{48} - 4q^{50} - 4q^{51} + 12q^{53} - 20q^{54} + 8q^{55} + 20q^{57} + 4q^{58} - 8q^{59} - 14q^{60} + 36q^{61} + 24q^{62} + 4q^{64} - 48q^{65} + 8q^{66} - 32q^{67} - 4q^{68} + 2q^{69} + 20q^{71} + 2q^{72} + 4q^{73} - 4q^{74} + 8q^{75} - 10q^{76} + 12q^{79} - 2q^{80} + 14q^{81} - 4q^{83} - 4q^{85} - 4q^{86} - 20q^{87} - 4q^{88} + 24q^{89} - 4q^{90} + 2q^{92} - 12q^{93} + 44q^{95} - 2q^{96} + 4q^{97} + 4q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-1 + \beta_{2}\) \(-1 + \beta_{2}\)

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
1.22474 + 0.707107i
−1.22474 0.707107i
1.22474 0.707107i
−1.22474 + 0.707107i
1.00000 −1.72474 0.158919i 1.00000 0.724745 + 1.25529i −1.72474 0.158919i 0 1.00000 2.94949 + 0.548188i 0.724745 + 1.25529i
373.2 1.00000 0.724745 1.57313i 1.00000 −1.72474 2.98735i 0.724745 1.57313i 0 1.00000 −1.94949 2.28024i −1.72474 2.98735i
655.1 1.00000 −1.72474 + 0.158919i 1.00000 0.724745 1.25529i −1.72474 + 0.158919i 0 1.00000 2.94949 0.548188i 0.724745 1.25529i
655.2 1.00000 0.724745 + 1.57313i 1.00000 −1.72474 + 2.98735i 0.724745 + 1.57313i 0 1.00000 −1.94949 + 2.28024i −1.72474 + 2.98735i
\(n\): 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.m 4
3.b odd 2 1 2646.2.e.l 4
7.b odd 2 1 882.2.e.n 4
7.c even 3 1 126.2.f.c 4
7.c even 3 1 882.2.h.k 4
7.d odd 6 1 882.2.f.j 4
7.d odd 6 1 882.2.h.l 4
9.c even 3 1 882.2.h.k 4
9.d odd 6 1 2646.2.h.m 4
21.c even 2 1 2646.2.e.k 4
21.g even 6 1 2646.2.f.k 4
21.g even 6 1 2646.2.h.n 4
21.h odd 6 1 378.2.f.d 4
21.h odd 6 1 2646.2.h.m 4
28.g odd 6 1 1008.2.r.e 4
63.g even 3 1 126.2.f.c 4
63.h even 3 1 inner 882.2.e.m 4
63.h even 3 1 1134.2.a.p 2
63.i even 6 1 2646.2.e.k 4
63.i even 6 1 7938.2.a.bm 2
63.j odd 6 1 1134.2.a.i 2
63.j odd 6 1 2646.2.e.l 4
63.k odd 6 1 882.2.f.j 4
63.l odd 6 1 882.2.h.l 4
63.n odd 6 1 378.2.f.d 4
63.o even 6 1 2646.2.h.n 4
63.s even 6 1 2646.2.f.k 4
63.t odd 6 1 882.2.e.n 4
63.t odd 6 1 7938.2.a.bn 2
84.n even 6 1 3024.2.r.e 4
252.o even 6 1 3024.2.r.e 4
252.u odd 6 1 9072.2.a.bk 2
252.bb even 6 1 9072.2.a.bd 2
252.bl odd 6 1 1008.2.r.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.c 4 7.c even 3 1
126.2.f.c 4 63.g even 3 1
378.2.f.d 4 21.h odd 6 1
378.2.f.d 4 63.n odd 6 1
882.2.e.m 4 1.a even 1 1 trivial
882.2.e.m 4 63.h even 3 1 inner
882.2.e.n 4 7.b odd 2 1
882.2.e.n 4 63.t odd 6 1
882.2.f.j 4 7.d odd 6 1
882.2.f.j 4 63.k odd 6 1
882.2.h.k 4 7.c even 3 1
882.2.h.k 4 9.c even 3 1
882.2.h.l 4 7.d odd 6 1
882.2.h.l 4 63.l odd 6 1
1008.2.r.e 4 28.g odd 6 1
1008.2.r.e 4 252.bl odd 6 1
1134.2.a.i 2 63.j odd 6 1
1134.2.a.p 2 63.h even 3 1
2646.2.e.k 4 21.c even 2 1
2646.2.e.k 4 63.i even 6 1
2646.2.e.l 4 3.b odd 2 1
2646.2.e.l 4 63.j odd 6 1
2646.2.f.k 4 21.g even 6 1
2646.2.f.k 4 63.s even 6 1
2646.2.h.m 4 9.d odd 6 1
2646.2.h.m 4 21.h odd 6 1
2646.2.h.n 4 21.g even 6 1
2646.2.h.n 4 63.o even 6 1
3024.2.r.e 4 84.n even 6 1
3024.2.r.e 4 252.o even 6 1
7938.2.a.bm 2 63.i even 6 1
7938.2.a.bn 2 63.t odd 6 1
9072.2.a.bd 2 252.bb even 6 1
9072.2.a.bk 2 252.u odd 6 1

Hecke kernels

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

\( T_{5}^{4} + 2 T_{5}^{3} + 9 T_{5}^{2} - 10 T_{5} + 25 \)
\( T_{11}^{2} + 2 T_{11} + 4 \)
\( T_{13}^{4} + 24 T_{13}^{2} + 576 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{4} \)
$3$ \( 1 + 2 T + T^{2} + 6 T^{3} + 9 T^{4} \)
$5$ \( 1 + 2 T - T^{2} - 10 T^{3} - 20 T^{4} - 50 T^{5} - 25 T^{6} + 250 T^{7} + 625 T^{8} \)
$7$ 1
$11$ \( ( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} )^{2} \)
$13$ \( 1 - 2 T^{2} - 165 T^{4} - 338 T^{6} + 28561 T^{8} \)
$17$ \( ( 1 + 2 T - 13 T^{2} + 34 T^{3} + 289 T^{4} )^{2} \)
$19$ \( 1 + 10 T + 43 T^{2} + 190 T^{3} + 988 T^{4} + 3610 T^{5} + 15523 T^{6} + 68590 T^{7} + 130321 T^{8} \)
$23$ \( ( 1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4} )^{2} \)
$29$ \( 1 - 4 T - 22 T^{2} + 80 T^{3} + 139 T^{4} + 2320 T^{5} - 18502 T^{6} - 97556 T^{7} + 707281 T^{8} \)
$31$ \( ( 1 - 6 T + 31 T^{2} )^{4} \)
$37$ \( 1 + 4 T + 34 T^{2} - 368 T^{3} - 1637 T^{4} - 13616 T^{5} + 46546 T^{6} + 202612 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 14 T^{2} - 1485 T^{4} + 23534 T^{6} + 2825761 T^{8} \)
$43$ \( 1 + 4 T - 50 T^{2} - 80 T^{3} + 1819 T^{4} - 3440 T^{5} - 92450 T^{6} + 318028 T^{7} + 3418801 T^{8} \)
$47$ \( ( 1 - 2 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( 1 - 12 T + 26 T^{2} - 144 T^{3} + 3483 T^{4} - 7632 T^{5} + 73034 T^{6} - 1786524 T^{7} + 7890481 T^{8} \)
$59$ \( ( 1 + 2 T + 59 T^{2} )^{4} \)
$61$ \( ( 1 - 18 T + 197 T^{2} - 1098 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 + 16 T + 174 T^{2} + 1072 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 10 T + 143 T^{2} - 710 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( 1 - 4 T - 110 T^{2} + 80 T^{3} + 9379 T^{4} + 5840 T^{5} - 586190 T^{6} - 1556068 T^{7} + 28398241 T^{8} \)
$79$ \( ( 1 - 6 T + 143 T^{2} - 474 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 + 2 T - 79 T^{2} + 166 T^{3} + 6889 T^{4} )^{2} \)
$89$ \( 1 - 24 T + 278 T^{2} - 2880 T^{3} + 29619 T^{4} - 256320 T^{5} + 2202038 T^{6} - 16919256 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - 4 T - 158 T^{2} + 80 T^{3} + 19315 T^{4} + 7760 T^{5} - 1486622 T^{6} - 3650692 T^{7} + 88529281 T^{8} \)
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