Properties

Label 882.2.e.k
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: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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.

\(f(q)\) \(=\) \( q - q^{2} + ( -\beta_{1} + \beta_{3} ) q^{3} + q^{4} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{5} + ( \beta_{1} - \beta_{3} ) q^{6} - q^{8} + ( -3 + \beta_{1} - \beta_{3} ) q^{9} +O(q^{10})\) \( q - q^{2} + ( -\beta_{1} + \beta_{3} ) q^{3} + q^{4} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{5} + ( \beta_{1} - \beta_{3} ) q^{6} - q^{8} + ( -3 + \beta_{1} - \beta_{3} ) q^{9} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{10} + ( -1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{11} + ( -\beta_{1} + \beta_{3} ) q^{12} + ( 2 - 2 \beta_{2} ) q^{13} + ( 6 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{15} + q^{16} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{17} + ( 3 - \beta_{1} + \beta_{3} ) q^{18} + ( 5 - 5 \beta_{2} ) q^{19} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{20} + ( 1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{22} + ( 1 - 2 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{23} + ( \beta_{1} - \beta_{3} ) q^{24} + ( -7 + 3 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} ) q^{25} + ( -2 + 2 \beta_{2} ) q^{26} + ( 3 + 2 \beta_{1} - 2 \beta_{3} ) q^{27} + ( -2 + 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{29} + ( -6 + \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{30} -2 q^{31} - q^{32} + ( -3 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{33} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{34} + ( -3 + \beta_{1} - \beta_{3} ) q^{36} + ( -2 + 2 \beta_{2} ) q^{37} + ( -5 + 5 \beta_{2} ) q^{38} -2 \beta_{1} q^{39} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{40} + ( 7 + \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{41} + ( -3 + 6 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{43} + ( -1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{44} + ( -3 - 5 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{45} + ( -1 + 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{46} + ( -\beta_{1} + \beta_{3} ) q^{48} + ( 7 - 3 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} ) q^{50} + ( 6 - \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{51} + ( 2 - 2 \beta_{2} ) q^{52} + ( -2 + 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{53} + ( -3 - 2 \beta_{1} + 2 \beta_{3} ) q^{54} + 6 q^{55} -5 \beta_{1} q^{57} + ( 2 - 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{58} + ( 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{59} + ( 6 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{60} + ( 7 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{61} + 2 q^{62} + q^{64} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{65} + ( 3 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{66} + ( 8 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{67} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{68} + ( -6 + \beta_{1} + 3 \beta_{2} + 4 \beta_{3} ) q^{69} + ( 3 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{71} + ( 3 - \beta_{1} + \beta_{3} ) q^{72} + ( -3 + 6 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} ) q^{73} + ( 2 - 2 \beta_{2} ) q^{74} + ( 9 + 4 \beta_{1} + 9 \beta_{2} + 3 \beta_{3} ) q^{75} + ( 5 - 5 \beta_{2} ) q^{76} + 2 \beta_{1} q^{78} + ( 5 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{79} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{80} + ( 6 - 5 \beta_{1} + 5 \beta_{3} ) q^{81} + ( -7 - \beta_{1} + 8 \beta_{2} + 2 \beta_{3} ) q^{82} + ( -4 + 8 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{83} + ( -6 + 6 \beta_{2} ) q^{85} + ( 3 - 6 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{86} + ( 12 - 2 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} ) q^{87} + ( 1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{88} + ( 10 - 2 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{89} + ( 3 + 5 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{90} + ( 1 - 2 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{92} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{93} + ( 5 + 5 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} ) q^{95} + ( \beta_{1} - \beta_{3} ) q^{96} + ( 3 - 6 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{97} + ( 6 + \beta_{1} - 3 \beta_{2} - 5 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} - 2q^{3} + 4q^{4} + 3q^{5} + 2q^{6} - 4q^{8} - 10q^{9} + O(q^{10}) \) \( 4q - 4q^{2} - 2q^{3} + 4q^{4} + 3q^{5} + 2q^{6} - 4q^{8} - 10q^{9} - 3q^{10} - 3q^{11} - 2q^{12} + 4q^{13} + 15q^{15} + 4q^{16} - 3q^{17} + 10q^{18} + 10q^{19} + 3q^{20} + 3q^{22} + 9q^{23} + 2q^{24} - 11q^{25} - 4q^{26} + 16q^{27} + 6q^{29} - 15q^{30} - 8q^{31} - 4q^{32} - 15q^{33} + 3q^{34} - 10q^{36} - 4q^{37} - 10q^{38} - 2q^{39} - 3q^{40} + 15q^{41} - q^{43} - 3q^{44} - 24q^{45} - 9q^{46} - 2q^{48} + 11q^{50} + 18q^{51} + 4q^{52} + 6q^{53} - 16q^{54} + 24q^{55} - 5q^{57} - 6q^{58} + 6q^{59} + 15q^{60} + 22q^{61} + 8q^{62} + 4q^{64} + 12q^{65} + 15q^{66} + 26q^{67} - 3q^{68} - 21q^{69} + 6q^{71} + 10q^{72} + 7q^{73} + 4q^{74} + 55q^{75} + 10q^{76} + 2q^{78} + 14q^{79} + 3q^{80} + 14q^{81} - 15q^{82} + 12q^{83} - 12q^{85} + q^{86} + 30q^{87} + 3q^{88} + 18q^{89} + 24q^{90} + 9q^{92} + 4q^{93} + 30q^{95} + 2q^{96} + q^{97} + 24q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \nu - 3 \)\()/6\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 2 \nu + 3 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(-2 \beta_{3} + 2 \beta_{1} + 3\)

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
1.68614 + 0.396143i
−1.18614 1.26217i
−1.18614 + 1.26217i
1.68614 0.396143i
−1.00000 −0.500000 1.65831i 1.00000 2.18614 + 3.78651i 0.500000 + 1.65831i 0 −1.00000 −2.50000 + 1.65831i −2.18614 3.78651i
373.2 −1.00000 −0.500000 + 1.65831i 1.00000 −0.686141 1.18843i 0.500000 1.65831i 0 −1.00000 −2.50000 1.65831i 0.686141 + 1.18843i
655.1 −1.00000 −0.500000 1.65831i 1.00000 −0.686141 + 1.18843i 0.500000 + 1.65831i 0 −1.00000 −2.50000 + 1.65831i 0.686141 1.18843i
655.2 −1.00000 −0.500000 + 1.65831i 1.00000 2.18614 3.78651i 0.500000 1.65831i 0 −1.00000 −2.50000 1.65831i −2.18614 + 3.78651i
\(n\): 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.k 4
3.b odd 2 1 2646.2.e.m 4
7.b odd 2 1 882.2.e.l 4
7.c even 3 1 882.2.f.k 4
7.c even 3 1 882.2.h.n 4
7.d odd 6 1 126.2.f.d 4
7.d odd 6 1 882.2.h.m 4
9.c even 3 1 882.2.h.n 4
9.d odd 6 1 2646.2.h.l 4
21.c even 2 1 2646.2.e.n 4
21.g even 6 1 378.2.f.c 4
21.g even 6 1 2646.2.h.k 4
21.h odd 6 1 2646.2.f.j 4
21.h odd 6 1 2646.2.h.l 4
28.f even 6 1 1008.2.r.f 4
63.g even 3 1 882.2.f.k 4
63.h even 3 1 inner 882.2.e.k 4
63.h even 3 1 7938.2.a.bh 2
63.i even 6 1 1134.2.a.n 2
63.i even 6 1 2646.2.e.n 4
63.j odd 6 1 2646.2.e.m 4
63.j odd 6 1 7938.2.a.bs 2
63.k odd 6 1 126.2.f.d 4
63.l odd 6 1 882.2.h.m 4
63.n odd 6 1 2646.2.f.j 4
63.o even 6 1 2646.2.h.k 4
63.s even 6 1 378.2.f.c 4
63.t odd 6 1 882.2.e.l 4
63.t odd 6 1 1134.2.a.k 2
84.j odd 6 1 3024.2.r.f 4
252.n even 6 1 1008.2.r.f 4
252.r odd 6 1 9072.2.a.bb 2
252.bj even 6 1 9072.2.a.bm 2
252.bn odd 6 1 3024.2.r.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.d 4 7.d odd 6 1
126.2.f.d 4 63.k odd 6 1
378.2.f.c 4 21.g even 6 1
378.2.f.c 4 63.s even 6 1
882.2.e.k 4 1.a even 1 1 trivial
882.2.e.k 4 63.h even 3 1 inner
882.2.e.l 4 7.b odd 2 1
882.2.e.l 4 63.t odd 6 1
882.2.f.k 4 7.c even 3 1
882.2.f.k 4 63.g even 3 1
882.2.h.m 4 7.d odd 6 1
882.2.h.m 4 63.l odd 6 1
882.2.h.n 4 7.c even 3 1
882.2.h.n 4 9.c even 3 1
1008.2.r.f 4 28.f even 6 1
1008.2.r.f 4 252.n even 6 1
1134.2.a.k 2 63.t odd 6 1
1134.2.a.n 2 63.i even 6 1
2646.2.e.m 4 3.b odd 2 1
2646.2.e.m 4 63.j odd 6 1
2646.2.e.n 4 21.c even 2 1
2646.2.e.n 4 63.i even 6 1
2646.2.f.j 4 21.h odd 6 1
2646.2.f.j 4 63.n odd 6 1
2646.2.h.k 4 21.g even 6 1
2646.2.h.k 4 63.o even 6 1
2646.2.h.l 4 9.d odd 6 1
2646.2.h.l 4 21.h odd 6 1
3024.2.r.f 4 84.j odd 6 1
3024.2.r.f 4 252.bn odd 6 1
7938.2.a.bh 2 63.h even 3 1
7938.2.a.bs 2 63.j odd 6 1
9072.2.a.bb 2 252.r odd 6 1
9072.2.a.bm 2 252.bj even 6 1

Hecke kernels

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

\( T_{5}^{4} - 3 T_{5}^{3} + 15 T_{5}^{2} + 18 T_{5} + 36 \)
\( T_{11}^{4} + 3 T_{11}^{3} + 15 T_{11}^{2} - 18 T_{11} + 36 \)
\( T_{13}^{2} - 2 T_{13} + 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( ( 1 + T + 3 T^{2} )^{2} \)
$5$ \( ( 1 - 3 T + 5 T^{2} )^{2}( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} ) \)
$7$ 1
$11$ \( 1 + 3 T - 7 T^{2} - 18 T^{3} + 36 T^{4} - 198 T^{5} - 847 T^{6} + 3993 T^{7} + 14641 T^{8} \)
$13$ \( ( 1 - 7 T + 13 T^{2} )^{2}( 1 + 5 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 3 T - 19 T^{2} - 18 T^{3} + 342 T^{4} - 306 T^{5} - 5491 T^{6} + 14739 T^{7} + 83521 T^{8} \)
$19$ \( ( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 9 T + 23 T^{2} )^{2}( 1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4} ) \)
$29$ \( 1 - 6 T + 2 T^{2} + 144 T^{3} - 729 T^{4} + 4176 T^{5} + 1682 T^{6} - 146334 T^{7} + 707281 T^{8} \)
$31$ \( ( 1 + 2 T + 31 T^{2} )^{4} \)
$37$ \( ( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( 1 - 15 T + 95 T^{2} - 720 T^{3} + 5994 T^{4} - 29520 T^{5} + 159695 T^{6} - 1033815 T^{7} + 2825761 T^{8} \)
$43$ \( 1 + T - 11 T^{2} - 74 T^{3} - 1748 T^{4} - 3182 T^{5} - 20339 T^{6} + 79507 T^{7} + 3418801 T^{8} \)
$47$ \( ( 1 + 47 T^{2} )^{4} \)
$53$ \( 1 - 6 T - 46 T^{2} + 144 T^{3} + 2007 T^{4} + 7632 T^{5} - 129214 T^{6} - 893262 T^{7} + 7890481 T^{8} \)
$59$ \( ( 1 - 3 T + 46 T^{2} - 177 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 11 T + 78 T^{2} - 671 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 3 T + 70 T^{2} - 213 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( 1 - 7 T - 35 T^{2} + 434 T^{3} - 1850 T^{4} + 31682 T^{5} - 186515 T^{6} - 2723119 T^{7} + 28398241 T^{8} \)
$79$ \( ( 1 - 7 T + 96 T^{2} - 553 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( 1 - 12 T + 74 T^{2} + 1152 T^{3} - 13941 T^{4} + 95616 T^{5} + 509786 T^{6} - 6861444 T^{7} + 47458321 T^{8} \)
$89$ \( 1 - 18 T + 98 T^{2} - 864 T^{3} + 14319 T^{4} - 76896 T^{5} + 776258 T^{6} - 12689442 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - T - 119 T^{2} + 74 T^{3} + 4894 T^{4} + 7178 T^{5} - 1119671 T^{6} - 912673 T^{7} + 88529281 T^{8} \)
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