Properties

Label 882.2.h.m
Level $882$
Weight $2$
Character orbit 882.h
Analytic conductor $7.043$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Defining polynomial: \(x^{4} - x^{3} - 2 x^{2} - 3 x + 9\)
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 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.

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

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 \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
1.68614 0.396143i
−1.18614 + 1.26217i
1.68614 + 0.396143i
−1.18614 1.26217i
0.500000 0.866025i −1.68614 + 0.396143i −0.500000 0.866025i −1.37228 −0.500000 + 1.65831i 0 −1.00000 2.68614 1.33591i −0.686141 + 1.18843i
67.2 0.500000 0.866025i 1.18614 1.26217i −0.500000 0.866025i 4.37228 −0.500000 1.65831i 0 −1.00000 −0.186141 2.99422i 2.18614 3.78651i
79.1 0.500000 + 0.866025i −1.68614 0.396143i −0.500000 + 0.866025i −1.37228 −0.500000 1.65831i 0 −1.00000 2.68614 + 1.33591i −0.686141 1.18843i
79.2 0.500000 + 0.866025i 1.18614 + 1.26217i −0.500000 + 0.866025i 4.37228 −0.500000 + 1.65831i 0 −1.00000 −0.186141 + 2.99422i 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.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.m 4
3.b odd 2 1 2646.2.h.k 4
7.b odd 2 1 882.2.h.n 4
7.c even 3 1 126.2.f.d 4
7.c even 3 1 882.2.e.l 4
7.d odd 6 1 882.2.e.k 4
7.d odd 6 1 882.2.f.k 4
9.c even 3 1 882.2.e.l 4
9.d odd 6 1 2646.2.e.n 4
21.c even 2 1 2646.2.h.l 4
21.g even 6 1 2646.2.e.m 4
21.g even 6 1 2646.2.f.j 4
21.h odd 6 1 378.2.f.c 4
21.h odd 6 1 2646.2.e.n 4
28.g odd 6 1 1008.2.r.f 4
63.g even 3 1 inner 882.2.h.m 4
63.g even 3 1 1134.2.a.k 2
63.h even 3 1 126.2.f.d 4
63.i even 6 1 2646.2.f.j 4
63.j odd 6 1 378.2.f.c 4
63.k odd 6 1 882.2.h.n 4
63.k odd 6 1 7938.2.a.bh 2
63.l odd 6 1 882.2.e.k 4
63.n odd 6 1 1134.2.a.n 2
63.n odd 6 1 2646.2.h.k 4
63.o even 6 1 2646.2.e.m 4
63.s even 6 1 2646.2.h.l 4
63.s even 6 1 7938.2.a.bs 2
63.t odd 6 1 882.2.f.k 4
84.n even 6 1 3024.2.r.f 4
252.o even 6 1 9072.2.a.bb 2
252.u odd 6 1 1008.2.r.f 4
252.bb even 6 1 3024.2.r.f 4
252.bl odd 6 1 9072.2.a.bm 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.d 4 7.c even 3 1
126.2.f.d 4 63.h even 3 1
378.2.f.c 4 21.h odd 6 1
378.2.f.c 4 63.j odd 6 1
882.2.e.k 4 7.d odd 6 1
882.2.e.k 4 63.l odd 6 1
882.2.e.l 4 7.c even 3 1
882.2.e.l 4 9.c even 3 1
882.2.f.k 4 7.d odd 6 1
882.2.f.k 4 63.t odd 6 1
882.2.h.m 4 1.a even 1 1 trivial
882.2.h.m 4 63.g even 3 1 inner
882.2.h.n 4 7.b odd 2 1
882.2.h.n 4 63.k odd 6 1
1008.2.r.f 4 28.g odd 6 1
1008.2.r.f 4 252.u odd 6 1
1134.2.a.k 2 63.g even 3 1
1134.2.a.n 2 63.n odd 6 1
2646.2.e.m 4 21.g even 6 1
2646.2.e.m 4 63.o even 6 1
2646.2.e.n 4 9.d odd 6 1
2646.2.e.n 4 21.h odd 6 1
2646.2.f.j 4 21.g even 6 1
2646.2.f.j 4 63.i even 6 1
2646.2.h.k 4 3.b odd 2 1
2646.2.h.k 4 63.n odd 6 1
2646.2.h.l 4 21.c even 2 1
2646.2.h.l 4 63.s even 6 1
3024.2.r.f 4 84.n even 6 1
3024.2.r.f 4 252.bb even 6 1
7938.2.a.bh 2 63.k odd 6 1
7938.2.a.bs 2 63.s even 6 1
9072.2.a.bb 2 252.o even 6 1
9072.2.a.bm 2 252.bl odd 6 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ \( 9 + 3 T - 2 T^{2} + T^{3} + T^{4} \)
$5$ \( ( -6 - 3 T + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( ( -6 - 3 T + T^{2} )^{2} \)
$13$ \( ( 4 + 2 T + T^{2} )^{2} \)
$17$ \( 36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4} \)
$19$ \( ( 25 + 5 T + T^{2} )^{2} \)
$23$ \( ( 12 + 9 T + T^{2} )^{2} \)
$29$ \( 576 + 144 T + 60 T^{2} - 6 T^{3} + T^{4} \)
$31$ \( ( 4 + 2 T + T^{2} )^{2} \)
$37$ \( ( 4 + 2 T + T^{2} )^{2} \)
$41$ \( 2304 + 720 T + 177 T^{2} + 15 T^{3} + T^{4} \)
$43$ \( 5476 - 74 T + 75 T^{2} + T^{3} + T^{4} \)
$47$ \( T^{4} \)
$53$ \( 576 + 144 T + 60 T^{2} - 6 T^{3} + T^{4} \)
$59$ \( 5184 + 216 T + 81 T^{2} - 3 T^{3} + T^{4} \)
$61$ \( 1936 + 484 T + 165 T^{2} - 11 T^{3} + T^{4} \)
$67$ \( 1024 - 416 T + 201 T^{2} + 13 T^{3} + T^{4} \)
$71$ \( ( -72 - 3 T + T^{2} )^{2} \)
$73$ \( 3844 - 434 T + 111 T^{2} + 7 T^{3} + T^{4} \)
$79$ \( 3844 - 434 T + 111 T^{2} + 7 T^{3} + T^{4} \)
$83$ \( 9216 - 1152 T + 240 T^{2} + 12 T^{3} + T^{4} \)
$89$ \( 2304 + 864 T + 276 T^{2} + 18 T^{3} + T^{4} \)
$97$ \( 5476 - 74 T + 75 T^{2} + T^{3} + T^{4} \)
show more
show less