# Properties

 Label 882.2.h.h 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.

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

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i 1.73205i −0.500000 0.866025i 2.00000 1.50000 + 0.866025i 0 −1.00000 −3.00000 1.00000 1.73205i
79.1 0.500000 + 0.866025i 1.73205i −0.500000 + 0.866025i 2.00000 1.50000 0.866025i 0 −1.00000 −3.00000 1.00000 + 1.73205i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.h 2
3.b odd 2 1 2646.2.h.b 2
7.b odd 2 1 882.2.h.g 2
7.c even 3 1 126.2.f.b 2
7.c even 3 1 882.2.e.a 2
7.d odd 6 1 882.2.e.e 2
7.d odd 6 1 882.2.f.f 2
9.c even 3 1 882.2.e.a 2
9.d odd 6 1 2646.2.e.i 2
21.c even 2 1 2646.2.h.c 2
21.g even 6 1 2646.2.e.h 2
21.g even 6 1 2646.2.f.b 2
21.h odd 6 1 378.2.f.b 2
21.h odd 6 1 2646.2.e.i 2
28.g odd 6 1 1008.2.r.a 2
63.g even 3 1 inner 882.2.h.h 2
63.g even 3 1 1134.2.a.c 1
63.h even 3 1 126.2.f.b 2
63.i even 6 1 2646.2.f.b 2
63.j odd 6 1 378.2.f.b 2
63.k odd 6 1 882.2.h.g 2
63.k odd 6 1 7938.2.a.e 1
63.l odd 6 1 882.2.e.e 2
63.n odd 6 1 1134.2.a.f 1
63.n odd 6 1 2646.2.h.b 2
63.o even 6 1 2646.2.e.h 2
63.s even 6 1 2646.2.h.c 2
63.s even 6 1 7938.2.a.bb 1
63.t odd 6 1 882.2.f.f 2
84.n even 6 1 3024.2.r.c 2
252.o even 6 1 9072.2.a.f 1
252.u odd 6 1 1008.2.r.a 2
252.bb even 6 1 3024.2.r.c 2
252.bl odd 6 1 9072.2.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.b 2 7.c even 3 1
126.2.f.b 2 63.h even 3 1
378.2.f.b 2 21.h odd 6 1
378.2.f.b 2 63.j odd 6 1
882.2.e.a 2 7.c even 3 1
882.2.e.a 2 9.c even 3 1
882.2.e.e 2 7.d odd 6 1
882.2.e.e 2 63.l odd 6 1
882.2.f.f 2 7.d odd 6 1
882.2.f.f 2 63.t odd 6 1
882.2.h.g 2 7.b odd 2 1
882.2.h.g 2 63.k odd 6 1
882.2.h.h 2 1.a even 1 1 trivial
882.2.h.h 2 63.g even 3 1 inner
1008.2.r.a 2 28.g odd 6 1
1008.2.r.a 2 252.u odd 6 1
1134.2.a.c 1 63.g even 3 1
1134.2.a.f 1 63.n odd 6 1
2646.2.e.h 2 21.g even 6 1
2646.2.e.h 2 63.o even 6 1
2646.2.e.i 2 9.d odd 6 1
2646.2.e.i 2 21.h odd 6 1
2646.2.f.b 2 21.g even 6 1
2646.2.f.b 2 63.i even 6 1
2646.2.h.b 2 3.b odd 2 1
2646.2.h.b 2 63.n odd 6 1
2646.2.h.c 2 21.c even 2 1
2646.2.h.c 2 63.s even 6 1
3024.2.r.c 2 84.n even 6 1
3024.2.r.c 2 252.bb even 6 1
7938.2.a.e 1 63.k odd 6 1
7938.2.a.bb 1 63.s even 6 1
9072.2.a.f 1 252.o even 6 1
9072.2.a.t 1 252.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 $$S_{2}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5} - 2$$ $$T_{11} - 1$$ $$T_{13}^{2} - 6 T_{13} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 + 3 T^{2}$$
$5$ $$( 1 - 2 T + 5 T^{2} )^{2}$$
$7$ 1
$11$ $$( 1 - T + 11 T^{2} )^{2}$$
$13$ $$1 - 6 T + 23 T^{2} - 78 T^{3} + 169 T^{4}$$
$17$ $$1 - 5 T + 8 T^{2} - 85 T^{3} + 289 T^{4}$$
$19$ $$( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} )$$
$23$ $$( 1 - 4 T + 23 T^{2} )^{2}$$
$29$ $$1 - 4 T - 13 T^{2} - 116 T^{3} + 841 T^{4}$$
$31$ $$1 - 6 T + 5 T^{2} - 186 T^{3} + 961 T^{4}$$
$37$ $$1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4}$$
$41$ $$1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4}$$
$43$ $$1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4}$$
$47$ $$1 - 47 T^{2} + 2209 T^{4}$$
$53$ $$1 + 12 T + 91 T^{2} + 636 T^{3} + 2809 T^{4}$$
$59$ $$1 - 7 T - 10 T^{2} - 413 T^{3} + 3481 T^{4}$$
$61$ $$1 - 12 T + 83 T^{2} - 732 T^{3} + 3721 T^{4}$$
$67$ $$1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 8 T + 71 T^{2} )^{2}$$
$73$ $$1 + T - 72 T^{2} + 73 T^{3} + 5329 T^{4}$$
$79$ $$1 - 6 T - 43 T^{2} - 474 T^{3} + 6241 T^{4}$$
$83$ $$1 + 16 T + 173 T^{2} + 1328 T^{3} + 6889 T^{4}$$
$89$ $$1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4}$$
$97$ $$( 1 - 19 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} )$$
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