# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 882.f (of order $$3$$, degree $$2$$, 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.

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
295.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i −1.50000 + 0.866025i −0.500000 0.866025i 1.50000 + 2.59808i 1.73205i 0 −1.00000 1.50000 2.59808i 3.00000
589.1 0.500000 + 0.866025i −1.50000 0.866025i −0.500000 + 0.866025i 1.50000 2.59808i 1.73205i 0 −1.00000 1.50000 + 2.59808i 3.00000
 $$n$$: 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
9.c 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.f.g 2
3.b odd 2 1 2646.2.f.a 2
7.b odd 2 1 882.2.f.i 2
7.c even 3 1 882.2.e.c 2
7.c even 3 1 882.2.h.i 2
7.d odd 6 1 126.2.e.a 2
7.d odd 6 1 126.2.h.b yes 2
9.c even 3 1 inner 882.2.f.g 2
9.c even 3 1 7938.2.a.b 1
9.d odd 6 1 2646.2.f.a 2
9.d odd 6 1 7938.2.a.be 1
21.c even 2 1 2646.2.f.d 2
21.g even 6 1 378.2.e.b 2
21.g even 6 1 378.2.h.a 2
21.h odd 6 1 2646.2.e.g 2
21.h odd 6 1 2646.2.h.d 2
28.f even 6 1 1008.2.q.a 2
28.f even 6 1 1008.2.t.f 2
63.g even 3 1 882.2.e.c 2
63.h even 3 1 882.2.h.i 2
63.i even 6 1 378.2.h.a 2
63.i even 6 1 1134.2.g.c 2
63.j odd 6 1 2646.2.h.d 2
63.k odd 6 1 126.2.e.a 2
63.k odd 6 1 1134.2.g.e 2
63.l odd 6 1 882.2.f.i 2
63.l odd 6 1 7938.2.a.m 1
63.n odd 6 1 2646.2.e.g 2
63.o even 6 1 2646.2.f.d 2
63.o even 6 1 7938.2.a.t 1
63.s even 6 1 378.2.e.b 2
63.s even 6 1 1134.2.g.c 2
63.t odd 6 1 126.2.h.b yes 2
63.t odd 6 1 1134.2.g.e 2
84.j odd 6 1 3024.2.q.f 2
84.j odd 6 1 3024.2.t.a 2
252.n even 6 1 1008.2.q.a 2
252.r odd 6 1 3024.2.t.a 2
252.bj even 6 1 1008.2.t.f 2
252.bn odd 6 1 3024.2.q.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.a 2 7.d odd 6 1
126.2.e.a 2 63.k odd 6 1
126.2.h.b yes 2 7.d odd 6 1
126.2.h.b yes 2 63.t odd 6 1
378.2.e.b 2 21.g even 6 1
378.2.e.b 2 63.s even 6 1
378.2.h.a 2 21.g even 6 1
378.2.h.a 2 63.i even 6 1
882.2.e.c 2 7.c even 3 1
882.2.e.c 2 63.g even 3 1
882.2.f.g 2 1.a even 1 1 trivial
882.2.f.g 2 9.c even 3 1 inner
882.2.f.i 2 7.b odd 2 1
882.2.f.i 2 63.l odd 6 1
882.2.h.i 2 7.c even 3 1
882.2.h.i 2 63.h even 3 1
1008.2.q.a 2 28.f even 6 1
1008.2.q.a 2 252.n even 6 1
1008.2.t.f 2 28.f even 6 1
1008.2.t.f 2 252.bj even 6 1
1134.2.g.c 2 63.i even 6 1
1134.2.g.c 2 63.s even 6 1
1134.2.g.e 2 63.k odd 6 1
1134.2.g.e 2 63.t odd 6 1
2646.2.e.g 2 21.h odd 6 1
2646.2.e.g 2 63.n odd 6 1
2646.2.f.a 2 3.b odd 2 1
2646.2.f.a 2 9.d odd 6 1
2646.2.f.d 2 21.c even 2 1
2646.2.f.d 2 63.o even 6 1
2646.2.h.d 2 21.h odd 6 1
2646.2.h.d 2 63.j odd 6 1
3024.2.q.f 2 84.j odd 6 1
3024.2.q.f 2 252.bn odd 6 1
3024.2.t.a 2 84.j odd 6 1
3024.2.t.a 2 252.r odd 6 1
7938.2.a.b 1 9.c even 3 1
7938.2.a.m 1 63.l odd 6 1
7938.2.a.t 1 63.o even 6 1
7938.2.a.be 1 9.d odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 3 T_{5} + 9$$ $$T_{11}^{2} - 3 T_{11} + 9$$ $$T_{13}^{2} + T_{13} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$3 + 3 T + T^{2}$$
$5$ $$9 - 3 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$9 - 3 T + T^{2}$$
$13$ $$1 + T + T^{2}$$
$17$ $$( 3 + T )^{2}$$
$19$ $$( -7 + T )^{2}$$
$23$ $$81 - 9 T + T^{2}$$
$29$ $$9 + 3 T + T^{2}$$
$31$ $$64 - 8 T + T^{2}$$
$37$ $$( 1 + T )^{2}$$
$41$ $$9 - 3 T + T^{2}$$
$43$ $$1 - T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$( -3 + T )^{2}$$
$59$ $$T^{2}$$
$61$ $$4 - 2 T + T^{2}$$
$67$ $$16 - 4 T + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$( 11 + T )^{2}$$
$79$ $$256 - 16 T + T^{2}$$
$83$ $$81 + 9 T + T^{2}$$
$89$ $$( 3 + T )^{2}$$
$97$ $$1 + T + T^{2}$$