# Properties

 Label 882.2.h.b Level $882$ Weight $2$ Character orbit 882.h Analytic conductor $7.043$ Analytic rank $1$ 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.h (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.04280545828$$ Analytic rank: $$1$$ 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 18) 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.

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

## 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 0 −1.50000 0.866025i 0 1.00000 −3.00000 0
79.1 −0.500000 0.866025i 1.73205i −0.500000 + 0.866025i 0 −1.50000 + 0.866025i 0 1.00000 −3.00000 0
 $$n$$: 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.b 2
3.b odd 2 1 2646.2.h.i 2
7.b odd 2 1 882.2.h.c 2
7.c even 3 1 882.2.e.g 2
7.c even 3 1 882.2.f.d 2
7.d odd 6 1 18.2.c.a 2
7.d odd 6 1 882.2.e.i 2
9.c even 3 1 882.2.e.g 2
9.d odd 6 1 2646.2.e.c 2
21.c even 2 1 2646.2.h.h 2
21.g even 6 1 54.2.c.a 2
21.g even 6 1 2646.2.e.b 2
21.h odd 6 1 2646.2.e.c 2
21.h odd 6 1 2646.2.f.g 2
28.f even 6 1 144.2.i.c 2
35.i odd 6 1 450.2.e.i 2
35.k even 12 2 450.2.j.e 4
56.j odd 6 1 576.2.i.g 2
56.m even 6 1 576.2.i.a 2
63.g even 3 1 inner 882.2.h.b 2
63.g even 3 1 7938.2.a.x 1
63.h even 3 1 882.2.f.d 2
63.i even 6 1 54.2.c.a 2
63.j odd 6 1 2646.2.f.g 2
63.k odd 6 1 162.2.a.c 1
63.k odd 6 1 882.2.h.c 2
63.l odd 6 1 882.2.e.i 2
63.n odd 6 1 2646.2.h.i 2
63.n odd 6 1 7938.2.a.i 1
63.o even 6 1 2646.2.e.b 2
63.s even 6 1 162.2.a.b 1
63.s even 6 1 2646.2.h.h 2
63.t odd 6 1 18.2.c.a 2
84.j odd 6 1 432.2.i.b 2
105.p even 6 1 1350.2.e.c 2
105.w odd 12 2 1350.2.j.a 4
168.ba even 6 1 1728.2.i.e 2
168.be odd 6 1 1728.2.i.f 2
252.n even 6 1 1296.2.a.g 1
252.r odd 6 1 432.2.i.b 2
252.bj even 6 1 144.2.i.c 2
252.bn odd 6 1 1296.2.a.f 1
315.q odd 6 1 450.2.e.i 2
315.u even 6 1 4050.2.a.v 1
315.bn odd 6 1 4050.2.a.c 1
315.bq even 6 1 1350.2.e.c 2
315.bs even 12 2 450.2.j.e 4
315.bu odd 12 2 1350.2.j.a 4
315.bw odd 12 2 4050.2.c.r 2
315.cg even 12 2 4050.2.c.c 2
504.u odd 6 1 5184.2.a.p 1
504.y even 6 1 5184.2.a.q 1
504.bf even 6 1 576.2.i.a 2
504.bp odd 6 1 576.2.i.g 2
504.ca even 6 1 1728.2.i.e 2
504.cm odd 6 1 1728.2.i.f 2
504.cw odd 6 1 5184.2.a.r 1
504.cz even 6 1 5184.2.a.o 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.2.c.a 2 7.d odd 6 1
18.2.c.a 2 63.t odd 6 1
54.2.c.a 2 21.g even 6 1
54.2.c.a 2 63.i even 6 1
144.2.i.c 2 28.f even 6 1
144.2.i.c 2 252.bj even 6 1
162.2.a.b 1 63.s even 6 1
162.2.a.c 1 63.k odd 6 1
432.2.i.b 2 84.j odd 6 1
432.2.i.b 2 252.r odd 6 1
450.2.e.i 2 35.i odd 6 1
450.2.e.i 2 315.q odd 6 1
450.2.j.e 4 35.k even 12 2
450.2.j.e 4 315.bs even 12 2
576.2.i.a 2 56.m even 6 1
576.2.i.a 2 504.bf even 6 1
576.2.i.g 2 56.j odd 6 1
576.2.i.g 2 504.bp odd 6 1
882.2.e.g 2 7.c even 3 1
882.2.e.g 2 9.c even 3 1
882.2.e.i 2 7.d odd 6 1
882.2.e.i 2 63.l odd 6 1
882.2.f.d 2 7.c even 3 1
882.2.f.d 2 63.h even 3 1
882.2.h.b 2 1.a even 1 1 trivial
882.2.h.b 2 63.g even 3 1 inner
882.2.h.c 2 7.b odd 2 1
882.2.h.c 2 63.k odd 6 1
1296.2.a.f 1 252.bn odd 6 1
1296.2.a.g 1 252.n even 6 1
1350.2.e.c 2 105.p even 6 1
1350.2.e.c 2 315.bq even 6 1
1350.2.j.a 4 105.w odd 12 2
1350.2.j.a 4 315.bu odd 12 2
1728.2.i.e 2 168.ba even 6 1
1728.2.i.e 2 504.ca even 6 1
1728.2.i.f 2 168.be odd 6 1
1728.2.i.f 2 504.cm odd 6 1
2646.2.e.b 2 21.g even 6 1
2646.2.e.b 2 63.o even 6 1
2646.2.e.c 2 9.d odd 6 1
2646.2.e.c 2 21.h odd 6 1
2646.2.f.g 2 21.h odd 6 1
2646.2.f.g 2 63.j odd 6 1
2646.2.h.h 2 21.c even 2 1
2646.2.h.h 2 63.s even 6 1
2646.2.h.i 2 3.b odd 2 1
2646.2.h.i 2 63.n odd 6 1
4050.2.a.c 1 315.bn odd 6 1
4050.2.a.v 1 315.u even 6 1
4050.2.c.c 2 315.cg even 12 2
4050.2.c.r 2 315.bw odd 12 2
5184.2.a.o 1 504.cz even 6 1
5184.2.a.p 1 504.u odd 6 1
5184.2.a.q 1 504.y even 6 1
5184.2.a.r 1 504.cw odd 6 1
7938.2.a.i 1 63.n odd 6 1
7938.2.a.x 1 63.g even 3 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{11} + 3$$ $$T_{13}^{2} - 2 T_{13} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$3 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 3 + T )^{2}$$
$13$ $$4 - 2 T + T^{2}$$
$17$ $$9 + 3 T + T^{2}$$
$19$ $$1 + T + T^{2}$$
$23$ $$( 6 + T )^{2}$$
$29$ $$36 + 6 T + T^{2}$$
$31$ $$16 + 4 T + T^{2}$$
$37$ $$16 - 4 T + T^{2}$$
$41$ $$81 - 9 T + T^{2}$$
$43$ $$1 - T + T^{2}$$
$47$ $$36 + 6 T + T^{2}$$
$53$ $$144 + 12 T + T^{2}$$
$59$ $$9 - 3 T + T^{2}$$
$61$ $$64 - 8 T + T^{2}$$
$67$ $$25 + 5 T + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$121 - 11 T + T^{2}$$
$79$ $$16 - 4 T + T^{2}$$
$83$ $$144 - 12 T + T^{2}$$
$89$ $$36 - 6 T + T^{2}$$
$97$ $$25 - 5 T + T^{2}$$