# Properties

 Label 882.2.g.h Level $882$ Weight $2$ Character orbit 882.g 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.g (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 42) 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} -2 \zeta_{6} q^{5} - q^{8} +O(q^{10})$$ $$q + \zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} -2 \zeta_{6} q^{5} - q^{8} + ( 2 - 2 \zeta_{6} ) q^{10} + ( -4 + 4 \zeta_{6} ) q^{11} + 6 q^{13} -\zeta_{6} q^{16} + ( 2 - 2 \zeta_{6} ) q^{17} + 4 \zeta_{6} q^{19} + 2 q^{20} -4 q^{22} + 8 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} + 6 \zeta_{6} q^{26} + 2 q^{29} + ( 1 - \zeta_{6} ) q^{32} + 2 q^{34} + 10 \zeta_{6} q^{37} + ( -4 + 4 \zeta_{6} ) q^{38} + 2 \zeta_{6} q^{40} + 6 q^{41} -4 q^{43} -4 \zeta_{6} q^{44} + ( -8 + 8 \zeta_{6} ) q^{46} + q^{50} + ( -6 + 6 \zeta_{6} ) q^{52} + ( 6 - 6 \zeta_{6} ) q^{53} + 8 q^{55} + 2 \zeta_{6} q^{58} + ( 4 - 4 \zeta_{6} ) q^{59} -6 \zeta_{6} q^{61} + q^{64} -12 \zeta_{6} q^{65} + ( -4 + 4 \zeta_{6} ) q^{67} + 2 \zeta_{6} q^{68} -8 q^{71} + ( -10 + 10 \zeta_{6} ) q^{73} + ( -10 + 10 \zeta_{6} ) q^{74} -4 q^{76} + ( -2 + 2 \zeta_{6} ) q^{80} + 6 \zeta_{6} q^{82} + 4 q^{83} -4 q^{85} -4 \zeta_{6} q^{86} + ( 4 - 4 \zeta_{6} ) q^{88} -6 \zeta_{6} q^{89} -8 q^{92} + ( 8 - 8 \zeta_{6} ) q^{95} -14 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} - 2 q^{5} - 2 q^{8} + O(q^{10})$$ $$2 q + q^{2} - q^{4} - 2 q^{5} - 2 q^{8} + 2 q^{10} - 4 q^{11} + 12 q^{13} - q^{16} + 2 q^{17} + 4 q^{19} + 4 q^{20} - 8 q^{22} + 8 q^{23} + q^{25} + 6 q^{26} + 4 q^{29} + q^{32} + 4 q^{34} + 10 q^{37} - 4 q^{38} + 2 q^{40} + 12 q^{41} - 8 q^{43} - 4 q^{44} - 8 q^{46} + 2 q^{50} - 6 q^{52} + 6 q^{53} + 16 q^{55} + 2 q^{58} + 4 q^{59} - 6 q^{61} + 2 q^{64} - 12 q^{65} - 4 q^{67} + 2 q^{68} - 16 q^{71} - 10 q^{73} - 10 q^{74} - 8 q^{76} - 2 q^{80} + 6 q^{82} + 8 q^{83} - 8 q^{85} - 4 q^{86} + 4 q^{88} - 6 q^{89} - 16 q^{92} + 8 q^{95} - 28 q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.00000 1.73205i 0 0 −1.00000 0 1.00000 1.73205i
667.1 0.500000 0.866025i 0 −0.500000 0.866025i −1.00000 + 1.73205i 0 0 −1.00000 0 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
7.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.g.h 2
3.b odd 2 1 294.2.e.c 2
7.b odd 2 1 882.2.g.j 2
7.c even 3 1 126.2.a.a 1
7.c even 3 1 inner 882.2.g.h 2
7.d odd 6 1 882.2.a.b 1
7.d odd 6 1 882.2.g.j 2
12.b even 2 1 2352.2.q.i 2
21.c even 2 1 294.2.e.a 2
21.g even 6 1 294.2.a.g 1
21.g even 6 1 294.2.e.a 2
21.h odd 6 1 42.2.a.a 1
21.h odd 6 1 294.2.e.c 2
28.f even 6 1 7056.2.a.k 1
28.g odd 6 1 1008.2.a.j 1
35.j even 6 1 3150.2.a.bo 1
35.l odd 12 2 3150.2.g.r 2
56.k odd 6 1 4032.2.a.m 1
56.p even 6 1 4032.2.a.e 1
63.g even 3 1 1134.2.f.j 2
63.h even 3 1 1134.2.f.j 2
63.j odd 6 1 1134.2.f.g 2
63.n odd 6 1 1134.2.f.g 2
84.h odd 2 1 2352.2.q.n 2
84.j odd 6 1 2352.2.a.l 1
84.j odd 6 1 2352.2.q.n 2
84.n even 6 1 336.2.a.d 1
84.n even 6 1 2352.2.q.i 2
105.o odd 6 1 1050.2.a.i 1
105.p even 6 1 7350.2.a.f 1
105.x even 12 2 1050.2.g.a 2
168.s odd 6 1 1344.2.a.q 1
168.v even 6 1 1344.2.a.i 1
168.ba even 6 1 9408.2.a.n 1
168.be odd 6 1 9408.2.a.bw 1
231.l even 6 1 5082.2.a.d 1
273.w odd 6 1 7098.2.a.f 1
336.bt odd 12 2 5376.2.c.bc 2
336.bu even 12 2 5376.2.c.e 2
420.ba even 6 1 8400.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.a.a 1 21.h odd 6 1
126.2.a.a 1 7.c even 3 1
294.2.a.g 1 21.g even 6 1
294.2.e.a 2 21.c even 2 1
294.2.e.a 2 21.g even 6 1
294.2.e.c 2 3.b odd 2 1
294.2.e.c 2 21.h odd 6 1
336.2.a.d 1 84.n even 6 1
882.2.a.b 1 7.d odd 6 1
882.2.g.h 2 1.a even 1 1 trivial
882.2.g.h 2 7.c even 3 1 inner
882.2.g.j 2 7.b odd 2 1
882.2.g.j 2 7.d odd 6 1
1008.2.a.j 1 28.g odd 6 1
1050.2.a.i 1 105.o odd 6 1
1050.2.g.a 2 105.x even 12 2
1134.2.f.g 2 63.j odd 6 1
1134.2.f.g 2 63.n odd 6 1
1134.2.f.j 2 63.g even 3 1
1134.2.f.j 2 63.h even 3 1
1344.2.a.i 1 168.v even 6 1
1344.2.a.q 1 168.s odd 6 1
2352.2.a.l 1 84.j odd 6 1
2352.2.q.i 2 12.b even 2 1
2352.2.q.i 2 84.n even 6 1
2352.2.q.n 2 84.h odd 2 1
2352.2.q.n 2 84.j odd 6 1
3150.2.a.bo 1 35.j even 6 1
3150.2.g.r 2 35.l odd 12 2
4032.2.a.e 1 56.p even 6 1
4032.2.a.m 1 56.k odd 6 1
5082.2.a.d 1 231.l even 6 1
5376.2.c.e 2 336.bu even 12 2
5376.2.c.bc 2 336.bt odd 12 2
7056.2.a.k 1 28.f even 6 1
7098.2.a.f 1 273.w odd 6 1
7350.2.a.f 1 105.p even 6 1
8400.2.a.k 1 420.ba even 6 1
9408.2.a.n 1 168.ba even 6 1
9408.2.a.bw 1 168.be odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} + 2 T_{5} + 4$$ $$T_{11}^{2} + 4 T_{11} + 16$$ $$T_{13} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$4 + 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$16 + 4 T + T^{2}$$
$13$ $$( -6 + T )^{2}$$
$17$ $$4 - 2 T + T^{2}$$
$19$ $$16 - 4 T + T^{2}$$
$23$ $$64 - 8 T + T^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$100 - 10 T + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$36 - 6 T + T^{2}$$
$59$ $$16 - 4 T + T^{2}$$
$61$ $$36 + 6 T + T^{2}$$
$67$ $$16 + 4 T + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$100 + 10 T + T^{2}$$
$79$ $$T^{2}$$
$83$ $$( -4 + T )^{2}$$
$89$ $$36 + 6 T + T^{2}$$
$97$ $$( 14 + T )^{2}$$