# Properties

 Label 882.2.g.e 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 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} + 3 \zeta_{6} q^{5} + q^{8} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} + 3 \zeta_{6} q^{5} + q^{8} + ( 3 - 3 \zeta_{6} ) q^{10} + ( 3 - 3 \zeta_{6} ) q^{11} -2 q^{13} -\zeta_{6} q^{16} + ( 6 - 6 \zeta_{6} ) q^{17} + 2 \zeta_{6} q^{19} -3 q^{20} -3 q^{22} + 6 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + 2 \zeta_{6} q^{26} + 9 q^{29} + ( -7 + 7 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} -6 q^{34} + 10 \zeta_{6} q^{37} + ( 2 - 2 \zeta_{6} ) q^{38} + 3 \zeta_{6} q^{40} -4 q^{43} + 3 \zeta_{6} q^{44} + ( 6 - 6 \zeta_{6} ) q^{46} + 12 \zeta_{6} q^{47} + 4 q^{50} + ( 2 - 2 \zeta_{6} ) q^{52} + ( 3 - 3 \zeta_{6} ) q^{53} + 9 q^{55} -9 \zeta_{6} q^{58} + ( -3 + 3 \zeta_{6} ) q^{59} -4 \zeta_{6} q^{61} + 7 q^{62} + q^{64} -6 \zeta_{6} q^{65} + ( -2 + 2 \zeta_{6} ) q^{67} + 6 \zeta_{6} q^{68} + ( 2 - 2 \zeta_{6} ) q^{73} + ( 10 - 10 \zeta_{6} ) q^{74} -2 q^{76} -5 \zeta_{6} q^{79} + ( 3 - 3 \zeta_{6} ) q^{80} -9 q^{83} + 18 q^{85} + 4 \zeta_{6} q^{86} + ( 3 - 3 \zeta_{6} ) q^{88} -6 \zeta_{6} q^{89} -6 q^{92} + ( 12 - 12 \zeta_{6} ) q^{94} + ( -6 + 6 \zeta_{6} ) q^{95} + 13 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} - q^{4} + 3q^{5} + 2q^{8} + O(q^{10})$$ $$2q - q^{2} - q^{4} + 3q^{5} + 2q^{8} + 3q^{10} + 3q^{11} - 4q^{13} - q^{16} + 6q^{17} + 2q^{19} - 6q^{20} - 6q^{22} + 6q^{23} - 4q^{25} + 2q^{26} + 18q^{29} - 7q^{31} - q^{32} - 12q^{34} + 10q^{37} + 2q^{38} + 3q^{40} - 8q^{43} + 3q^{44} + 6q^{46} + 12q^{47} + 8q^{50} + 2q^{52} + 3q^{53} + 18q^{55} - 9q^{58} - 3q^{59} - 4q^{61} + 14q^{62} + 2q^{64} - 6q^{65} - 2q^{67} + 6q^{68} + 2q^{73} + 10q^{74} - 4q^{76} - 5q^{79} + 3q^{80} - 18q^{83} + 36q^{85} + 4q^{86} + 3q^{88} - 6q^{89} - 12q^{92} + 12q^{94} - 6q^{95} + 26q^{97} + O(q^{100})$$

## 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 1.50000 + 2.59808i 0 0 1.00000 0 1.50000 2.59808i
667.1 −0.500000 + 0.866025i 0 −0.500000 0.866025i 1.50000 2.59808i 0 0 1.00000 0 1.50000 + 2.59808i
 $$n$$: 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.e 2
3.b odd 2 1 882.2.g.g 2
7.b odd 2 1 126.2.g.a 2
7.c even 3 1 882.2.a.h 1
7.c even 3 1 inner 882.2.g.e 2
7.d odd 6 1 126.2.g.a 2
7.d odd 6 1 882.2.a.j 1
21.c even 2 1 126.2.g.d yes 2
21.g even 6 1 126.2.g.d yes 2
21.g even 6 1 882.2.a.a 1
21.h odd 6 1 882.2.a.e 1
21.h odd 6 1 882.2.g.g 2
28.d even 2 1 1008.2.s.b 2
28.f even 6 1 1008.2.s.b 2
28.f even 6 1 7056.2.a.by 1
28.g odd 6 1 7056.2.a.h 1
63.i even 6 1 1134.2.h.j 2
63.k odd 6 1 1134.2.e.k 2
63.l odd 6 1 1134.2.e.k 2
63.l odd 6 1 1134.2.h.f 2
63.o even 6 1 1134.2.e.g 2
63.o even 6 1 1134.2.h.j 2
63.s even 6 1 1134.2.e.g 2
63.t odd 6 1 1134.2.h.f 2
84.h odd 2 1 1008.2.s.o 2
84.j odd 6 1 1008.2.s.o 2
84.j odd 6 1 7056.2.a.e 1
84.n even 6 1 7056.2.a.bx 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.g.a 2 7.b odd 2 1
126.2.g.a 2 7.d odd 6 1
126.2.g.d yes 2 21.c even 2 1
126.2.g.d yes 2 21.g even 6 1
882.2.a.a 1 21.g even 6 1
882.2.a.e 1 21.h odd 6 1
882.2.a.h 1 7.c even 3 1
882.2.a.j 1 7.d odd 6 1
882.2.g.e 2 1.a even 1 1 trivial
882.2.g.e 2 7.c even 3 1 inner
882.2.g.g 2 3.b odd 2 1
882.2.g.g 2 21.h odd 6 1
1008.2.s.b 2 28.d even 2 1
1008.2.s.b 2 28.f even 6 1
1008.2.s.o 2 84.h odd 2 1
1008.2.s.o 2 84.j odd 6 1
1134.2.e.g 2 63.o even 6 1
1134.2.e.g 2 63.s even 6 1
1134.2.e.k 2 63.k odd 6 1
1134.2.e.k 2 63.l odd 6 1
1134.2.h.f 2 63.l odd 6 1
1134.2.h.f 2 63.t odd 6 1
1134.2.h.j 2 63.i even 6 1
1134.2.h.j 2 63.o even 6 1
7056.2.a.e 1 84.j odd 6 1
7056.2.a.h 1 28.g odd 6 1
7056.2.a.bx 1 84.n even 6 1
7056.2.a.by 1 28.f even 6 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ 1
$5$ $$1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4}$$
$7$ 1
$11$ $$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$
$13$ $$( 1 + 2 T + 13 T^{2} )^{2}$$
$17$ $$1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4}$$
$19$ $$1 - 2 T - 15 T^{2} - 38 T^{3} + 361 T^{4}$$
$23$ $$1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4}$$
$29$ $$( 1 - 9 T + 29 T^{2} )^{2}$$
$31$ $$( 1 - 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} )$$
$37$ $$( 1 - 11 T + 37 T^{2} )( 1 + T + 37 T^{2} )$$
$41$ $$( 1 + 41 T^{2} )^{2}$$
$43$ $$( 1 + 4 T + 43 T^{2} )^{2}$$
$47$ $$1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4}$$
$53$ $$1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4}$$
$59$ $$1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4}$$
$61$ $$1 + 4 T - 45 T^{2} + 244 T^{3} + 3721 T^{4}$$
$67$ $$1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$1 - 2 T - 69 T^{2} - 146 T^{3} + 5329 T^{4}$$
$79$ $$1 + 5 T - 54 T^{2} + 395 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 9 T + 83 T^{2} )^{2}$$
$89$ $$1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4}$$
$97$ $$( 1 - 13 T + 97 T^{2} )^{2}$$