# Properties

 Label 882.2.f.d 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 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{2} + ( 1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + ( -2 + \zeta_{6} ) q^{6} + q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{2} + ( 1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{4} + ( -2 + \zeta_{6} ) q^{6} + q^{8} + 3 \zeta_{6} q^{9} + ( 3 - 3 \zeta_{6} ) q^{11} + ( 1 - 2 \zeta_{6} ) q^{12} + 2 \zeta_{6} q^{13} + ( -1 + \zeta_{6} ) q^{16} + 3 q^{17} -3 q^{18} + q^{19} + 3 \zeta_{6} q^{22} + 6 \zeta_{6} q^{23} + ( 1 + \zeta_{6} ) q^{24} + ( 5 - 5 \zeta_{6} ) q^{25} -2 q^{26} + ( -3 + 6 \zeta_{6} ) q^{27} + ( -6 + 6 \zeta_{6} ) q^{29} -4 \zeta_{6} q^{31} -\zeta_{6} q^{32} + ( 6 - 3 \zeta_{6} ) q^{33} + ( -3 + 3 \zeta_{6} ) q^{34} + ( 3 - 3 \zeta_{6} ) q^{36} -4 q^{37} + ( -1 + \zeta_{6} ) q^{38} + ( -2 + 4 \zeta_{6} ) q^{39} + 9 \zeta_{6} q^{41} + ( 1 - \zeta_{6} ) q^{43} -3 q^{44} -6 q^{46} + ( -6 + 6 \zeta_{6} ) q^{47} + ( -2 + \zeta_{6} ) q^{48} + 5 \zeta_{6} q^{50} + ( 3 + 3 \zeta_{6} ) q^{51} + ( 2 - 2 \zeta_{6} ) q^{52} + 12 q^{53} + ( -3 - 3 \zeta_{6} ) q^{54} + ( 1 + \zeta_{6} ) q^{57} -6 \zeta_{6} q^{58} + 3 \zeta_{6} q^{59} + ( 8 - 8 \zeta_{6} ) q^{61} + 4 q^{62} + q^{64} + ( -3 + 6 \zeta_{6} ) q^{66} -5 \zeta_{6} q^{67} -3 \zeta_{6} q^{68} + ( -6 + 12 \zeta_{6} ) q^{69} -12 q^{71} + 3 \zeta_{6} q^{72} -11 q^{73} + ( 4 - 4 \zeta_{6} ) q^{74} + ( 10 - 5 \zeta_{6} ) q^{75} -\zeta_{6} q^{76} + ( -2 - 2 \zeta_{6} ) q^{78} + ( 4 - 4 \zeta_{6} ) q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} -9 q^{82} + ( 12 - 12 \zeta_{6} ) q^{83} + \zeta_{6} q^{86} + ( -12 + 6 \zeta_{6} ) q^{87} + ( 3 - 3 \zeta_{6} ) q^{88} -6 q^{89} + ( 6 - 6 \zeta_{6} ) q^{92} + ( 4 - 8 \zeta_{6} ) q^{93} -6 \zeta_{6} q^{94} + ( 1 - 2 \zeta_{6} ) q^{96} + ( 5 - 5 \zeta_{6} ) q^{97} + 9 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + 3q^{3} - q^{4} - 3q^{6} + 2q^{8} + 3q^{9} + O(q^{10})$$ $$2q - q^{2} + 3q^{3} - q^{4} - 3q^{6} + 2q^{8} + 3q^{9} + 3q^{11} + 2q^{13} - q^{16} + 6q^{17} - 6q^{18} + 2q^{19} + 3q^{22} + 6q^{23} + 3q^{24} + 5q^{25} - 4q^{26} - 6q^{29} - 4q^{31} - q^{32} + 9q^{33} - 3q^{34} + 3q^{36} - 8q^{37} - q^{38} + 9q^{41} + q^{43} - 6q^{44} - 12q^{46} - 6q^{47} - 3q^{48} + 5q^{50} + 9q^{51} + 2q^{52} + 24q^{53} - 9q^{54} + 3q^{57} - 6q^{58} + 3q^{59} + 8q^{61} + 8q^{62} + 2q^{64} - 5q^{67} - 3q^{68} - 24q^{71} + 3q^{72} - 22q^{73} + 4q^{74} + 15q^{75} - q^{76} - 6q^{78} + 4q^{79} - 9q^{81} - 18q^{82} + 12q^{83} + q^{86} - 18q^{87} + 3q^{88} - 12q^{89} + 6q^{92} - 6q^{94} + 5q^{97} + 18q^{99} + O(q^{100})$$

## Character values

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

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

## 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 0 −1.50000 + 0.866025i 0 1.00000 1.50000 + 2.59808i 0
589.1 −0.500000 0.866025i 1.50000 0.866025i −0.500000 + 0.866025i 0 −1.50000 0.866025i 0 1.00000 1.50000 2.59808i 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
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.d 2
3.b odd 2 1 2646.2.f.g 2
7.b odd 2 1 18.2.c.a 2
7.c even 3 1 882.2.e.g 2
7.c even 3 1 882.2.h.b 2
7.d odd 6 1 882.2.e.i 2
7.d odd 6 1 882.2.h.c 2
9.c even 3 1 inner 882.2.f.d 2
9.c even 3 1 7938.2.a.x 1
9.d odd 6 1 2646.2.f.g 2
9.d odd 6 1 7938.2.a.i 1
21.c even 2 1 54.2.c.a 2
21.g even 6 1 2646.2.e.b 2
21.g even 6 1 2646.2.h.h 2
21.h odd 6 1 2646.2.e.c 2
21.h odd 6 1 2646.2.h.i 2
28.d even 2 1 144.2.i.c 2
35.c odd 2 1 450.2.e.i 2
35.f even 4 2 450.2.j.e 4
56.e even 2 1 576.2.i.a 2
56.h odd 2 1 576.2.i.g 2
63.g even 3 1 882.2.e.g 2
63.h even 3 1 882.2.h.b 2
63.i even 6 1 2646.2.h.h 2
63.j odd 6 1 2646.2.h.i 2
63.k odd 6 1 882.2.e.i 2
63.l odd 6 1 18.2.c.a 2
63.l odd 6 1 162.2.a.c 1
63.n odd 6 1 2646.2.e.c 2
63.o even 6 1 54.2.c.a 2
63.o even 6 1 162.2.a.b 1
63.s even 6 1 2646.2.e.b 2
63.t odd 6 1 882.2.h.c 2
84.h odd 2 1 432.2.i.b 2
105.g even 2 1 1350.2.e.c 2
105.k odd 4 2 1350.2.j.a 4
168.e odd 2 1 1728.2.i.f 2
168.i even 2 1 1728.2.i.e 2
252.s odd 6 1 432.2.i.b 2
252.s odd 6 1 1296.2.a.f 1
252.bi even 6 1 144.2.i.c 2
252.bi even 6 1 1296.2.a.g 1
315.z even 6 1 1350.2.e.c 2
315.z even 6 1 4050.2.a.v 1
315.bg odd 6 1 450.2.e.i 2
315.bg odd 6 1 4050.2.a.c 1
315.cb even 12 2 450.2.j.e 4
315.cb even 12 2 4050.2.c.c 2
315.cf odd 12 2 1350.2.j.a 4
315.cf odd 12 2 4050.2.c.r 2
504.be even 6 1 576.2.i.a 2
504.be even 6 1 5184.2.a.o 1
504.bn odd 6 1 576.2.i.g 2
504.bn odd 6 1 5184.2.a.r 1
504.cc even 6 1 1728.2.i.e 2
504.cc even 6 1 5184.2.a.q 1
504.co odd 6 1 1728.2.i.f 2
504.co odd 6 1 5184.2.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.2.c.a 2 7.b odd 2 1
18.2.c.a 2 63.l odd 6 1
54.2.c.a 2 21.c even 2 1
54.2.c.a 2 63.o even 6 1
144.2.i.c 2 28.d even 2 1
144.2.i.c 2 252.bi even 6 1
162.2.a.b 1 63.o even 6 1
162.2.a.c 1 63.l odd 6 1
432.2.i.b 2 84.h odd 2 1
432.2.i.b 2 252.s odd 6 1
450.2.e.i 2 35.c odd 2 1
450.2.e.i 2 315.bg odd 6 1
450.2.j.e 4 35.f even 4 2
450.2.j.e 4 315.cb even 12 2
576.2.i.a 2 56.e even 2 1
576.2.i.a 2 504.be even 6 1
576.2.i.g 2 56.h odd 2 1
576.2.i.g 2 504.bn odd 6 1
882.2.e.g 2 7.c even 3 1
882.2.e.g 2 63.g even 3 1
882.2.e.i 2 7.d odd 6 1
882.2.e.i 2 63.k odd 6 1
882.2.f.d 2 1.a even 1 1 trivial
882.2.f.d 2 9.c even 3 1 inner
882.2.h.b 2 7.c even 3 1
882.2.h.b 2 63.h even 3 1
882.2.h.c 2 7.d odd 6 1
882.2.h.c 2 63.t odd 6 1
1296.2.a.f 1 252.s odd 6 1
1296.2.a.g 1 252.bi even 6 1
1350.2.e.c 2 105.g even 2 1
1350.2.e.c 2 315.z even 6 1
1350.2.j.a 4 105.k odd 4 2
1350.2.j.a 4 315.cf odd 12 2
1728.2.i.e 2 168.i even 2 1
1728.2.i.e 2 504.cc even 6 1
1728.2.i.f 2 168.e odd 2 1
1728.2.i.f 2 504.co odd 6 1
2646.2.e.b 2 21.g even 6 1
2646.2.e.b 2 63.s even 6 1
2646.2.e.c 2 21.h odd 6 1
2646.2.e.c 2 63.n odd 6 1
2646.2.f.g 2 3.b odd 2 1
2646.2.f.g 2 9.d odd 6 1
2646.2.h.h 2 21.g even 6 1
2646.2.h.h 2 63.i even 6 1
2646.2.h.i 2 21.h odd 6 1
2646.2.h.i 2 63.j odd 6 1
4050.2.a.c 1 315.bg odd 6 1
4050.2.a.v 1 315.z even 6 1
4050.2.c.c 2 315.cb even 12 2
4050.2.c.r 2 315.cf odd 12 2
5184.2.a.o 1 504.be even 6 1
5184.2.a.p 1 504.co odd 6 1
5184.2.a.q 1 504.cc even 6 1
5184.2.a.r 1 504.bn odd 6 1
7938.2.a.i 1 9.d odd 6 1
7938.2.a.x 1 9.c even 3 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{11}^{2} - 3 T_{11} + 9$$ $$T_{13}^{2} - 2 T_{13} + 4$$

## Hecke characteristic polynomials

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