# Properties

 Label 882.2.e.m Level $882$ Weight $2$ Character orbit 882.e Analytic conductor $7.043$ Analytic rank $0$ Dimension $4$ 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.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.04280545828$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{3} + q^{4} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{6} + q^{8} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{9} +O(q^{10})$$ $$q + q^{2} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{3} + q^{4} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{6} + q^{8} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{9} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{10} + ( -2 + 2 \beta_{2} ) q^{11} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{12} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{13} + ( -3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{15} + q^{16} -2 \beta_{2} q^{17} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{18} + ( -5 + \beta_{1} + 5 \beta_{2} - 2 \beta_{3} ) q^{19} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{20} + ( -2 + 2 \beta_{2} ) q^{22} + \beta_{2} q^{23} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{24} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{25} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{26} + ( -5 - \beta_{3} ) q^{27} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{29} + ( -3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{30} + 6 q^{31} + q^{32} + ( 2 - 2 \beta_{3} ) q^{33} -2 \beta_{2} q^{34} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{36} + ( -2 - 4 \beta_{1} + 2 \beta_{2} + 8 \beta_{3} ) q^{37} + ( -5 + \beta_{1} + 5 \beta_{2} - 2 \beta_{3} ) q^{38} + ( 4 + 4 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{39} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{40} + ( -4 \beta_{1} + 8 \beta_{3} ) q^{41} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{43} + ( -2 + 2 \beta_{2} ) q^{44} + ( -5 + 2 \beta_{1} + 8 \beta_{2} - 3 \beta_{3} ) q^{45} + \beta_{2} q^{46} + ( 8 \beta_{1} - 4 \beta_{3} ) q^{47} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{48} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{50} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{51} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{52} + ( 2 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{53} + ( -5 - \beta_{3} ) q^{54} + ( 2 - 4 \beta_{1} + 2 \beta_{3} ) q^{55} + ( 3 - 2 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{57} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{58} -2 q^{59} + ( -3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{60} + ( 9 - 2 \beta_{1} + \beta_{3} ) q^{61} + 6 q^{62} + q^{64} + ( -12 + 4 \beta_{1} - 2 \beta_{3} ) q^{65} + ( 2 - 2 \beta_{3} ) q^{66} + ( -8 - 4 \beta_{1} + 2 \beta_{3} ) q^{67} -2 \beta_{2} q^{68} + ( 1 - \beta_{1} - \beta_{2} ) q^{69} + ( 5 - 4 \beta_{1} + 2 \beta_{3} ) q^{71} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{72} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{73} + ( -2 - 4 \beta_{1} + 2 \beta_{2} + 8 \beta_{3} ) q^{74} + ( -2 - 4 \beta_{1} + 8 \beta_{2} ) q^{75} + ( -5 + \beta_{1} + 5 \beta_{2} - 2 \beta_{3} ) q^{76} + ( 4 + 4 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{78} + ( 3 - 4 \beta_{1} + 2 \beta_{3} ) q^{79} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{80} + ( 4 \beta_{1} + 7 \beta_{2} - 4 \beta_{3} ) q^{81} + ( -4 \beta_{1} + 8 \beta_{3} ) q^{82} -2 \beta_{2} q^{83} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{85} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{86} + ( -2 - 6 \beta_{2} - 4 \beta_{3} ) q^{87} + ( -2 + 2 \beta_{2} ) q^{88} + ( 12 + 2 \beta_{1} - 12 \beta_{2} - 4 \beta_{3} ) q^{89} + ( -5 + 2 \beta_{1} + 8 \beta_{2} - 3 \beta_{3} ) q^{90} + \beta_{2} q^{92} + ( -6 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} ) q^{93} + ( 8 \beta_{1} - 4 \beta_{3} ) q^{94} + ( 11 - 12 \beta_{1} + 6 \beta_{3} ) q^{95} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{96} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{97} + ( -4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} - 2q^{3} + 4q^{4} - 2q^{5} - 2q^{6} + 4q^{8} + 2q^{9} + O(q^{10})$$ $$4q + 4q^{2} - 2q^{3} + 4q^{4} - 2q^{5} - 2q^{6} + 4q^{8} + 2q^{9} - 2q^{10} - 4q^{11} - 2q^{12} - 14q^{15} + 4q^{16} - 4q^{17} + 2q^{18} - 10q^{19} - 2q^{20} - 4q^{22} + 2q^{23} - 2q^{24} - 4q^{25} - 20q^{27} + 4q^{29} - 14q^{30} + 24q^{31} + 4q^{32} + 8q^{33} - 4q^{34} + 2q^{36} - 4q^{37} - 10q^{38} - 2q^{40} - 4q^{43} - 4q^{44} - 4q^{45} + 2q^{46} - 2q^{48} - 4q^{50} - 4q^{51} + 12q^{53} - 20q^{54} + 8q^{55} + 20q^{57} + 4q^{58} - 8q^{59} - 14q^{60} + 36q^{61} + 24q^{62} + 4q^{64} - 48q^{65} + 8q^{66} - 32q^{67} - 4q^{68} + 2q^{69} + 20q^{71} + 2q^{72} + 4q^{73} - 4q^{74} + 8q^{75} - 10q^{76} + 12q^{79} - 2q^{80} + 14q^{81} - 4q^{83} - 4q^{85} - 4q^{86} - 20q^{87} - 4q^{88} + 24q^{89} - 4q^{90} + 2q^{92} - 12q^{93} + 44q^{95} - 2q^{96} + 4q^{97} + 4q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

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

 $$n$$ $$199$$ $$785$$ $$\chi(n)$$ $$-1 + \beta_{2}$$ $$-1 + \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
373.1
 1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 − 0.707107i −1.22474 + 0.707107i
1.00000 −1.72474 0.158919i 1.00000 0.724745 + 1.25529i −1.72474 0.158919i 0 1.00000 2.94949 + 0.548188i 0.724745 + 1.25529i
373.2 1.00000 0.724745 1.57313i 1.00000 −1.72474 2.98735i 0.724745 1.57313i 0 1.00000 −1.94949 2.28024i −1.72474 2.98735i
655.1 1.00000 −1.72474 + 0.158919i 1.00000 0.724745 1.25529i −1.72474 + 0.158919i 0 1.00000 2.94949 0.548188i 0.724745 1.25529i
655.2 1.00000 0.724745 + 1.57313i 1.00000 −1.72474 + 2.98735i 0.724745 + 1.57313i 0 1.00000 −1.94949 + 2.28024i −1.72474 + 2.98735i
 $$n$$: 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.h 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.e.m 4
3.b odd 2 1 2646.2.e.l 4
7.b odd 2 1 882.2.e.n 4
7.c even 3 1 126.2.f.c 4
7.c even 3 1 882.2.h.k 4
7.d odd 6 1 882.2.f.j 4
7.d odd 6 1 882.2.h.l 4
9.c even 3 1 882.2.h.k 4
9.d odd 6 1 2646.2.h.m 4
21.c even 2 1 2646.2.e.k 4
21.g even 6 1 2646.2.f.k 4
21.g even 6 1 2646.2.h.n 4
21.h odd 6 1 378.2.f.d 4
21.h odd 6 1 2646.2.h.m 4
28.g odd 6 1 1008.2.r.e 4
63.g even 3 1 126.2.f.c 4
63.h even 3 1 inner 882.2.e.m 4
63.h even 3 1 1134.2.a.p 2
63.i even 6 1 2646.2.e.k 4
63.i even 6 1 7938.2.a.bm 2
63.j odd 6 1 1134.2.a.i 2
63.j odd 6 1 2646.2.e.l 4
63.k odd 6 1 882.2.f.j 4
63.l odd 6 1 882.2.h.l 4
63.n odd 6 1 378.2.f.d 4
63.o even 6 1 2646.2.h.n 4
63.s even 6 1 2646.2.f.k 4
63.t odd 6 1 882.2.e.n 4
63.t odd 6 1 7938.2.a.bn 2
84.n even 6 1 3024.2.r.e 4
252.o even 6 1 3024.2.r.e 4
252.u odd 6 1 9072.2.a.bk 2
252.bb even 6 1 9072.2.a.bd 2
252.bl odd 6 1 1008.2.r.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.c 4 7.c even 3 1
126.2.f.c 4 63.g even 3 1
378.2.f.d 4 21.h odd 6 1
378.2.f.d 4 63.n odd 6 1
882.2.e.m 4 1.a even 1 1 trivial
882.2.e.m 4 63.h even 3 1 inner
882.2.e.n 4 7.b odd 2 1
882.2.e.n 4 63.t odd 6 1
882.2.f.j 4 7.d odd 6 1
882.2.f.j 4 63.k odd 6 1
882.2.h.k 4 7.c even 3 1
882.2.h.k 4 9.c even 3 1
882.2.h.l 4 7.d odd 6 1
882.2.h.l 4 63.l odd 6 1
1008.2.r.e 4 28.g odd 6 1
1008.2.r.e 4 252.bl odd 6 1
1134.2.a.i 2 63.j odd 6 1
1134.2.a.p 2 63.h even 3 1
2646.2.e.k 4 21.c even 2 1
2646.2.e.k 4 63.i even 6 1
2646.2.e.l 4 3.b odd 2 1
2646.2.e.l 4 63.j odd 6 1
2646.2.f.k 4 21.g even 6 1
2646.2.f.k 4 63.s even 6 1
2646.2.h.m 4 9.d odd 6 1
2646.2.h.m 4 21.h odd 6 1
2646.2.h.n 4 21.g even 6 1
2646.2.h.n 4 63.o even 6 1
3024.2.r.e 4 84.n even 6 1
3024.2.r.e 4 252.o even 6 1
7938.2.a.bm 2 63.i even 6 1
7938.2.a.bn 2 63.t odd 6 1
9072.2.a.bd 2 252.bb even 6 1
9072.2.a.bk 2 252.u odd 6 1

## Hecke kernels

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

 $$T_{5}^{4} + 2 T_{5}^{3} + 9 T_{5}^{2} - 10 T_{5} + 25$$ $$T_{11}^{2} + 2 T_{11} + 4$$ $$T_{13}^{4} + 24 T_{13}^{2} + 576$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{4}$$
$3$ $$9 + 6 T + T^{2} + 2 T^{3} + T^{4}$$
$5$ $$25 - 10 T + 9 T^{2} + 2 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 4 + 2 T + T^{2} )^{2}$$
$13$ $$576 + 24 T^{2} + T^{4}$$
$17$ $$( 4 + 2 T + T^{2} )^{2}$$
$19$ $$361 + 190 T + 81 T^{2} + 10 T^{3} + T^{4}$$
$23$ $$( 1 - T + T^{2} )^{2}$$
$29$ $$400 + 80 T + 36 T^{2} - 4 T^{3} + T^{4}$$
$31$ $$( -6 + T )^{4}$$
$37$ $$8464 - 368 T + 108 T^{2} + 4 T^{3} + T^{4}$$
$41$ $$9216 + 96 T^{2} + T^{4}$$
$43$ $$400 - 80 T + 36 T^{2} + 4 T^{3} + T^{4}$$
$47$ $$( -96 + T^{2} )^{2}$$
$53$ $$144 - 144 T + 132 T^{2} - 12 T^{3} + T^{4}$$
$59$ $$( 2 + T )^{4}$$
$61$ $$( 75 - 18 T + T^{2} )^{2}$$
$67$ $$( 40 + 16 T + T^{2} )^{2}$$
$71$ $$( 1 - 10 T + T^{2} )^{2}$$
$73$ $$400 + 80 T + 36 T^{2} - 4 T^{3} + T^{4}$$
$79$ $$( -15 - 6 T + T^{2} )^{2}$$
$83$ $$( 4 + 2 T + T^{2} )^{2}$$
$89$ $$14400 - 2880 T + 456 T^{2} - 24 T^{3} + T^{4}$$
$97$ $$400 + 80 T + 36 T^{2} - 4 T^{3} + T^{4}$$