# Properties

 Label 882.2.e.l Level $882$ Weight $2$ Character orbit 882.e Analytic conductor $7.043$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

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## 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: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + ( \beta_{1} - \beta_{3} ) q^{3} + q^{4} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{5} + ( -\beta_{1} + \beta_{3} ) q^{6} - q^{8} + ( -3 + \beta_{1} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q - q^{2} + ( \beta_{1} - \beta_{3} ) q^{3} + q^{4} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{5} + ( -\beta_{1} + \beta_{3} ) q^{6} - q^{8} + ( -3 + \beta_{1} - \beta_{3} ) q^{9} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{10} + ( -1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{11} + ( \beta_{1} - \beta_{3} ) q^{12} + ( -2 + 2 \beta_{2} ) q^{13} + ( 6 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{15} + q^{16} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{17} + ( 3 - \beta_{1} + \beta_{3} ) q^{18} + ( -5 + 5 \beta_{2} ) q^{19} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{20} + ( 1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{22} + ( 1 - 2 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{23} + ( -\beta_{1} + \beta_{3} ) q^{24} + ( -7 + 3 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} ) q^{25} + ( 2 - 2 \beta_{2} ) q^{26} + ( -3 - 2 \beta_{1} + 2 \beta_{3} ) q^{27} + ( -2 + 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{29} + ( -6 + \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{30} + 2 q^{31} - q^{32} + ( 3 - 2 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{33} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{34} + ( -3 + \beta_{1} - \beta_{3} ) q^{36} + ( -2 + 2 \beta_{2} ) q^{37} + ( 5 - 5 \beta_{2} ) q^{38} -2 \beta_{1} q^{39} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{40} + ( -7 - \beta_{1} + 8 \beta_{2} + 2 \beta_{3} ) q^{41} + ( -3 + 6 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{43} + ( -1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{44} + ( 3 + 5 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{45} + ( -1 + 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{46} + ( \beta_{1} - \beta_{3} ) q^{48} + ( 7 - 3 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} ) q^{50} + ( 6 - \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{51} + ( -2 + 2 \beta_{2} ) q^{52} + ( -2 + 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{53} + ( 3 + 2 \beta_{1} - 2 \beta_{3} ) q^{54} -6 q^{55} -5 \beta_{1} q^{57} + ( 2 - 4 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{58} + ( -3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{59} + ( 6 - \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{60} + ( -7 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{61} -2 q^{62} + q^{64} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{65} + ( -3 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{66} + ( 8 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{67} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{68} + ( 6 - \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{69} + ( 3 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{71} + ( 3 - \beta_{1} + \beta_{3} ) q^{72} + ( 3 - 6 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} ) q^{73} + ( 2 - 2 \beta_{2} ) q^{74} + ( -9 - 4 \beta_{1} - 9 \beta_{2} - 3 \beta_{3} ) q^{75} + ( -5 + 5 \beta_{2} ) q^{76} + 2 \beta_{1} q^{78} + ( 5 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{79} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{80} + ( 6 - 5 \beta_{1} + 5 \beta_{3} ) q^{81} + ( 7 + \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{82} + ( 4 - 8 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{83} + ( -6 + 6 \beta_{2} ) q^{85} + ( 3 - 6 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{86} + ( -12 + 2 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} ) q^{87} + ( 1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{88} + ( -10 + 2 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{89} + ( -3 - 5 \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{90} + ( 1 - 2 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{92} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{93} + ( 5 + 5 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} ) q^{95} + ( -\beta_{1} + \beta_{3} ) q^{96} + ( -3 + 6 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{97} + ( 6 + \beta_{1} - 3 \beta_{2} - 5 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} + 2q^{3} + 4q^{4} - 3q^{5} - 2q^{6} - 4q^{8} - 10q^{9} + O(q^{10})$$ $$4q - 4q^{2} + 2q^{3} + 4q^{4} - 3q^{5} - 2q^{6} - 4q^{8} - 10q^{9} + 3q^{10} - 3q^{11} + 2q^{12} - 4q^{13} + 15q^{15} + 4q^{16} + 3q^{17} + 10q^{18} - 10q^{19} - 3q^{20} + 3q^{22} + 9q^{23} - 2q^{24} - 11q^{25} + 4q^{26} - 16q^{27} + 6q^{29} - 15q^{30} + 8q^{31} - 4q^{32} + 15q^{33} - 3q^{34} - 10q^{36} - 4q^{37} + 10q^{38} - 2q^{39} + 3q^{40} - 15q^{41} - q^{43} - 3q^{44} + 24q^{45} - 9q^{46} + 2q^{48} + 11q^{50} + 18q^{51} - 4q^{52} + 6q^{53} + 16q^{54} - 24q^{55} - 5q^{57} - 6q^{58} - 6q^{59} + 15q^{60} - 22q^{61} - 8q^{62} + 4q^{64} + 12q^{65} - 15q^{66} + 26q^{67} + 3q^{68} + 21q^{69} + 6q^{71} + 10q^{72} - 7q^{73} + 4q^{74} - 55q^{75} - 10q^{76} + 2q^{78} + 14q^{79} - 3q^{80} + 14q^{81} + 15q^{82} - 12q^{83} - 12q^{85} + q^{86} - 30q^{87} + 3q^{88} - 18q^{89} - 24q^{90} + 9q^{92} + 4q^{93} + 30q^{95} - 2q^{96} - q^{97} + 24q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 3$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu + 3$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{3} + 2 \beta_{1} + 3$$

## 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
 −1.18614 − 1.26217i 1.68614 + 0.396143i 1.68614 − 0.396143i −1.18614 + 1.26217i
−1.00000 0.500000 1.65831i 1.00000 0.686141 + 1.18843i −0.500000 + 1.65831i 0 −1.00000 −2.50000 1.65831i −0.686141 1.18843i
373.2 −1.00000 0.500000 + 1.65831i 1.00000 −2.18614 3.78651i −0.500000 1.65831i 0 −1.00000 −2.50000 + 1.65831i 2.18614 + 3.78651i
655.1 −1.00000 0.500000 1.65831i 1.00000 −2.18614 + 3.78651i −0.500000 + 1.65831i 0 −1.00000 −2.50000 1.65831i 2.18614 3.78651i
655.2 −1.00000 0.500000 + 1.65831i 1.00000 0.686141 1.18843i −0.500000 1.65831i 0 −1.00000 −2.50000 + 1.65831i −0.686141 + 1.18843i
 $$n$$: 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.l 4
3.b odd 2 1 2646.2.e.n 4
7.b odd 2 1 882.2.e.k 4
7.c even 3 1 126.2.f.d 4
7.c even 3 1 882.2.h.m 4
7.d odd 6 1 882.2.f.k 4
7.d odd 6 1 882.2.h.n 4
9.c even 3 1 882.2.h.m 4
9.d odd 6 1 2646.2.h.k 4
21.c even 2 1 2646.2.e.m 4
21.g even 6 1 2646.2.f.j 4
21.g even 6 1 2646.2.h.l 4
21.h odd 6 1 378.2.f.c 4
21.h odd 6 1 2646.2.h.k 4
28.g odd 6 1 1008.2.r.f 4
63.g even 3 1 126.2.f.d 4
63.h even 3 1 inner 882.2.e.l 4
63.h even 3 1 1134.2.a.k 2
63.i even 6 1 2646.2.e.m 4
63.i even 6 1 7938.2.a.bs 2
63.j odd 6 1 1134.2.a.n 2
63.j odd 6 1 2646.2.e.n 4
63.k odd 6 1 882.2.f.k 4
63.l odd 6 1 882.2.h.n 4
63.n odd 6 1 378.2.f.c 4
63.o even 6 1 2646.2.h.l 4
63.s even 6 1 2646.2.f.j 4
63.t odd 6 1 882.2.e.k 4
63.t odd 6 1 7938.2.a.bh 2
84.n even 6 1 3024.2.r.f 4
252.o even 6 1 3024.2.r.f 4
252.u odd 6 1 9072.2.a.bm 2
252.bb even 6 1 9072.2.a.bb 2
252.bl odd 6 1 1008.2.r.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.d 4 7.c even 3 1
126.2.f.d 4 63.g even 3 1
378.2.f.c 4 21.h odd 6 1
378.2.f.c 4 63.n odd 6 1
882.2.e.k 4 7.b odd 2 1
882.2.e.k 4 63.t odd 6 1
882.2.e.l 4 1.a even 1 1 trivial
882.2.e.l 4 63.h even 3 1 inner
882.2.f.k 4 7.d odd 6 1
882.2.f.k 4 63.k odd 6 1
882.2.h.m 4 7.c even 3 1
882.2.h.m 4 9.c even 3 1
882.2.h.n 4 7.d odd 6 1
882.2.h.n 4 63.l odd 6 1
1008.2.r.f 4 28.g odd 6 1
1008.2.r.f 4 252.bl odd 6 1
1134.2.a.k 2 63.h even 3 1
1134.2.a.n 2 63.j odd 6 1
2646.2.e.m 4 21.c even 2 1
2646.2.e.m 4 63.i even 6 1
2646.2.e.n 4 3.b odd 2 1
2646.2.e.n 4 63.j odd 6 1
2646.2.f.j 4 21.g even 6 1
2646.2.f.j 4 63.s even 6 1
2646.2.h.k 4 9.d odd 6 1
2646.2.h.k 4 21.h odd 6 1
2646.2.h.l 4 21.g even 6 1
2646.2.h.l 4 63.o even 6 1
3024.2.r.f 4 84.n even 6 1
3024.2.r.f 4 252.o even 6 1
7938.2.a.bh 2 63.t odd 6 1
7938.2.a.bs 2 63.i even 6 1
9072.2.a.bb 2 252.bb even 6 1
9072.2.a.bm 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 $$S_{2}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{4} + 3 T_{5}^{3} + 15 T_{5}^{2} - 18 T_{5} + 36$$ $$T_{11}^{4} + 3 T_{11}^{3} + 15 T_{11}^{2} - 18 T_{11} + 36$$ $$T_{13}^{2} + 2 T_{13} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{4}$$
$3$ $$( 3 - T + T^{2} )^{2}$$
$5$ $$36 - 18 T + 15 T^{2} + 3 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$36 - 18 T + 15 T^{2} + 3 T^{3} + T^{4}$$
$13$ $$( 4 + 2 T + T^{2} )^{2}$$
$17$ $$36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4}$$
$19$ $$( 25 + 5 T + T^{2} )^{2}$$
$23$ $$144 - 108 T + 69 T^{2} - 9 T^{3} + T^{4}$$
$29$ $$576 + 144 T + 60 T^{2} - 6 T^{3} + T^{4}$$
$31$ $$( -2 + T )^{4}$$
$37$ $$( 4 + 2 T + T^{2} )^{2}$$
$41$ $$2304 + 720 T + 177 T^{2} + 15 T^{3} + T^{4}$$
$43$ $$5476 - 74 T + 75 T^{2} + T^{3} + T^{4}$$
$47$ $$T^{4}$$
$53$ $$576 + 144 T + 60 T^{2} - 6 T^{3} + T^{4}$$
$59$ $$( -72 + 3 T + T^{2} )^{2}$$
$61$ $$( -44 + 11 T + T^{2} )^{2}$$
$67$ $$( -32 - 13 T + T^{2} )^{2}$$
$71$ $$( -72 - 3 T + T^{2} )^{2}$$
$73$ $$3844 - 434 T + 111 T^{2} + 7 T^{3} + T^{4}$$
$79$ $$( -62 - 7 T + T^{2} )^{2}$$
$83$ $$9216 - 1152 T + 240 T^{2} + 12 T^{3} + T^{4}$$
$89$ $$2304 + 864 T + 276 T^{2} + 18 T^{3} + T^{4}$$
$97$ $$5476 - 74 T + 75 T^{2} + T^{3} + T^{4}$$
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