Properties

 Label 882.2.h.n Level $882$ Weight $2$ Character orbit 882.h 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.h (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: $$\Z[a_1, a_2, a_3]$$ 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.

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

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 $$\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$$.

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 −1.18614 − 1.26217i 1.68614 + 0.396143i −1.18614 + 1.26217i 1.68614 − 0.396143i
0.500000 0.866025i −1.18614 + 1.26217i −0.500000 0.866025i −4.37228 0.500000 + 1.65831i 0 −1.00000 −0.186141 2.99422i −2.18614 + 3.78651i
67.2 0.500000 0.866025i 1.68614 0.396143i −0.500000 0.866025i 1.37228 0.500000 1.65831i 0 −1.00000 2.68614 1.33591i 0.686141 1.18843i
79.1 0.500000 + 0.866025i −1.18614 1.26217i −0.500000 + 0.866025i −4.37228 0.500000 1.65831i 0 −1.00000 −0.186141 + 2.99422i −2.18614 3.78651i
79.2 0.500000 + 0.866025i 1.68614 + 0.396143i −0.500000 + 0.866025i 1.37228 0.500000 + 1.65831i 0 −1.00000 2.68614 + 1.33591i 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.g 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.h.n 4
3.b odd 2 1 2646.2.h.l 4
7.b odd 2 1 882.2.h.m 4
7.c even 3 1 882.2.e.k 4
7.c even 3 1 882.2.f.k 4
7.d odd 6 1 126.2.f.d 4
7.d odd 6 1 882.2.e.l 4
9.c even 3 1 882.2.e.k 4
9.d odd 6 1 2646.2.e.m 4
21.c even 2 1 2646.2.h.k 4
21.g even 6 1 378.2.f.c 4
21.g even 6 1 2646.2.e.n 4
21.h odd 6 1 2646.2.e.m 4
21.h odd 6 1 2646.2.f.j 4
28.f even 6 1 1008.2.r.f 4
63.g even 3 1 inner 882.2.h.n 4
63.g even 3 1 7938.2.a.bh 2
63.h even 3 1 882.2.f.k 4
63.i even 6 1 378.2.f.c 4
63.j odd 6 1 2646.2.f.j 4
63.k odd 6 1 882.2.h.m 4
63.k odd 6 1 1134.2.a.k 2
63.l odd 6 1 882.2.e.l 4
63.n odd 6 1 2646.2.h.l 4
63.n odd 6 1 7938.2.a.bs 2
63.o even 6 1 2646.2.e.n 4
63.s even 6 1 1134.2.a.n 2
63.s even 6 1 2646.2.h.k 4
63.t odd 6 1 126.2.f.d 4
84.j odd 6 1 3024.2.r.f 4
252.n even 6 1 9072.2.a.bm 2
252.r odd 6 1 3024.2.r.f 4
252.bj even 6 1 1008.2.r.f 4
252.bn odd 6 1 9072.2.a.bb 2

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

Hecke kernels

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

 $$T_{5}^{2} + 3 T_{5} - 6$$ $$T_{11}^{2} - 3 T_{11} - 6$$ $$T_{13}^{2} - 2 T_{13} + 4$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )^{2}$$
$3$ $$9 - 3 T - 2 T^{2} - T^{3} + T^{4}$$
$5$ $$( -6 + 3 T + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$( -6 - 3 T + T^{2} )^{2}$$
$13$ $$( 4 - 2 T + T^{2} )^{2}$$
$17$ $$36 - 18 T + 15 T^{2} + 3 T^{3} + T^{4}$$
$19$ $$( 25 - 5 T + T^{2} )^{2}$$
$23$ $$( 12 + 9 T + T^{2} )^{2}$$
$29$ $$576 + 144 T + 60 T^{2} - 6 T^{3} + T^{4}$$
$31$ $$( 4 - 2 T + T^{2} )^{2}$$
$37$ $$( 4 + 2 T + T^{2} )^{2}$$
$41$ $$2304 - 720 T + 177 T^{2} - 15 T^{3} + T^{4}$$
$43$ $$5476 - 74 T + 75 T^{2} + T^{3} + T^{4}$$
$47$ $$T^{4}$$
$53$ $$576 + 144 T + 60 T^{2} - 6 T^{3} + T^{4}$$
$59$ $$5184 - 216 T + 81 T^{2} + 3 T^{3} + T^{4}$$
$61$ $$1936 - 484 T + 165 T^{2} + 11 T^{3} + T^{4}$$
$67$ $$1024 - 416 T + 201 T^{2} + 13 T^{3} + T^{4}$$
$71$ $$( -72 - 3 T + T^{2} )^{2}$$
$73$ $$3844 + 434 T + 111 T^{2} - 7 T^{3} + T^{4}$$
$79$ $$3844 - 434 T + 111 T^{2} + 7 T^{3} + T^{4}$$
$83$ $$9216 + 1152 T + 240 T^{2} - 12 T^{3} + T^{4}$$
$89$ $$2304 - 864 T + 276 T^{2} - 18 T^{3} + T^{4}$$
$97$ $$5476 + 74 T + 75 T^{2} - T^{3} + T^{4}$$