# Properties

 Label 882.3.s.e Level $882$ Weight $3$ Character orbit 882.s Analytic conductor $24.033$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 882.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.0327593166$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.12745506816.5 Defining polynomial: $$x^{8} - 8 x^{6} + 55 x^{4} - 72 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 126) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{2} + \beta_{6} ) q^{2} + 2 \beta_{1} q^{4} + ( -\beta_{2} - \beta_{6} + \beta_{7} ) q^{5} + 2 \beta_{6} q^{8} +O(q^{10})$$ $$q + ( \beta_{2} + \beta_{6} ) q^{2} + 2 \beta_{1} q^{4} + ( -\beta_{2} - \beta_{6} + \beta_{7} ) q^{5} + 2 \beta_{6} q^{8} + ( -2 \beta_{1} + \beta_{5} ) q^{10} + ( 2 \beta_{2} - 2 \beta_{4} ) q^{11} + ( -8 + \beta_{3} ) q^{13} + ( -4 + 4 \beta_{1} ) q^{16} + ( -13 \beta_{2} + \beta_{4} ) q^{17} + ( -20 + 20 \beta_{1} ) q^{19} + ( -2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{20} + ( 4 + 2 \beta_{3} ) q^{22} + ( -2 \beta_{2} - 2 \beta_{6} + 2 \beta_{7} ) q^{23} + ( 33 \beta_{1} - 2 \beta_{5} ) q^{25} + ( -8 \beta_{2} - 8 \beta_{6} - 2 \beta_{7} ) q^{26} + ( -2 \beta_{4} + 19 \beta_{6} + 2 \beta_{7} ) q^{29} + ( -4 \beta_{1} - 2 \beta_{5} ) q^{31} -4 \beta_{2} q^{32} + ( -26 - \beta_{3} ) q^{34} + ( -38 + 38 \beta_{1} ) q^{37} -20 \beta_{2} q^{38} + ( 4 - 4 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} ) q^{40} + ( 3 \beta_{4} - 27 \beta_{6} - 3 \beta_{7} ) q^{41} + ( 20 - 6 \beta_{3} ) q^{43} + ( 4 \beta_{2} + 4 \beta_{6} - 4 \beta_{7} ) q^{44} + ( -4 \beta_{1} + 2 \beta_{5} ) q^{46} + ( 12 \beta_{2} + 12 \beta_{6} ) q^{47} + ( 4 \beta_{4} + 33 \beta_{6} - 4 \beta_{7} ) q^{50} + ( -16 \beta_{1} - 2 \beta_{5} ) q^{52} + ( 3 \beta_{2} + 12 \beta_{4} ) q^{53} + ( -116 - 4 \beta_{3} ) q^{55} + ( -38 + 38 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} ) q^{58} + ( -20 \beta_{2} - 4 \beta_{4} ) q^{59} + ( -58 + 58 \beta_{1} - 4 \beta_{3} - 4 \beta_{5} ) q^{61} + ( 4 \beta_{4} - 4 \beta_{6} - 4 \beta_{7} ) q^{62} -8 q^{64} + ( -48 \beta_{2} - 48 \beta_{6} - 6 \beta_{7} ) q^{65} + ( 48 \beta_{1} + 8 \beta_{5} ) q^{67} + ( -26 \beta_{2} - 26 \beta_{6} + 2 \beta_{7} ) q^{68} + ( 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{71} + ( 24 \beta_{1} + 5 \beta_{5} ) q^{73} -38 \beta_{2} q^{74} -40 q^{76} + ( -76 + 76 \beta_{1} - 4 \beta_{3} - 4 \beta_{5} ) q^{79} + ( 4 \beta_{2} - 4 \beta_{4} ) q^{80} + ( 54 - 54 \beta_{1} - 3 \beta_{3} - 3 \beta_{5} ) q^{82} + ( -4 \beta_{4} - 64 \beta_{6} + 4 \beta_{7} ) q^{83} + ( 82 + 14 \beta_{3} ) q^{85} + ( 20 \beta_{2} + 20 \beta_{6} + 12 \beta_{7} ) q^{86} + ( 8 \beta_{1} - 4 \beta_{5} ) q^{88} + ( 51 \beta_{2} + 51 \beta_{6} - 9 \beta_{7} ) q^{89} + ( -4 \beta_{4} - 4 \beta_{6} + 4 \beta_{7} ) q^{92} + 24 \beta_{1} q^{94} + ( 20 \beta_{2} - 20 \beta_{4} ) q^{95} + ( -72 + 11 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{4} + O(q^{10})$$ $$8q + 8q^{4} - 8q^{10} - 64q^{13} - 16q^{16} - 80q^{19} + 32q^{22} + 132q^{25} - 16q^{31} - 208q^{34} - 152q^{37} + 16q^{40} + 160q^{43} - 16q^{46} - 64q^{52} - 928q^{55} - 152q^{58} - 232q^{61} - 64q^{64} + 192q^{67} + 96q^{73} - 320q^{76} - 304q^{79} + 216q^{82} + 656q^{85} + 32q^{88} + 96q^{94} - 576q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 8 x^{6} + 55 x^{4} - 72 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-8 \nu^{6} + 55 \nu^{4} - 440 \nu^{2} + 576$$$$)/495$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 203 \nu$$$$)/165$$ $$\beta_{3}$$ $$=$$ $$($$$$-4 \nu^{6} - 592$$$$)/55$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{7} + 1066 \nu$$$$)/165$$ $$\beta_{5}$$ $$=$$ $$($$$$-92 \nu^{6} + 880 \nu^{4} - 5060 \nu^{2} + 6624$$$$)/495$$ $$\beta_{6}$$ $$=$$ $$($$$$-8 \nu^{7} + 55 \nu^{5} - 341 \nu^{3} + 81 \nu$$$$)/297$$ $$\beta_{7}$$ $$=$$ $$($$$$-158 \nu^{7} + 1210 \nu^{5} - 8690 \nu^{3} + 11376 \nu$$$$)/1485$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} - 2 \beta_{2}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + \beta_{3} - 16 \beta_{1} + 16$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{7} + 22 \beta_{6} + 5 \beta_{4}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{5} - 23 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$($$$$-31 \beta_{7} + 158 \beta_{6} + 158 \beta_{2}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-55 \beta_{3} - 592$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-203 \beta_{4} + 1066 \beta_{2}$$$$)/4$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
557.1
 −1.00781 − 0.581861i 2.23256 + 1.28897i −2.23256 − 1.28897i 1.00781 + 0.581861i −1.00781 + 0.581861i 2.23256 − 1.28897i −2.23256 + 1.28897i 1.00781 − 0.581861i
−1.22474 + 0.707107i 0 1.00000 1.73205i −5.25600 + 3.03455i 0 0 2.82843i 0 4.29150 7.43310i
557.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 7.70549 4.44876i 0 0 2.82843i 0 −6.29150 + 10.8972i
557.3 1.22474 0.707107i 0 1.00000 1.73205i −7.70549 + 4.44876i 0 0 2.82843i 0 −6.29150 + 10.8972i
557.4 1.22474 0.707107i 0 1.00000 1.73205i 5.25600 3.03455i 0 0 2.82843i 0 4.29150 7.43310i
863.1 −1.22474 0.707107i 0 1.00000 + 1.73205i −5.25600 3.03455i 0 0 2.82843i 0 4.29150 + 7.43310i
863.2 −1.22474 0.707107i 0 1.00000 + 1.73205i 7.70549 + 4.44876i 0 0 2.82843i 0 −6.29150 10.8972i
863.3 1.22474 + 0.707107i 0 1.00000 + 1.73205i −7.70549 4.44876i 0 0 2.82843i 0 −6.29150 10.8972i
863.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 5.25600 + 3.03455i 0 0 2.82843i 0 4.29150 + 7.43310i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 863.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.3.s.e 8
3.b odd 2 1 inner 882.3.s.e 8
7.b odd 2 1 882.3.s.i 8
7.c even 3 1 126.3.b.a 4
7.c even 3 1 inner 882.3.s.e 8
7.d odd 6 1 882.3.b.f 4
7.d odd 6 1 882.3.s.i 8
21.c even 2 1 882.3.s.i 8
21.g even 6 1 882.3.b.f 4
21.g even 6 1 882.3.s.i 8
21.h odd 6 1 126.3.b.a 4
21.h odd 6 1 inner 882.3.s.e 8
28.g odd 6 1 1008.3.d.a 4
35.j even 6 1 3150.3.e.e 4
35.l odd 12 2 3150.3.c.b 8
56.k odd 6 1 4032.3.d.j 4
56.p even 6 1 4032.3.d.i 4
63.g even 3 1 1134.3.q.c 8
63.h even 3 1 1134.3.q.c 8
63.j odd 6 1 1134.3.q.c 8
63.n odd 6 1 1134.3.q.c 8
84.n even 6 1 1008.3.d.a 4
105.o odd 6 1 3150.3.e.e 4
105.x even 12 2 3150.3.c.b 8
168.s odd 6 1 4032.3.d.i 4
168.v even 6 1 4032.3.d.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.b.a 4 7.c even 3 1
126.3.b.a 4 21.h odd 6 1
882.3.b.f 4 7.d odd 6 1
882.3.b.f 4 21.g even 6 1
882.3.s.e 8 1.a even 1 1 trivial
882.3.s.e 8 3.b odd 2 1 inner
882.3.s.e 8 7.c even 3 1 inner
882.3.s.e 8 21.h odd 6 1 inner
882.3.s.i 8 7.b odd 2 1
882.3.s.i 8 7.d odd 6 1
882.3.s.i 8 21.c even 2 1
882.3.s.i 8 21.g even 6 1
1008.3.d.a 4 28.g odd 6 1
1008.3.d.a 4 84.n even 6 1
1134.3.q.c 8 63.g even 3 1
1134.3.q.c 8 63.h even 3 1
1134.3.q.c 8 63.j odd 6 1
1134.3.q.c 8 63.n odd 6 1
3150.3.c.b 8 35.l odd 12 2
3150.3.c.b 8 105.x even 12 2
3150.3.e.e 4 35.j even 6 1
3150.3.e.e 4 105.o odd 6 1
4032.3.d.i 4 56.p even 6 1
4032.3.d.i 4 168.s odd 6 1
4032.3.d.j 4 56.k odd 6 1
4032.3.d.j 4 168.v even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{8} - 116 T_{5}^{6} + 10540 T_{5}^{4} - 338256 T_{5}^{2} + 8503056$$ $$T_{11}^{8} - 464 T_{11}^{6} + 168640 T_{11}^{4} - 21648384 T_{11}^{2} + 2176782336$$ $$T_{13}^{2} + 16 T_{13} - 48$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 - 2 T^{2} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$8503056 - 338256 T^{2} + 10540 T^{4} - 116 T^{6} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$2176782336 - 21648384 T^{2} + 168640 T^{4} - 464 T^{6} + T^{8}$$
$13$ $$( -48 + 16 T + T^{2} )^{4}$$
$17$ $$6324066576 - 62664912 T^{2} + 541420 T^{4} - 788 T^{6} + T^{8}$$
$19$ $$( 400 + 20 T + T^{2} )^{4}$$
$23$ $$2176782336 - 21648384 T^{2} + 168640 T^{4} - 464 T^{6} + T^{8}$$
$29$ $$( 248004 + 1892 T^{2} + T^{4} )^{2}$$
$31$ $$( 186624 - 3456 T + 496 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$37$ $$( 1444 + 38 T + T^{2} )^{4}$$
$41$ $$( 910116 + 3924 T^{2} + T^{4} )^{2}$$
$43$ $$( -3632 - 40 T + T^{2} )^{4}$$
$47$ $$( 82944 - 288 T^{2} + T^{4} )^{2}$$
$53$ $$4191023663229456 - 1046426907024 T^{2} + 196536780 T^{4} - 16164 T^{6} + T^{8}$$
$59$ $$84934656 - 31260672 T^{2} + 11496448 T^{4} - 3392 T^{6} + T^{8}$$
$61$ $$( 2471184 + 182352 T + 11884 T^{2} + 116 T^{3} + T^{4} )^{2}$$
$67$ $$( 23658496 + 466944 T + 14080 T^{2} - 96 T^{3} + T^{4} )^{2}$$
$71$ $$( 46656 + 464 T^{2} + T^{4} )^{2}$$
$73$ $$( 4946176 + 106752 T + 4528 T^{2} - 48 T^{3} + T^{4} )^{2}$$
$79$ $$( 15872256 + 605568 T + 19120 T^{2} + 152 T^{3} + T^{4} )^{2}$$
$83$ $$( 53231616 + 18176 T^{2} + T^{4} )^{2}$$
$89$ $$196741925136 - 8638696656 T^{2} + 378871020 T^{4} - 19476 T^{6} + T^{8}$$
$97$ $$( -8368 + 144 T + T^{2} )^{4}$$