# Properties

 Label 882.2.d.a Level 882 Weight 2 Character orbit 882.d Analytic conductor 7.043 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: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 882.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.04280545828$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 126) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{24}^{6} q^{2} - q^{4} + ( -\zeta_{24} - 2 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{5} -\zeta_{24}^{6} q^{8} +O(q^{10})$$ $$q + \zeta_{24}^{6} q^{2} - q^{4} + ( -\zeta_{24} - 2 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{5} -\zeta_{24}^{6} q^{8} + ( 1 - \zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{4} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{10} + 3 \zeta_{24}^{6} q^{11} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{13} + q^{16} + ( -\zeta_{24} + 4 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{17} + ( 2 + \zeta_{24} - \zeta_{24}^{3} - 4 \zeta_{24}^{4} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{19} + ( \zeta_{24} + 2 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{20} -3 q^{22} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{23} + ( 4 + 6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{25} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{26} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{29} + ( -1 - 3 \zeta_{24} + 3 \zeta_{24}^{3} + 2 \zeta_{24}^{4} - 3 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{31} + \zeta_{24}^{6} q^{32} + ( -2 - \zeta_{24} + \zeta_{24}^{3} + 4 \zeta_{24}^{4} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{34} + ( 4 + 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{37} + ( -\zeta_{24} + 4 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{38} + ( -1 + \zeta_{24} - \zeta_{24}^{3} + 2 \zeta_{24}^{4} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{40} + ( -2 \zeta_{24} + 8 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{41} + ( 4 + 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{43} -3 \zeta_{24}^{6} q^{44} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{46} + ( -\zeta_{24} + 4 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{47} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{50} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{52} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{53} + ( 3 - 3 \zeta_{24} + 3 \zeta_{24}^{3} - 6 \zeta_{24}^{4} - 3 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{55} + ( -3 + 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{58} + ( 4 \zeta_{24} + 2 \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{59} + ( -2 + \zeta_{24} - \zeta_{24}^{3} + 4 \zeta_{24}^{4} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{61} + ( 3 \zeta_{24} - 2 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{62} - q^{64} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 6 \zeta_{24}^{6} ) q^{65} -10 q^{67} + ( \zeta_{24} - 4 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{68} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{71} + ( 2 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{73} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{74} + ( -2 - \zeta_{24} + \zeta_{24}^{3} + 4 \zeta_{24}^{4} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{76} + ( -7 - 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{79} + ( -\zeta_{24} - 2 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{80} + ( -4 - 2 \zeta_{24} + 2 \zeta_{24}^{3} + 8 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{82} + ( -2 \zeta_{24} + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{83} + ( -3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{85} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{86} + 3 q^{88} + ( -12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{89} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{92} + ( -2 - \zeta_{24} + \zeta_{24}^{3} + 4 \zeta_{24}^{4} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{94} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{95} + ( 5 + 2 \zeta_{24} - 2 \zeta_{24}^{3} - 10 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{4} + O(q^{10})$$ $$8q - 8q^{4} + 8q^{16} - 24q^{22} + 32q^{25} + 32q^{37} + 32q^{43} - 24q^{58} - 8q^{64} - 80q^{67} - 56q^{79} + 24q^{88} + O(q^{100})$$

## Character values

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

 $$n$$ $$199$$ $$785$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
881.1
 0.965926 − 0.258819i 0.258819 − 0.965926i −0.965926 + 0.258819i −0.258819 + 0.965926i 0.965926 + 0.258819i 0.258819 + 0.965926i −0.965926 − 0.258819i −0.258819 − 0.965926i
1.00000i 0 −1.00000 −4.18154 0 0 1.00000i 0 4.18154i
881.2 1.00000i 0 −1.00000 −0.717439 0 0 1.00000i 0 0.717439i
881.3 1.00000i 0 −1.00000 0.717439 0 0 1.00000i 0 0.717439i
881.4 1.00000i 0 −1.00000 4.18154 0 0 1.00000i 0 4.18154i
881.5 1.00000i 0 −1.00000 −4.18154 0 0 1.00000i 0 4.18154i
881.6 1.00000i 0 −1.00000 −0.717439 0 0 1.00000i 0 0.717439i
881.7 1.00000i 0 −1.00000 0.717439 0 0 1.00000i 0 0.717439i
881.8 1.00000i 0 −1.00000 4.18154 0 0 1.00000i 0 4.18154i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 881.8 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.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.2.d.a 8
3.b odd 2 1 inner 882.2.d.a 8
4.b odd 2 1 7056.2.k.f 8
7.b odd 2 1 inner 882.2.d.a 8
7.c even 3 1 126.2.k.a 8
7.c even 3 1 882.2.k.a 8
7.d odd 6 1 126.2.k.a 8
7.d odd 6 1 882.2.k.a 8
12.b even 2 1 7056.2.k.f 8
21.c even 2 1 inner 882.2.d.a 8
21.g even 6 1 126.2.k.a 8
21.g even 6 1 882.2.k.a 8
21.h odd 6 1 126.2.k.a 8
21.h odd 6 1 882.2.k.a 8
28.d even 2 1 7056.2.k.f 8
28.f even 6 1 1008.2.bt.c 8
28.g odd 6 1 1008.2.bt.c 8
35.i odd 6 1 3150.2.bf.a 8
35.j even 6 1 3150.2.bf.a 8
35.k even 12 1 3150.2.bp.b 8
35.k even 12 1 3150.2.bp.e 8
35.l odd 12 1 3150.2.bp.b 8
35.l odd 12 1 3150.2.bp.e 8
63.g even 3 1 1134.2.t.e 8
63.h even 3 1 1134.2.l.f 8
63.i even 6 1 1134.2.l.f 8
63.j odd 6 1 1134.2.l.f 8
63.k odd 6 1 1134.2.t.e 8
63.n odd 6 1 1134.2.t.e 8
63.s even 6 1 1134.2.t.e 8
63.t odd 6 1 1134.2.l.f 8
84.h odd 2 1 7056.2.k.f 8
84.j odd 6 1 1008.2.bt.c 8
84.n even 6 1 1008.2.bt.c 8
105.o odd 6 1 3150.2.bf.a 8
105.p even 6 1 3150.2.bf.a 8
105.w odd 12 1 3150.2.bp.b 8
105.w odd 12 1 3150.2.bp.e 8
105.x even 12 1 3150.2.bp.b 8
105.x even 12 1 3150.2.bp.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.k.a 8 7.c even 3 1
126.2.k.a 8 7.d odd 6 1
126.2.k.a 8 21.g even 6 1
126.2.k.a 8 21.h odd 6 1
882.2.d.a 8 1.a even 1 1 trivial
882.2.d.a 8 3.b odd 2 1 inner
882.2.d.a 8 7.b odd 2 1 inner
882.2.d.a 8 21.c even 2 1 inner
882.2.k.a 8 7.c even 3 1
882.2.k.a 8 7.d odd 6 1
882.2.k.a 8 21.g even 6 1
882.2.k.a 8 21.h odd 6 1
1008.2.bt.c 8 28.f even 6 1
1008.2.bt.c 8 28.g odd 6 1
1008.2.bt.c 8 84.j odd 6 1
1008.2.bt.c 8 84.n even 6 1
1134.2.l.f 8 63.h even 3 1
1134.2.l.f 8 63.i even 6 1
1134.2.l.f 8 63.j odd 6 1
1134.2.l.f 8 63.t odd 6 1
1134.2.t.e 8 63.g even 3 1
1134.2.t.e 8 63.k odd 6 1
1134.2.t.e 8 63.n odd 6 1
1134.2.t.e 8 63.s even 6 1
3150.2.bf.a 8 35.i odd 6 1
3150.2.bf.a 8 35.j even 6 1
3150.2.bf.a 8 105.o odd 6 1
3150.2.bf.a 8 105.p even 6 1
3150.2.bp.b 8 35.k even 12 1
3150.2.bp.b 8 35.l odd 12 1
3150.2.bp.b 8 105.w odd 12 1
3150.2.bp.b 8 105.x even 12 1
3150.2.bp.e 8 35.k even 12 1
3150.2.bp.e 8 35.l odd 12 1
3150.2.bp.e 8 105.w odd 12 1
3150.2.bp.e 8 105.x even 12 1
7056.2.k.f 8 4.b odd 2 1
7056.2.k.f 8 12.b even 2 1
7056.2.k.f 8 28.d even 2 1
7056.2.k.f 8 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 18 T_{5}^{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(882, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{4}$$
$3$ 
$5$ $$( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} )^{2}$$
$7$ 
$11$ $$( 1 - 13 T^{2} + 121 T^{4} )^{4}$$
$13$ $$( 1 - 20 T^{2} + 169 T^{4} )^{4}$$
$17$ $$( 1 + 32 T^{2} + 546 T^{4} + 9248 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 40 T^{2} + 834 T^{4} - 14440 T^{6} + 130321 T^{8} )^{2}$$
$23$ $$( 1 - 28 T^{2} + 529 T^{4} )^{4}$$
$29$ $$( 1 - 62 T^{2} + 1995 T^{4} - 52142 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 - 10 T^{2} + 1299 T^{4} - 9610 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$( 1 - 8 T + 72 T^{2} - 296 T^{3} + 1369 T^{4} )^{4}$$
$41$ $$( 1 + 20 T^{2} - 1146 T^{4} + 33620 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 - 8 T + 84 T^{2} - 344 T^{3} + 1849 T^{4} )^{4}$$
$47$ $$( 1 + 152 T^{2} + 9906 T^{4} + 335768 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 - 158 T^{2} + 11211 T^{4} - 443822 T^{6} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 + 38 T^{2} + 6171 T^{4} + 132278 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 208 T^{2} + 17970 T^{4} - 773968 T^{6} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 + 10 T + 67 T^{2} )^{8}$$
$71$ $$( 1 - 176 T^{2} + 15234 T^{4} - 887216 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 - 220 T^{2} + 21606 T^{4} - 1172380 T^{6} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 + 14 T + 189 T^{2} + 1106 T^{3} + 6241 T^{4} )^{4}$$
$83$ $$( 1 + 278 T^{2} + 32811 T^{4} + 1915142 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 + 70 T^{2} + 7921 T^{4} )^{4}$$
$97$ $$( 1 - 190 T^{2} + 20643 T^{4} - 1787710 T^{6} + 88529281 T^{8} )^{2}$$