# Properties

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

# 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: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{2} + 2 \zeta_{24}^{4} q^{4} -4 \zeta_{24}^{2} q^{5} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{8} +O(q^{10})$$ $$q + ( \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{2} + 2 \zeta_{24}^{4} q^{4} -4 \zeta_{24}^{2} q^{5} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{8} + ( -4 \zeta_{24} - 4 \zeta_{24}^{7} ) q^{10} + ( 2 \zeta_{24} - 2 \zeta_{24}^{7} ) q^{11} + ( -9 \zeta_{24} - 9 \zeta_{24}^{3} + 9 \zeta_{24}^{5} ) q^{13} + ( -4 + 4 \zeta_{24}^{4} ) q^{16} + ( 4 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{17} + ( 16 \zeta_{24}^{3} - 16 \zeta_{24}^{5} - 16 \zeta_{24}^{7} ) q^{19} -8 \zeta_{24}^{6} q^{20} + 4 q^{22} + ( 26 \zeta_{24}^{3} + 26 \zeta_{24}^{5} - 26 \zeta_{24}^{7} ) q^{23} -9 \zeta_{24}^{4} q^{25} -18 \zeta_{24}^{2} q^{26} + ( 23 \zeta_{24} - 23 \zeta_{24}^{3} - 23 \zeta_{24}^{5} ) q^{29} + ( -36 \zeta_{24} - 36 \zeta_{24}^{7} ) q^{31} + ( -4 \zeta_{24} + 4 \zeta_{24}^{7} ) q^{32} + ( 4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} ) q^{34} + ( 32 - 32 \zeta_{24}^{4} ) q^{37} + ( 32 \zeta_{24}^{2} - 32 \zeta_{24}^{6} ) q^{38} + ( 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{40} -38 \zeta_{24}^{6} q^{41} + 20 q^{43} + ( 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{44} + 52 \zeta_{24}^{4} q^{46} + 20 \zeta_{24}^{2} q^{47} + ( 9 \zeta_{24} - 9 \zeta_{24}^{3} - 9 \zeta_{24}^{5} ) q^{50} + ( -18 \zeta_{24} - 18 \zeta_{24}^{7} ) q^{52} + ( -67 \zeta_{24} + 67 \zeta_{24}^{7} ) q^{53} + ( -8 \zeta_{24} - 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} ) q^{55} + ( 46 - 46 \zeta_{24}^{4} ) q^{58} + ( 4 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{59} + ( 59 \zeta_{24}^{3} - 59 \zeta_{24}^{5} - 59 \zeta_{24}^{7} ) q^{61} -72 \zeta_{24}^{6} q^{62} -8 q^{64} + ( 36 \zeta_{24}^{3} + 36 \zeta_{24}^{5} - 36 \zeta_{24}^{7} ) q^{65} + 48 \zeta_{24}^{4} q^{67} + 8 \zeta_{24}^{2} q^{68} + ( 54 \zeta_{24} - 54 \zeta_{24}^{3} - 54 \zeta_{24}^{5} ) q^{71} + ( -85 \zeta_{24} - 85 \zeta_{24}^{7} ) q^{73} + ( 32 \zeta_{24} - 32 \zeta_{24}^{7} ) q^{74} + ( 32 \zeta_{24} + 32 \zeta_{24}^{3} - 32 \zeta_{24}^{5} ) q^{76} + ( 148 - 148 \zeta_{24}^{4} ) q^{79} + ( 16 \zeta_{24}^{2} - 16 \zeta_{24}^{6} ) q^{80} + ( 38 \zeta_{24}^{3} - 38 \zeta_{24}^{5} - 38 \zeta_{24}^{7} ) q^{82} + 80 \zeta_{24}^{6} q^{83} -16 q^{85} + ( 20 \zeta_{24}^{3} + 20 \zeta_{24}^{5} - 20 \zeta_{24}^{7} ) q^{86} + 8 \zeta_{24}^{4} q^{88} + 106 \zeta_{24}^{2} q^{89} + ( -52 \zeta_{24} + 52 \zeta_{24}^{3} + 52 \zeta_{24}^{5} ) q^{92} + ( 20 \zeta_{24} + 20 \zeta_{24}^{7} ) q^{94} + ( -64 \zeta_{24} + 64 \zeta_{24}^{7} ) q^{95} + ( -109 \zeta_{24} - 109 \zeta_{24}^{3} + 109 \zeta_{24}^{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{4} + O(q^{10})$$ $$8 q + 8 q^{4} - 16 q^{16} + 32 q^{22} - 36 q^{25} + 128 q^{37} + 160 q^{43} + 208 q^{46} + 184 q^{58} - 64 q^{64} + 192 q^{67} + 592 q^{79} - 128 q^{85} + 32 q^{88} + O(q^{100})$$

## Character values

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

 $$n$$ $$199$$ $$785$$ $$\chi(n)$$ $$-1 + \zeta_{24}^{2}$$ $$-1$$

## 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
 −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
−1.22474 + 0.707107i 0 1.00000 1.73205i −3.46410 + 2.00000i 0 0 2.82843i 0 2.82843 4.89898i
557.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 3.46410 2.00000i 0 0 2.82843i 0 −2.82843 + 4.89898i
557.3 1.22474 0.707107i 0 1.00000 1.73205i −3.46410 + 2.00000i 0 0 2.82843i 0 −2.82843 + 4.89898i
557.4 1.22474 0.707107i 0 1.00000 1.73205i 3.46410 2.00000i 0 0 2.82843i 0 2.82843 4.89898i
863.1 −1.22474 0.707107i 0 1.00000 + 1.73205i −3.46410 2.00000i 0 0 2.82843i 0 2.82843 + 4.89898i
863.2 −1.22474 0.707107i 0 1.00000 + 1.73205i 3.46410 + 2.00000i 0 0 2.82843i 0 −2.82843 4.89898i
863.3 1.22474 + 0.707107i 0 1.00000 + 1.73205i −3.46410 2.00000i 0 0 2.82843i 0 −2.82843 4.89898i
863.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 3.46410 + 2.00000i 0 0 2.82843i 0 2.82843 + 4.89898i
 $$n$$: 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.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 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.g 8
3.b odd 2 1 inner 882.3.s.g 8
7.b odd 2 1 inner 882.3.s.g 8
7.c even 3 1 882.3.b.h 4
7.c even 3 1 inner 882.3.s.g 8
7.d odd 6 1 882.3.b.h 4
7.d odd 6 1 inner 882.3.s.g 8
21.c even 2 1 inner 882.3.s.g 8
21.g even 6 1 882.3.b.h 4
21.g even 6 1 inner 882.3.s.g 8
21.h odd 6 1 882.3.b.h 4
21.h odd 6 1 inner 882.3.s.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.3.b.h 4 7.c even 3 1
882.3.b.h 4 7.d odd 6 1
882.3.b.h 4 21.g even 6 1
882.3.b.h 4 21.h odd 6 1
882.3.s.g 8 1.a even 1 1 trivial
882.3.s.g 8 3.b odd 2 1 inner
882.3.s.g 8 7.b odd 2 1 inner
882.3.s.g 8 7.c even 3 1 inner
882.3.s.g 8 7.d odd 6 1 inner
882.3.s.g 8 21.c even 2 1 inner
882.3.s.g 8 21.g even 6 1 inner
882.3.s.g 8 21.h odd 6 1 inner

## Hecke kernels

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

 $$T_{5}^{4} - 16 T_{5}^{2} + 256$$ $$T_{11}^{4} - 8 T_{11}^{2} + 64$$ $$T_{13}^{2} - 162$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 - 2 T^{2} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$( 256 - 16 T^{2} + T^{4} )^{2}$$
$7$ $$T^{8}$$
$11$ $$( 64 - 8 T^{2} + T^{4} )^{2}$$
$13$ $$( -162 + T^{2} )^{4}$$
$17$ $$( 256 - 16 T^{2} + T^{4} )^{2}$$
$19$ $$( 262144 + 512 T^{2} + T^{4} )^{2}$$
$23$ $$( 1827904 - 1352 T^{2} + T^{4} )^{2}$$
$29$ $$( 1058 + T^{2} )^{4}$$
$31$ $$( 6718464 + 2592 T^{2} + T^{4} )^{2}$$
$37$ $$( 1024 - 32 T + T^{2} )^{4}$$
$41$ $$( 1444 + T^{2} )^{4}$$
$43$ $$( -20 + T )^{8}$$
$47$ $$( 160000 - 400 T^{2} + T^{4} )^{2}$$
$53$ $$( 80604484 - 8978 T^{2} + T^{4} )^{2}$$
$59$ $$( 256 - 16 T^{2} + T^{4} )^{2}$$
$61$ $$( 48469444 + 6962 T^{2} + T^{4} )^{2}$$
$67$ $$( 2304 - 48 T + T^{2} )^{4}$$
$71$ $$( 5832 + T^{2} )^{4}$$
$73$ $$( 208802500 + 14450 T^{2} + T^{4} )^{2}$$
$79$ $$( 21904 - 148 T + T^{2} )^{4}$$
$83$ $$( 6400 + T^{2} )^{4}$$
$89$ $$( 126247696 - 11236 T^{2} + T^{4} )^{2}$$
$97$ $$( -23762 + T^{2} )^{4}$$