# Properties

 Label 882.3.s.f 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.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{2} + 2 \zeta_{24}^{4} q^{4} -2 \zeta_{24}^{2} q^{5} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{8} +O(q^{10})$$ $$q + ( \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{2} + 2 \zeta_{24}^{4} q^{4} -2 \zeta_{24}^{2} q^{5} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{8} + ( -2 \zeta_{24} - 2 \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} + ( -22 \zeta_{24}^{2} + 22 \zeta_{24}^{6} ) q^{17} + ( -4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{19} -4 \zeta_{24}^{6} q^{20} + 4 q^{22} + ( 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{23} -21 \zeta_{24}^{4} q^{25} -18 \zeta_{24}^{2} q^{26} + ( -25 \zeta_{24} + 25 \zeta_{24}^{3} + 25 \zeta_{24}^{5} ) q^{29} + ( -24 \zeta_{24} - 24 \zeta_{24}^{7} ) q^{31} + ( -4 \zeta_{24} + 4 \zeta_{24}^{7} ) q^{32} + ( -22 \zeta_{24} - 22 \zeta_{24}^{3} + 22 \zeta_{24}^{5} ) q^{34} + ( -64 + 64 \zeta_{24}^{4} ) q^{37} + ( -8 \zeta_{24}^{2} + 8 \zeta_{24}^{6} ) q^{38} + ( 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{40} + 20 \zeta_{24}^{6} q^{41} + 44 q^{43} + ( 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{44} + 4 \zeta_{24}^{4} q^{46} -68 \zeta_{24}^{2} q^{47} + ( 21 \zeta_{24} - 21 \zeta_{24}^{3} - 21 \zeta_{24}^{5} ) q^{50} + ( -18 \zeta_{24} - 18 \zeta_{24}^{7} ) q^{52} + ( -13 \zeta_{24} + 13 \zeta_{24}^{7} ) q^{53} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{55} + ( -50 + 50 \zeta_{24}^{4} ) q^{58} + ( -100 \zeta_{24}^{2} + 100 \zeta_{24}^{6} ) q^{59} + ( 37 \zeta_{24}^{3} - 37 \zeta_{24}^{5} - 37 \zeta_{24}^{7} ) q^{61} -48 \zeta_{24}^{6} q^{62} -8 q^{64} + ( 18 \zeta_{24}^{3} + 18 \zeta_{24}^{5} - 18 \zeta_{24}^{7} ) q^{65} -120 \zeta_{24}^{4} q^{67} -44 \zeta_{24}^{2} q^{68} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{71} + ( -53 \zeta_{24} - 53 \zeta_{24}^{7} ) q^{73} + ( -64 \zeta_{24} + 64 \zeta_{24}^{7} ) q^{74} + ( -8 \zeta_{24} - 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} ) q^{76} + ( -92 + 92 \zeta_{24}^{4} ) q^{79} + ( 8 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{80} + ( -20 \zeta_{24}^{3} + 20 \zeta_{24}^{5} + 20 \zeta_{24}^{7} ) q^{82} + 112 \zeta_{24}^{6} q^{83} + 44 q^{85} + ( 44 \zeta_{24}^{3} + 44 \zeta_{24}^{5} - 44 \zeta_{24}^{7} ) q^{86} + 8 \zeta_{24}^{4} q^{88} + 20 \zeta_{24}^{2} q^{89} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{92} + ( -68 \zeta_{24} - 68 \zeta_{24}^{7} ) q^{94} + ( 8 \zeta_{24} - 8 \zeta_{24}^{7} ) q^{95} + ( 19 \zeta_{24} + 19 \zeta_{24}^{3} - 19 \zeta_{24}^{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{4} + O(q^{10})$$ $$8 q + 8 q^{4} - 16 q^{16} + 32 q^{22} - 84 q^{25} - 256 q^{37} + 352 q^{43} + 16 q^{46} - 200 q^{58} - 64 q^{64} - 480 q^{67} - 368 q^{79} + 352 q^{85} + 32 q^{88} + O(q^{100})$$

## 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 −1.73205 + 1.00000i 0 0 2.82843i 0 1.41421 2.44949i
557.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 1.73205 1.00000i 0 0 2.82843i 0 −1.41421 + 2.44949i
557.3 1.22474 0.707107i 0 1.00000 1.73205i −1.73205 + 1.00000i 0 0 2.82843i 0 −1.41421 + 2.44949i
557.4 1.22474 0.707107i 0 1.00000 1.73205i 1.73205 1.00000i 0 0 2.82843i 0 1.41421 2.44949i
863.1 −1.22474 0.707107i 0 1.00000 + 1.73205i −1.73205 1.00000i 0 0 2.82843i 0 1.41421 + 2.44949i
863.2 −1.22474 0.707107i 0 1.00000 + 1.73205i 1.73205 + 1.00000i 0 0 2.82843i 0 −1.41421 2.44949i
863.3 1.22474 + 0.707107i 0 1.00000 + 1.73205i −1.73205 1.00000i 0 0 2.82843i 0 −1.41421 2.44949i
863.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 1.73205 + 1.00000i 0 0 2.82843i 0 1.41421 + 2.44949i
 $$n$$: 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.f 8
3.b odd 2 1 inner 882.3.s.f 8
7.b odd 2 1 inner 882.3.s.f 8
7.c even 3 1 882.3.b.i 4
7.c even 3 1 inner 882.3.s.f 8
7.d odd 6 1 882.3.b.i 4
7.d odd 6 1 inner 882.3.s.f 8
21.c even 2 1 inner 882.3.s.f 8
21.g even 6 1 882.3.b.i 4
21.g even 6 1 inner 882.3.s.f 8
21.h odd 6 1 882.3.b.i 4
21.h odd 6 1 inner 882.3.s.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.3.b.i 4 7.c even 3 1
882.3.b.i 4 7.d odd 6 1
882.3.b.i 4 21.g even 6 1
882.3.b.i 4 21.h odd 6 1
882.3.s.f 8 1.a even 1 1 trivial
882.3.s.f 8 3.b odd 2 1 inner
882.3.s.f 8 7.b odd 2 1 inner
882.3.s.f 8 7.c even 3 1 inner
882.3.s.f 8 7.d odd 6 1 inner
882.3.s.f 8 21.c even 2 1 inner
882.3.s.f 8 21.g even 6 1 inner
882.3.s.f 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 $$S_{3}^{\mathrm{new}}(882, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 - 2 T^{2} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$( 16 - 4 T^{2} + T^{4} )^{2}$$
$7$ $$T^{8}$$
$11$ $$( 64 - 8 T^{2} + T^{4} )^{2}$$
$13$ $$( -162 + T^{2} )^{4}$$
$17$ $$( 234256 - 484 T^{2} + T^{4} )^{2}$$
$19$ $$( 1024 + 32 T^{2} + T^{4} )^{2}$$
$23$ $$( 64 - 8 T^{2} + T^{4} )^{2}$$
$29$ $$( 1250 + T^{2} )^{4}$$
$31$ $$( 1327104 + 1152 T^{2} + T^{4} )^{2}$$
$37$ $$( 4096 + 64 T + T^{2} )^{4}$$
$41$ $$( 400 + T^{2} )^{4}$$
$43$ $$( -44 + T )^{8}$$
$47$ $$( 21381376 - 4624 T^{2} + T^{4} )^{2}$$
$53$ $$( 114244 - 338 T^{2} + T^{4} )^{2}$$
$59$ $$( 100000000 - 10000 T^{2} + T^{4} )^{2}$$
$61$ $$( 7496644 + 2738 T^{2} + T^{4} )^{2}$$
$67$ $$( 14400 + 120 T + T^{2} )^{4}$$
$71$ $$( 72 + T^{2} )^{4}$$
$73$ $$( 31561924 + 5618 T^{2} + T^{4} )^{2}$$
$79$ $$( 8464 + 92 T + T^{2} )^{4}$$
$83$ $$( 12544 + T^{2} )^{4}$$
$89$ $$( 160000 - 400 T^{2} + T^{4} )^{2}$$
$97$ $$( -722 + T^{2} )^{4}$$