# Properties

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

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## 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + 2 \beta_{2} q^{4} + \beta_{1} q^{5} -2 \beta_{3} q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + 2 \beta_{2} q^{4} + \beta_{1} q^{5} -2 \beta_{3} q^{8} -2 \beta_{2} q^{10} + ( 5 \beta_{1} - 5 \beta_{3} ) q^{11} -15 q^{13} + ( -4 + 4 \beta_{2} ) q^{16} + ( -8 \beta_{1} + 8 \beta_{3} ) q^{17} + ( -13 + 13 \beta_{2} ) q^{19} + 2 \beta_{3} q^{20} -10 q^{22} + 16 \beta_{1} q^{23} -23 \beta_{2} q^{25} + 15 \beta_{1} q^{26} + 16 \beta_{3} q^{29} + 3 \beta_{2} q^{31} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{32} + 16 q^{34} + ( -17 + 17 \beta_{2} ) q^{37} + ( 13 \beta_{1} - 13 \beta_{3} ) q^{38} + ( 4 - 4 \beta_{2} ) q^{40} -57 \beta_{3} q^{41} -85 q^{43} + 10 \beta_{1} q^{44} -32 \beta_{2} q^{46} + 51 \beta_{1} q^{47} + 23 \beta_{3} q^{50} -30 \beta_{2} q^{52} + ( -24 \beta_{1} + 24 \beta_{3} ) q^{53} + 10 q^{55} + ( 32 - 32 \beta_{2} ) q^{58} + ( -64 \beta_{1} + 64 \beta_{3} ) q^{59} + ( -72 + 72 \beta_{2} ) q^{61} -3 \beta_{3} q^{62} -8 q^{64} -15 \beta_{1} q^{65} -43 \beta_{2} q^{67} -16 \beta_{1} q^{68} -37 \beta_{3} q^{71} -95 \beta_{2} q^{73} + ( 17 \beta_{1} - 17 \beta_{3} ) q^{74} -26 q^{76} + ( -69 + 69 \beta_{2} ) q^{79} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{80} + ( -114 + 114 \beta_{2} ) q^{82} + 43 \beta_{3} q^{83} -16 q^{85} + 85 \beta_{1} q^{86} -20 \beta_{2} q^{88} + 96 \beta_{1} q^{89} + 32 \beta_{3} q^{92} -102 \beta_{2} q^{94} + ( -13 \beta_{1} + 13 \beta_{3} ) q^{95} -16 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4} + O(q^{10})$$ $$4 q + 4 q^{4} - 4 q^{10} - 60 q^{13} - 8 q^{16} - 26 q^{19} - 40 q^{22} - 46 q^{25} + 6 q^{31} + 64 q^{34} - 34 q^{37} + 8 q^{40} - 340 q^{43} - 64 q^{46} - 60 q^{52} + 40 q^{55} + 64 q^{58} - 144 q^{61} - 32 q^{64} - 86 q^{67} - 190 q^{73} - 104 q^{76} - 138 q^{79} - 228 q^{82} - 64 q^{85} - 40 q^{88} - 204 q^{94} - 64 q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

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

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

## 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
 1.22474 − 0.707107i −1.22474 + 0.707107i 1.22474 + 0.707107i −1.22474 − 0.707107i
−1.22474 + 0.707107i 0 1.00000 1.73205i 1.22474 0.707107i 0 0 2.82843i 0 −1.00000 + 1.73205i
557.2 1.22474 0.707107i 0 1.00000 1.73205i −1.22474 + 0.707107i 0 0 2.82843i 0 −1.00000 + 1.73205i
863.1 −1.22474 0.707107i 0 1.00000 + 1.73205i 1.22474 + 0.707107i 0 0 2.82843i 0 −1.00000 1.73205i
863.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i −1.22474 0.707107i 0 0 2.82843i 0 −1.00000 1.73205i
 $$n$$: 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
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.c 4
3.b odd 2 1 inner 882.3.s.c 4
7.b odd 2 1 126.3.s.b 4
7.c even 3 1 882.3.b.d 2
7.c even 3 1 inner 882.3.s.c 4
7.d odd 6 1 126.3.s.b 4
7.d odd 6 1 882.3.b.c 2
21.c even 2 1 126.3.s.b 4
21.g even 6 1 126.3.s.b 4
21.g even 6 1 882.3.b.c 2
21.h odd 6 1 882.3.b.d 2
21.h odd 6 1 inner 882.3.s.c 4
28.d even 2 1 1008.3.dc.a 4
28.f even 6 1 1008.3.dc.a 4
84.h odd 2 1 1008.3.dc.a 4
84.j odd 6 1 1008.3.dc.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.s.b 4 7.b odd 2 1
126.3.s.b 4 7.d odd 6 1
126.3.s.b 4 21.c even 2 1
126.3.s.b 4 21.g even 6 1
882.3.b.c 2 7.d odd 6 1
882.3.b.c 2 21.g even 6 1
882.3.b.d 2 7.c even 3 1
882.3.b.d 2 21.h odd 6 1
882.3.s.c 4 1.a even 1 1 trivial
882.3.s.c 4 3.b odd 2 1 inner
882.3.s.c 4 7.c even 3 1 inner
882.3.s.c 4 21.h odd 6 1 inner
1008.3.dc.a 4 28.d even 2 1
1008.3.dc.a 4 28.f even 6 1
1008.3.dc.a 4 84.h odd 2 1
1008.3.dc.a 4 84.j odd 6 1

## Hecke kernels

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

 $$T_{5}^{4} - 2 T_{5}^{2} + 4$$ $$T_{11}^{4} - 50 T_{11}^{2} + 2500$$ $$T_{13} + 15$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 2 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$4 - 2 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$2500 - 50 T^{2} + T^{4}$$
$13$ $$( 15 + T )^{4}$$
$17$ $$16384 - 128 T^{2} + T^{4}$$
$19$ $$( 169 + 13 T + T^{2} )^{2}$$
$23$ $$262144 - 512 T^{2} + T^{4}$$
$29$ $$( 512 + T^{2} )^{2}$$
$31$ $$( 9 - 3 T + T^{2} )^{2}$$
$37$ $$( 289 + 17 T + T^{2} )^{2}$$
$41$ $$( 6498 + T^{2} )^{2}$$
$43$ $$( 85 + T )^{4}$$
$47$ $$27060804 - 5202 T^{2} + T^{4}$$
$53$ $$1327104 - 1152 T^{2} + T^{4}$$
$59$ $$67108864 - 8192 T^{2} + T^{4}$$
$61$ $$( 5184 + 72 T + T^{2} )^{2}$$
$67$ $$( 1849 + 43 T + T^{2} )^{2}$$
$71$ $$( 2738 + T^{2} )^{2}$$
$73$ $$( 9025 + 95 T + T^{2} )^{2}$$
$79$ $$( 4761 + 69 T + T^{2} )^{2}$$
$83$ $$( 3698 + T^{2} )^{2}$$
$89$ $$339738624 - 18432 T^{2} + T^{4}$$
$97$ $$( 16 + T )^{4}$$
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