# Properties

 Label 882.3.n.c Level $882$ Weight $3$ Character orbit 882.n Analytic conductor $24.033$ Analytic rank $0$ Dimension $4$ 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.n (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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} - \beta_{3} ) q^{2} + ( -2 - 2 \beta_{2} ) q^{4} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{5} + 2 \beta_{3} q^{8} +O(q^{10})$$ $$q + ( -\beta_{1} - \beta_{3} ) q^{2} + ( -2 - 2 \beta_{2} ) q^{4} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{5} + 2 \beta_{3} q^{8} + ( 4 - 4 \beta_{2} ) q^{10} -12 \beta_{1} q^{11} + ( -1 - 2 \beta_{2} ) q^{13} + 4 \beta_{2} q^{16} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{17} + ( -34 - 17 \beta_{2} ) q^{19} + ( -8 \beta_{1} - 4 \beta_{3} ) q^{20} -24 q^{22} + ( 6 \beta_{1} + 6 \beta_{3} ) q^{23} + ( -1 - \beta_{2} ) q^{25} + ( -\beta_{1} + \beta_{3} ) q^{26} -24 \beta_{3} q^{29} + ( 7 - 7 \beta_{2} ) q^{31} + 4 \beta_{1} q^{32} + ( 4 + 8 \beta_{2} ) q^{34} -47 \beta_{2} q^{37} + ( 17 \beta_{1} + 34 \beta_{3} ) q^{38} + ( -16 - 8 \beta_{2} ) q^{40} + ( -56 \beta_{1} - 28 \beta_{3} ) q^{41} + 31 q^{43} + ( 24 \beta_{1} + 24 \beta_{3} ) q^{44} + ( 12 + 12 \beta_{2} ) q^{46} + ( -34 \beta_{1} + 34 \beta_{3} ) q^{47} + \beta_{3} q^{50} + ( -2 + 2 \beta_{2} ) q^{52} -54 \beta_{1} q^{53} + ( -48 - 96 \beta_{2} ) q^{55} -48 \beta_{2} q^{58} + ( 34 \beta_{1} + 68 \beta_{3} ) q^{59} + ( 96 + 48 \beta_{2} ) q^{61} + ( -14 \beta_{1} - 7 \beta_{3} ) q^{62} + 8 q^{64} + ( -6 \beta_{1} - 6 \beta_{3} ) q^{65} + ( 31 + 31 \beta_{2} ) q^{67} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{68} + 42 \beta_{3} q^{71} + ( 47 - 47 \beta_{2} ) q^{73} -47 \beta_{1} q^{74} + ( 34 + 68 \beta_{2} ) q^{76} + 41 \beta_{2} q^{79} + ( 8 \beta_{1} + 16 \beta_{3} ) q^{80} + ( -112 - 56 \beta_{2} ) q^{82} + ( 4 \beta_{1} + 2 \beta_{3} ) q^{83} -24 q^{85} + ( -31 \beta_{1} - 31 \beta_{3} ) q^{86} + ( 48 + 48 \beta_{2} ) q^{88} + ( 24 \beta_{1} - 24 \beta_{3} ) q^{89} -12 \beta_{3} q^{92} + ( -68 + 68 \beta_{2} ) q^{94} -102 \beta_{1} q^{95} + ( 24 + 48 \beta_{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + O(q^{10})$$ $$4q - 4q^{4} + 24q^{10} - 8q^{16} - 102q^{19} - 96q^{22} - 2q^{25} + 42q^{31} + 94q^{37} - 48q^{40} + 124q^{43} + 24q^{46} - 12q^{52} + 96q^{58} + 288q^{61} + 32q^{64} + 62q^{67} + 282q^{73} - 82q^{79} - 336q^{82} - 96q^{85} + 96q^{88} - 408q^{94} + O(q^{100})$$

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 $$\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
−0.707107 + 1.22474i 0 −1.00000 1.73205i −4.24264 2.44949i 0 0 2.82843 0 6.00000 3.46410i
19.2 0.707107 1.22474i 0 −1.00000 1.73205i 4.24264 + 2.44949i 0 0 −2.82843 0 6.00000 3.46410i
325.1 −0.707107 1.22474i 0 −1.00000 + 1.73205i −4.24264 + 2.44949i 0 0 2.82843 0 6.00000 + 3.46410i
325.2 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 4.24264 2.44949i 0 0 −2.82843 0 6.00000 + 3.46410i
 $$n$$: 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.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.3.n.c 4
3.b odd 2 1 inner 882.3.n.c 4
7.b odd 2 1 126.3.n.b 4
7.c even 3 1 126.3.n.b 4
7.c even 3 1 882.3.c.c 4
7.d odd 6 1 882.3.c.c 4
7.d odd 6 1 inner 882.3.n.c 4
21.c even 2 1 126.3.n.b 4
21.g even 6 1 882.3.c.c 4
21.g even 6 1 inner 882.3.n.c 4
21.h odd 6 1 126.3.n.b 4
21.h odd 6 1 882.3.c.c 4
28.d even 2 1 1008.3.cg.i 4
28.g odd 6 1 1008.3.cg.i 4
84.h odd 2 1 1008.3.cg.i 4
84.n even 6 1 1008.3.cg.i 4

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

## Hecke kernels

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

 $$T_{5}^{4} - 24 T_{5}^{2} + 576$$ $$T_{23}^{4} + 72 T_{23}^{2} + 5184$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$576 - 24 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$82944 + 288 T^{2} + T^{4}$$
$13$ $$( 3 + T^{2} )^{2}$$
$17$ $$576 - 24 T^{2} + T^{4}$$
$19$ $$( 867 + 51 T + T^{2} )^{2}$$
$23$ $$5184 + 72 T^{2} + T^{4}$$
$29$ $$( -1152 + T^{2} )^{2}$$
$31$ $$( 147 - 21 T + T^{2} )^{2}$$
$37$ $$( 2209 - 47 T + T^{2} )^{2}$$
$41$ $$( 4704 + T^{2} )^{2}$$
$43$ $$( -31 + T )^{4}$$
$47$ $$48108096 - 6936 T^{2} + T^{4}$$
$53$ $$34012224 + 5832 T^{2} + T^{4}$$
$59$ $$48108096 - 6936 T^{2} + T^{4}$$
$61$ $$( 6912 - 144 T + T^{2} )^{2}$$
$67$ $$( 961 - 31 T + T^{2} )^{2}$$
$71$ $$( -3528 + T^{2} )^{2}$$
$73$ $$( 6627 - 141 T + T^{2} )^{2}$$
$79$ $$( 1681 + 41 T + T^{2} )^{2}$$
$83$ $$( 24 + T^{2} )^{2}$$
$89$ $$11943936 - 3456 T^{2} + T^{4}$$
$97$ $$( 1728 + T^{2} )^{2}$$