# Properties

 Label 882.3.c.c Level $882$ Weight $3$ Character orbit 882.c 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.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.0327593166$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ 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 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + 2 q^{4} + 2 \beta_{3} q^{5} -2 \beta_{1} q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + 2 q^{4} + 2 \beta_{3} q^{5} -2 \beta_{1} q^{8} + 4 \beta_{2} q^{10} + 12 \beta_{1} q^{11} -\beta_{2} q^{13} + 4 q^{16} + 2 \beta_{3} q^{17} + 17 \beta_{2} q^{19} + 4 \beta_{3} q^{20} -24 q^{22} + 6 \beta_{1} q^{23} + q^{25} -\beta_{3} q^{26} + 24 \beta_{1} q^{29} + 7 \beta_{2} q^{31} -4 \beta_{1} q^{32} + 4 \beta_{2} q^{34} -47 q^{37} + 17 \beta_{3} q^{38} + 8 \beta_{2} q^{40} + 28 \beta_{3} q^{41} + 31 q^{43} + 24 \beta_{1} q^{44} -12 q^{46} -34 \beta_{3} q^{47} -\beta_{1} q^{50} -2 \beta_{2} q^{52} + 54 \beta_{1} q^{53} -48 \beta_{2} q^{55} -48 q^{58} + 34 \beta_{3} q^{59} -48 \beta_{2} q^{61} + 7 \beta_{3} q^{62} + 8 q^{64} -6 \beta_{1} q^{65} -31 q^{67} + 4 \beta_{3} q^{68} -42 \beta_{1} q^{71} + 47 \beta_{2} q^{73} + 47 \beta_{1} q^{74} + 34 \beta_{2} q^{76} + 41 q^{79} + 8 \beta_{3} q^{80} + 56 \beta_{2} q^{82} -2 \beta_{3} q^{83} -24 q^{85} -31 \beta_{1} q^{86} -48 q^{88} + 24 \beta_{3} q^{89} + 12 \beta_{1} q^{92} -68 \beta_{2} q^{94} + 102 \beta_{1} q^{95} + 24 \beta_{2} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{4} + O(q^{10})$$ $$4q + 8q^{4} + 16q^{16} - 96q^{22} + 4q^{25} - 188q^{37} + 124q^{43} - 48q^{46} - 192q^{58} + 32q^{64} - 124q^{67} + 164q^{79} - 96q^{85} - 192q^{88} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
685.1
 −0.707107 − 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i 0.707107 + 1.22474i
−1.41421 0 2.00000 4.89898i 0 0 −2.82843 0 6.92820i
685.2 −1.41421 0 2.00000 4.89898i 0 0 −2.82843 0 6.92820i
685.3 1.41421 0 2.00000 4.89898i 0 0 2.82843 0 6.92820i
685.4 1.41421 0 2.00000 4.89898i 0 0 2.82843 0 6.92820i
 $$n$$: 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.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 882.3.c.c 4
3.b odd 2 1 inner 882.3.c.c 4
7.b odd 2 1 inner 882.3.c.c 4
7.c even 3 1 126.3.n.b 4
7.c even 3 1 882.3.n.c 4
7.d odd 6 1 126.3.n.b 4
7.d odd 6 1 882.3.n.c 4
21.c even 2 1 inner 882.3.c.c 4
21.g even 6 1 126.3.n.b 4
21.g even 6 1 882.3.n.c 4
21.h odd 6 1 126.3.n.b 4
21.h odd 6 1 882.3.n.c 4
28.f even 6 1 1008.3.cg.i 4
28.g odd 6 1 1008.3.cg.i 4
84.j odd 6 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.c even 3 1
126.3.n.b 4 7.d odd 6 1
126.3.n.b 4 21.g even 6 1
126.3.n.b 4 21.h odd 6 1
882.3.c.c 4 1.a even 1 1 trivial
882.3.c.c 4 3.b odd 2 1 inner
882.3.c.c 4 7.b odd 2 1 inner
882.3.c.c 4 21.c even 2 1 inner
882.3.n.c 4 7.c even 3 1
882.3.n.c 4 7.d odd 6 1
882.3.n.c 4 21.g even 6 1
882.3.n.c 4 21.h odd 6 1
1008.3.cg.i 4 28.f even 6 1
1008.3.cg.i 4 28.g odd 6 1
1008.3.cg.i 4 84.j odd 6 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 $$S_{3}^{\mathrm{new}}(882, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$( 24 + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$( -288 + T^{2} )^{2}$$
$13$ $$( 3 + T^{2} )^{2}$$
$17$ $$( 24 + T^{2} )^{2}$$
$19$ $$( 867 + T^{2} )^{2}$$
$23$ $$( -72 + T^{2} )^{2}$$
$29$ $$( -1152 + T^{2} )^{2}$$
$31$ $$( 147 + T^{2} )^{2}$$
$37$ $$( 47 + T )^{4}$$
$41$ $$( 4704 + T^{2} )^{2}$$
$43$ $$( -31 + T )^{4}$$
$47$ $$( 6936 + T^{2} )^{2}$$
$53$ $$( -5832 + T^{2} )^{2}$$
$59$ $$( 6936 + T^{2} )^{2}$$
$61$ $$( 6912 + T^{2} )^{2}$$
$67$ $$( 31 + T )^{4}$$
$71$ $$( -3528 + T^{2} )^{2}$$
$73$ $$( 6627 + T^{2} )^{2}$$
$79$ $$( -41 + T )^{4}$$
$83$ $$( 24 + T^{2} )^{2}$$
$89$ $$( 3456 + T^{2} )^{2}$$
$97$ $$( 1728 + T^{2} )^{2}$$