# Properties

 Label 882.4.g.bc Level $882$ Weight $4$ Character orbit 882.g Analytic conductor $52.040$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 882.g (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$52.0396846251$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -2 - 2 \beta_{2} ) q^{2} + 4 \beta_{2} q^{4} + 5 \beta_{1} q^{5} + 8 q^{8} +O(q^{10})$$ $$q + ( -2 - 2 \beta_{2} ) q^{2} + 4 \beta_{2} q^{4} + 5 \beta_{1} q^{5} + 8 q^{8} + ( -10 \beta_{1} - 10 \beta_{3} ) q^{10} -40 \beta_{2} q^{11} -45 \beta_{3} q^{13} + ( -16 - 16 \beta_{2} ) q^{16} + ( \beta_{1} + \beta_{3} ) q^{17} + 8 \beta_{1} q^{19} + 20 \beta_{3} q^{20} -80 q^{22} + ( 68 + 68 \beta_{2} ) q^{23} -75 \beta_{2} q^{25} -90 \beta_{1} q^{26} -110 q^{29} + ( 84 \beta_{1} + 84 \beta_{3} ) q^{31} + 32 \beta_{2} q^{32} -2 \beta_{3} q^{34} + ( 20 + 20 \beta_{2} ) q^{37} + ( -16 \beta_{1} - 16 \beta_{3} ) q^{38} + 40 \beta_{1} q^{40} + 35 \beta_{3} q^{41} -340 q^{43} + ( 160 + 160 \beta_{2} ) q^{44} -136 \beta_{2} q^{46} -64 \beta_{1} q^{47} -150 q^{50} + ( 180 \beta_{1} + 180 \beta_{3} ) q^{52} -628 \beta_{2} q^{53} -200 \beta_{3} q^{55} + ( 220 + 220 \beta_{2} ) q^{58} + ( -620 \beta_{1} - 620 \beta_{3} ) q^{59} + 649 \beta_{1} q^{61} -168 \beta_{3} q^{62} + 64 q^{64} + ( 450 + 450 \beta_{2} ) q^{65} + 540 \beta_{2} q^{67} -4 \beta_{1} q^{68} + 420 q^{71} + ( -205 \beta_{1} - 205 \beta_{3} ) q^{73} -40 \beta_{2} q^{74} + 32 \beta_{3} q^{76} + ( 760 + 760 \beta_{2} ) q^{79} + ( -80 \beta_{1} - 80 \beta_{3} ) q^{80} + 70 \beta_{1} q^{82} -668 \beta_{3} q^{83} -10 q^{85} + ( 680 + 680 \beta_{2} ) q^{86} -320 \beta_{2} q^{88} -815 \beta_{1} q^{89} -272 q^{92} + ( 128 \beta_{1} + 128 \beta_{3} ) q^{94} + 80 \beta_{2} q^{95} -355 \beta_{3} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} - 8q^{4} + 32q^{8} + O(q^{10})$$ $$4q - 4q^{2} - 8q^{4} + 32q^{8} + 80q^{11} - 32q^{16} - 320q^{22} + 136q^{23} + 150q^{25} - 440q^{29} - 64q^{32} + 40q^{37} - 1360q^{43} + 320q^{44} + 272q^{46} - 600q^{50} + 1256q^{53} + 440q^{58} + 256q^{64} + 900q^{65} - 1080q^{67} + 1680q^{71} + 80q^{74} + 1520q^{79} - 40q^{85} + 1360q^{86} + 640q^{88} - 1088q^{92} - 160q^{95} + O(q^{100})$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
−1.00000 1.73205i 0 −2.00000 + 3.46410i −3.53553 6.12372i 0 0 8.00000 0 −7.07107 + 12.2474i
361.2 −1.00000 1.73205i 0 −2.00000 + 3.46410i 3.53553 + 6.12372i 0 0 8.00000 0 7.07107 12.2474i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i −3.53553 + 6.12372i 0 0 8.00000 0 −7.07107 12.2474i
667.2 −1.00000 + 1.73205i 0 −2.00000 3.46410i 3.53553 6.12372i 0 0 8.00000 0 7.07107 + 12.2474i
 $$n$$: 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
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.4.g.bc 4
3.b odd 2 1 882.4.g.bg 4
7.b odd 2 1 inner 882.4.g.bc 4
7.c even 3 1 882.4.a.be yes 2
7.c even 3 1 inner 882.4.g.bc 4
7.d odd 6 1 882.4.a.be yes 2
7.d odd 6 1 inner 882.4.g.bc 4
21.c even 2 1 882.4.g.bg 4
21.g even 6 1 882.4.a.y 2
21.g even 6 1 882.4.g.bg 4
21.h odd 6 1 882.4.a.y 2
21.h odd 6 1 882.4.g.bg 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.4.a.y 2 21.g even 6 1
882.4.a.y 2 21.h odd 6 1
882.4.a.be yes 2 7.c even 3 1
882.4.a.be yes 2 7.d odd 6 1
882.4.g.bc 4 1.a even 1 1 trivial
882.4.g.bc 4 7.b odd 2 1 inner
882.4.g.bc 4 7.c even 3 1 inner
882.4.g.bc 4 7.d odd 6 1 inner
882.4.g.bg 4 3.b odd 2 1
882.4.g.bg 4 21.c even 2 1
882.4.g.bg 4 21.g even 6 1
882.4.g.bg 4 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{4} + 50 T_{5}^{2} + 2500$$ $$T_{11}^{2} - 40 T_{11} + 1600$$ $$T_{13}^{2} - 4050$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 + 2 T + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$2500 + 50 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 1600 - 40 T + T^{2} )^{2}$$
$13$ $$( -4050 + T^{2} )^{2}$$
$17$ $$4 + 2 T^{2} + T^{4}$$
$19$ $$16384 + 128 T^{2} + T^{4}$$
$23$ $$( 4624 - 68 T + T^{2} )^{2}$$
$29$ $$( 110 + T )^{4}$$
$31$ $$199148544 + 14112 T^{2} + T^{4}$$
$37$ $$( 400 - 20 T + T^{2} )^{2}$$
$41$ $$( -2450 + T^{2} )^{2}$$
$43$ $$( 340 + T )^{4}$$
$47$ $$67108864 + 8192 T^{2} + T^{4}$$
$53$ $$( 394384 - 628 T + T^{2} )^{2}$$
$59$ $$591053440000 + 768800 T^{2} + T^{4}$$
$61$ $$709641129604 + 842402 T^{2} + T^{4}$$
$67$ $$( 291600 + 540 T + T^{2} )^{2}$$
$71$ $$( -420 + T )^{4}$$
$73$ $$7064402500 + 84050 T^{2} + T^{4}$$
$79$ $$( 577600 - 760 T + T^{2} )^{2}$$
$83$ $$( -892448 + T^{2} )^{2}$$
$89$ $$1764779402500 + 1328450 T^{2} + T^{4}$$
$97$ $$( -252050 + T^{2} )^{2}$$