# Properties

 Label 882.4.g.ba 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: no (minimal twist has level 98) 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.

 $$f(q)$$ $$=$$ $$q + ( -2 - 2 \beta_{2} ) q^{2} + 4 \beta_{2} q^{4} + 14 \beta_{1} q^{5} + 8 q^{8} +O(q^{10})$$ $$q + ( -2 - 2 \beta_{2} ) q^{2} + 4 \beta_{2} q^{4} + 14 \beta_{1} q^{5} + 8 q^{8} + ( -28 \beta_{1} - 28 \beta_{3} ) q^{10} + 14 \beta_{2} q^{11} + 36 \beta_{3} q^{13} + ( -16 - 16 \beta_{2} ) q^{16} + ( \beta_{1} + \beta_{3} ) q^{17} -\beta_{1} q^{19} + 56 \beta_{3} q^{20} + 28 q^{22} + ( 140 + 140 \beta_{2} ) q^{23} + 267 \beta_{2} q^{25} + 72 \beta_{1} q^{26} + 286 q^{29} + ( 66 \beta_{1} + 66 \beta_{3} ) q^{31} + 32 \beta_{2} q^{32} -2 \beta_{3} q^{34} + ( 38 + 38 \beta_{2} ) q^{37} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{38} + 112 \beta_{1} q^{40} + 89 \beta_{3} q^{41} -34 q^{43} + ( -56 - 56 \beta_{2} ) q^{44} -280 \beta_{2} q^{46} -370 \beta_{1} q^{47} + 534 q^{50} + ( -144 \beta_{1} - 144 \beta_{3} ) q^{52} + 74 \beta_{2} q^{53} + 196 \beta_{3} q^{55} + ( -572 - 572 \beta_{2} ) q^{58} + ( 307 \beta_{1} + 307 \beta_{3} ) q^{59} + 10 \beta_{1} q^{61} -132 \beta_{3} q^{62} + 64 q^{64} + ( -1008 - 1008 \beta_{2} ) q^{65} + 684 \beta_{2} q^{67} -4 \beta_{1} q^{68} -588 q^{71} + ( 191 \beta_{1} + 191 \beta_{3} ) q^{73} -76 \beta_{2} q^{74} -4 \beta_{3} q^{76} + ( -1220 - 1220 \beta_{2} ) q^{79} + ( -224 \beta_{1} - 224 \beta_{3} ) q^{80} + 178 \beta_{1} q^{82} -299 \beta_{3} q^{83} -28 q^{85} + ( 68 + 68 \beta_{2} ) q^{86} + 112 \beta_{2} q^{88} -437 \beta_{1} q^{89} -560 q^{92} + ( 740 \beta_{1} + 740 \beta_{3} ) q^{94} -28 \beta_{2} q^{95} + 1049 \beta_{3} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} - 8q^{4} + 32q^{8} + O(q^{10})$$ $$4q - 4q^{2} - 8q^{4} + 32q^{8} - 28q^{11} - 32q^{16} + 112q^{22} + 280q^{23} - 534q^{25} + 1144q^{29} - 64q^{32} + 76q^{37} - 136q^{43} - 112q^{44} + 560q^{46} + 2136q^{50} - 148q^{53} - 1144q^{58} + 256q^{64} - 2016q^{65} - 1368q^{67} - 2352q^{71} + 152q^{74} - 2440q^{79} - 112q^{85} + 136q^{86} - 224q^{88} - 2240q^{92} + 56q^{95} + O(q^{100})$$

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 −9.89949 17.1464i 0 0 8.00000 0 −19.7990 + 34.2929i
361.2 −1.00000 1.73205i 0 −2.00000 + 3.46410i 9.89949 + 17.1464i 0 0 8.00000 0 19.7990 34.2929i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i −9.89949 + 17.1464i 0 0 8.00000 0 −19.7990 34.2929i
667.2 −1.00000 + 1.73205i 0 −2.00000 3.46410i 9.89949 17.1464i 0 0 8.00000 0 19.7990 + 34.2929i
 $$n$$: 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.ba 4
3.b odd 2 1 98.4.c.h 4
7.b odd 2 1 inner 882.4.g.ba 4
7.c even 3 1 882.4.a.bg 2
7.c even 3 1 inner 882.4.g.ba 4
7.d odd 6 1 882.4.a.bg 2
7.d odd 6 1 inner 882.4.g.ba 4
21.c even 2 1 98.4.c.h 4
21.g even 6 1 98.4.a.g 2
21.g even 6 1 98.4.c.h 4
21.h odd 6 1 98.4.a.g 2
21.h odd 6 1 98.4.c.h 4
84.j odd 6 1 784.4.a.y 2
84.n even 6 1 784.4.a.y 2
105.o odd 6 1 2450.4.a.bx 2
105.p even 6 1 2450.4.a.bx 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.a.g 2 21.g even 6 1
98.4.a.g 2 21.h odd 6 1
98.4.c.h 4 3.b odd 2 1
98.4.c.h 4 21.c even 2 1
98.4.c.h 4 21.g even 6 1
98.4.c.h 4 21.h odd 6 1
784.4.a.y 2 84.j odd 6 1
784.4.a.y 2 84.n even 6 1
882.4.a.bg 2 7.c even 3 1
882.4.a.bg 2 7.d odd 6 1
882.4.g.ba 4 1.a even 1 1 trivial
882.4.g.ba 4 7.b odd 2 1 inner
882.4.g.ba 4 7.c even 3 1 inner
882.4.g.ba 4 7.d odd 6 1 inner
2450.4.a.bx 2 105.o odd 6 1
2450.4.a.bx 2 105.p even 6 1

## Hecke kernels

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

 $$T_{5}^{4} + 392 T_{5}^{2} + 153664$$ $$T_{11}^{2} + 14 T_{11} + 196$$ $$T_{13}^{2} - 2592$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 + 2 T + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$153664 + 392 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 196 + 14 T + T^{2} )^{2}$$
$13$ $$( -2592 + T^{2} )^{2}$$
$17$ $$4 + 2 T^{2} + T^{4}$$
$19$ $$4 + 2 T^{2} + T^{4}$$
$23$ $$( 19600 - 140 T + T^{2} )^{2}$$
$29$ $$( -286 + T )^{4}$$
$31$ $$75898944 + 8712 T^{2} + T^{4}$$
$37$ $$( 1444 - 38 T + T^{2} )^{2}$$
$41$ $$( -15842 + T^{2} )^{2}$$
$43$ $$( 34 + T )^{4}$$
$47$ $$74966440000 + 273800 T^{2} + T^{4}$$
$53$ $$( 5476 + 74 T + T^{2} )^{2}$$
$59$ $$35531496004 + 188498 T^{2} + T^{4}$$
$61$ $$40000 + 200 T^{2} + T^{4}$$
$67$ $$( 467856 + 684 T + T^{2} )^{2}$$
$71$ $$( 588 + T )^{4}$$
$73$ $$5323453444 + 72962 T^{2} + T^{4}$$
$79$ $$( 1488400 + 1220 T + T^{2} )^{2}$$
$83$ $$( -178802 + T^{2} )^{2}$$
$89$ $$145876635844 + 381938 T^{2} + T^{4}$$
$97$ $$( -2200802 + T^{2} )^{2}$$