# Properties

 Label 882.4.g.bd Level $882$ Weight $4$ Character orbit 882.g Analytic conductor $52.040$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# 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: $$7^{2}$$ Twist minimal: no (minimal twist has level 294) 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.

 $$f(q)$$ $$=$$ $$q + ( -2 - 2 \beta_{2} ) q^{2} + 4 \beta_{2} q^{4} + ( 6 + \beta_{1} + 6 \beta_{2} ) q^{5} + 8 q^{8} +O(q^{10})$$ $$q + ( -2 - 2 \beta_{2} ) q^{2} + 4 \beta_{2} q^{4} + ( 6 + \beta_{1} + 6 \beta_{2} ) q^{5} + 8 q^{8} + ( -2 \beta_{1} - 12 \beta_{2} - 2 \beta_{3} ) q^{10} + ( -6 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} ) q^{11} + ( 24 + 3 \beta_{3} ) q^{13} + ( -16 - 16 \beta_{2} ) q^{16} + ( -\beta_{1} - 42 \beta_{2} - \beta_{3} ) q^{17} + ( 36 - 2 \beta_{1} + 36 \beta_{2} ) q^{19} + ( -24 + 4 \beta_{3} ) q^{20} + ( 4 + 12 \beta_{3} ) q^{22} + ( -154 - 6 \beta_{1} - 154 \beta_{2} ) q^{23} + ( 12 \beta_{1} + 9 \beta_{2} + 12 \beta_{3} ) q^{25} + ( -48 + 6 \beta_{1} - 48 \beta_{2} ) q^{26} + ( 40 - 18 \beta_{3} ) q^{29} + ( 6 \beta_{1} + 192 \beta_{2} + 6 \beta_{3} ) q^{31} + 32 \beta_{2} q^{32} + ( -84 + 2 \beta_{3} ) q^{34} + ( -268 - 12 \beta_{1} - 268 \beta_{2} ) q^{37} + ( 4 \beta_{1} - 72 \beta_{2} + 4 \beta_{3} ) q^{38} + ( 48 + 8 \beta_{1} + 48 \beta_{2} ) q^{40} + ( -378 - 5 \beta_{3} ) q^{41} + ( 200 + 24 \beta_{3} ) q^{43} + ( -8 + 24 \beta_{1} - 8 \beta_{2} ) q^{44} + ( 12 \beta_{1} + 308 \beta_{2} + 12 \beta_{3} ) q^{46} + ( 156 + 10 \beta_{1} + 156 \beta_{2} ) q^{47} + ( 18 - 24 \beta_{3} ) q^{50} + ( -12 \beta_{1} + 96 \beta_{2} - 12 \beta_{3} ) q^{52} + ( -24 \beta_{1} + 26 \beta_{2} - 24 \beta_{3} ) q^{53} + ( 576 - 34 \beta_{3} ) q^{55} + ( -80 - 36 \beta_{1} - 80 \beta_{2} ) q^{58} + ( 2 \beta_{1} - 432 \beta_{2} + 2 \beta_{3} ) q^{59} + ( -708 - 13 \beta_{1} - 708 \beta_{2} ) q^{61} + ( 384 - 12 \beta_{3} ) q^{62} + 64 q^{64} + ( -150 + 6 \beta_{1} - 150 \beta_{2} ) q^{65} + ( -24 \beta_{1} + 72 \beta_{2} - 24 \beta_{3} ) q^{67} + ( 168 + 4 \beta_{1} + 168 \beta_{2} ) q^{68} + ( 762 + 30 \beta_{3} ) q^{71} + ( -83 \beta_{1} + 372 \beta_{2} - 83 \beta_{3} ) q^{73} + ( 24 \beta_{1} + 536 \beta_{2} + 24 \beta_{3} ) q^{74} + ( -144 - 8 \beta_{3} ) q^{76} + ( -488 + 84 \beta_{1} - 488 \beta_{2} ) q^{79} + ( -16 \beta_{1} - 96 \beta_{2} - 16 \beta_{3} ) q^{80} + ( 756 - 10 \beta_{1} + 756 \beta_{2} ) q^{82} + ( 156 - 136 \beta_{3} ) q^{83} + ( 350 - 48 \beta_{3} ) q^{85} + ( -400 + 48 \beta_{1} - 400 \beta_{2} ) q^{86} + ( -48 \beta_{1} + 16 \beta_{2} - 48 \beta_{3} ) q^{88} + ( 54 + 29 \beta_{1} + 54 \beta_{2} ) q^{89} + ( 616 - 24 \beta_{3} ) q^{92} + ( -20 \beta_{1} - 312 \beta_{2} - 20 \beta_{3} ) q^{94} + ( 24 \beta_{1} + 20 \beta_{2} + 24 \beta_{3} ) q^{95} + ( -372 - 125 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} - 8q^{4} + 12q^{5} + 32q^{8} + O(q^{10})$$ $$4q - 4q^{2} - 8q^{4} + 12q^{5} + 32q^{8} + 24q^{10} - 4q^{11} + 96q^{13} - 32q^{16} + 84q^{17} + 72q^{19} - 96q^{20} + 16q^{22} - 308q^{23} - 18q^{25} - 96q^{26} + 160q^{29} - 384q^{31} - 64q^{32} - 336q^{34} - 536q^{37} + 144q^{38} + 96q^{40} - 1512q^{41} + 800q^{43} - 16q^{44} - 616q^{46} + 312q^{47} + 72q^{50} - 192q^{52} - 52q^{53} + 2304q^{55} - 160q^{58} + 864q^{59} - 1416q^{61} + 1536q^{62} + 256q^{64} - 300q^{65} - 144q^{67} + 336q^{68} + 3048q^{71} - 744q^{73} - 1072q^{74} - 576q^{76} - 976q^{79} + 192q^{80} + 1512q^{82} + 624q^{83} + 1400q^{85} - 800q^{86} - 32q^{88} + 108q^{89} + 2464q^{92} + 624q^{94} - 40q^{95} - 1488q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$7 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$7 \nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/7$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$$$/7$$

## 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 −1.94975 3.37706i 0 0 8.00000 0 −3.89949 + 6.75412i
361.2 −1.00000 1.73205i 0 −2.00000 + 3.46410i 7.94975 + 13.7694i 0 0 8.00000 0 15.8995 27.5387i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i −1.94975 + 3.37706i 0 0 8.00000 0 −3.89949 6.75412i
667.2 −1.00000 + 1.73205i 0 −2.00000 3.46410i 7.94975 13.7694i 0 0 8.00000 0 15.8995 + 27.5387i
 $$n$$: 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.4.g.bd 4
3.b odd 2 1 294.4.e.o 4
7.b odd 2 1 882.4.g.y 4
7.c even 3 1 882.4.a.bc 2
7.c even 3 1 inner 882.4.g.bd 4
7.d odd 6 1 882.4.a.bi 2
7.d odd 6 1 882.4.g.y 4
21.c even 2 1 294.4.e.n 4
21.g even 6 1 294.4.a.k yes 2
21.g even 6 1 294.4.e.n 4
21.h odd 6 1 294.4.a.j 2
21.h odd 6 1 294.4.e.o 4
84.j odd 6 1 2352.4.a.bn 2
84.n even 6 1 2352.4.a.cd 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.4.a.j 2 21.h odd 6 1
294.4.a.k yes 2 21.g even 6 1
294.4.e.n 4 21.c even 2 1
294.4.e.n 4 21.g even 6 1
294.4.e.o 4 3.b odd 2 1
294.4.e.o 4 21.h odd 6 1
882.4.a.bc 2 7.c even 3 1
882.4.a.bi 2 7.d odd 6 1
882.4.g.y 4 7.b odd 2 1
882.4.g.y 4 7.d odd 6 1
882.4.g.bd 4 1.a even 1 1 trivial
882.4.g.bd 4 7.c even 3 1 inner
2352.4.a.bn 2 84.j odd 6 1
2352.4.a.cd 2 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 $$S_{4}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{4} - 12 T_{5}^{3} + 206 T_{5}^{2} + 744 T_{5} + 3844$$ $$T_{11}^{4} + 4 T_{11}^{3} + 3540 T_{11}^{2} - 14096 T_{11} + 12418576$$ $$T_{13}^{2} - 48 T_{13} - 306$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 + 2 T + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$3844 + 744 T + 206 T^{2} - 12 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$12418576 - 14096 T + 3540 T^{2} + 4 T^{3} + T^{4}$$
$13$ $$( -306 - 48 T + T^{2} )^{2}$$
$17$ $$2775556 - 139944 T + 5390 T^{2} - 84 T^{3} + T^{4}$$
$19$ $$817216 - 65088 T + 4280 T^{2} - 72 T^{3} + T^{4}$$
$23$ $$407555344 + 6217904 T + 74676 T^{2} + 308 T^{3} + T^{4}$$
$29$ $$( -30152 - 80 T + T^{2} )^{2}$$
$31$ $$1111288896 + 12801024 T + 114120 T^{2} + 384 T^{3} + T^{4}$$
$37$ $$3330674944 + 30933632 T + 229584 T^{2} + 536 T^{3} + T^{4}$$
$41$ $$( 140434 + 756 T + T^{2} )^{2}$$
$43$ $$( -16448 - 400 T + T^{2} )^{2}$$
$47$ $$211295296 - 4535232 T + 82808 T^{2} - 312 T^{3} + T^{4}$$
$53$ $$3110515984 - 2900144 T + 58476 T^{2} + 52 T^{3} + T^{4}$$
$59$ $$34682357824 - 160904448 T + 560264 T^{2} - 864 T^{3} + T^{4}$$
$61$ $$234936028804 + 686338032 T + 1520354 T^{2} + 1416 T^{3} + T^{4}$$
$67$ $$2627997696 - 7382016 T + 72000 T^{2} + 144 T^{3} + T^{4}$$
$71$ $$( 492444 - 1524 T + T^{2} )^{2}$$
$73$ $$288087680644 - 399333072 T + 1090274 T^{2} + 744 T^{3} + T^{4}$$
$79$ $$205520782336 - 442463744 T + 1405920 T^{2} + 976 T^{3} + T^{4}$$
$83$ $$( -1788272 - 312 T + T^{2} )^{2}$$
$89$ $$6320568004 + 8586216 T + 91166 T^{2} - 108 T^{3} + T^{4}$$
$97$ $$( -1392866 + 744 T + T^{2} )^{2}$$