# Properties

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

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## 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: $$x^{4} + 2x^{2} + 4$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \beta_{2} - 2) q^{2} + 4 \beta_{2} q^{4} + ( - 6 \beta_{2} - \beta_1 - 6) q^{5} + 8 q^{8}+O(q^{10})$$ q + (-2*b2 - 2) * q^2 + 4*b2 * q^4 + (-6*b2 - b1 - 6) * q^5 + 8 * q^8 $$q + ( - 2 \beta_{2} - 2) q^{2} + 4 \beta_{2} q^{4} + ( - 6 \beta_{2} - \beta_1 - 6) q^{5} + 8 q^{8} + (2 \beta_{3} + 12 \beta_{2} + 2 \beta_1) q^{10} + ( - 6 \beta_{3} + 2 \beta_{2} - 6 \beta_1) q^{11} + ( - 3 \beta_{3} - 24) q^{13} + ( - 16 \beta_{2} - 16) q^{16} + (\beta_{3} + 42 \beta_{2} + \beta_1) q^{17} + ( - 36 \beta_{2} + 2 \beta_1 - 36) q^{19} + ( - 4 \beta_{3} + 24) q^{20} + (12 \beta_{3} + 4) q^{22} + ( - 154 \beta_{2} - 6 \beta_1 - 154) q^{23} + (12 \beta_{3} + 9 \beta_{2} + 12 \beta_1) q^{25} + (48 \beta_{2} - 6 \beta_1 + 48) q^{26} + ( - 18 \beta_{3} + 40) q^{29} + ( - 6 \beta_{3} - 192 \beta_{2} - 6 \beta_1) q^{31} + 32 \beta_{2} q^{32} + ( - 2 \beta_{3} + 84) q^{34} + ( - 268 \beta_{2} - 12 \beta_1 - 268) q^{37} + ( - 4 \beta_{3} + 72 \beta_{2} - 4 \beta_1) q^{38} + ( - 48 \beta_{2} - 8 \beta_1 - 48) q^{40} + (5 \beta_{3} + 378) q^{41} + (24 \beta_{3} + 200) q^{43} + ( - 8 \beta_{2} + 24 \beta_1 - 8) q^{44} + (12 \beta_{3} + 308 \beta_{2} + 12 \beta_1) q^{46} + ( - 156 \beta_{2} - 10 \beta_1 - 156) q^{47} + ( - 24 \beta_{3} + 18) q^{50} + (12 \beta_{3} - 96 \beta_{2} + 12 \beta_1) q^{52} + ( - 24 \beta_{3} + 26 \beta_{2} - 24 \beta_1) q^{53} + (34 \beta_{3} - 576) q^{55} + ( - 80 \beta_{2} - 36 \beta_1 - 80) q^{58} + ( - 2 \beta_{3} + 432 \beta_{2} - 2 \beta_1) q^{59} + (708 \beta_{2} + 13 \beta_1 + 708) q^{61} + (12 \beta_{3} - 384) q^{62} + 64 q^{64} + ( - 150 \beta_{2} + 6 \beta_1 - 150) q^{65} + ( - 24 \beta_{3} + 72 \beta_{2} - 24 \beta_1) q^{67} + ( - 168 \beta_{2} - 4 \beta_1 - 168) q^{68} + (30 \beta_{3} + 762) q^{71} + (83 \beta_{3} - 372 \beta_{2} + 83 \beta_1) q^{73} + (24 \beta_{3} + 536 \beta_{2} + 24 \beta_1) q^{74} + (8 \beta_{3} + 144) q^{76} + ( - 488 \beta_{2} + 84 \beta_1 - 488) q^{79} + (16 \beta_{3} + 96 \beta_{2} + 16 \beta_1) q^{80} + ( - 756 \beta_{2} + 10 \beta_1 - 756) q^{82} + (136 \beta_{3} - 156) q^{83} + ( - 48 \beta_{3} + 350) q^{85} + ( - 400 \beta_{2} + 48 \beta_1 - 400) q^{86} + ( - 48 \beta_{3} + 16 \beta_{2} - 48 \beta_1) q^{88} + ( - 54 \beta_{2} - 29 \beta_1 - 54) q^{89} + ( - 24 \beta_{3} + 616) q^{92} + (20 \beta_{3} + 312 \beta_{2} + 20 \beta_1) q^{94} + (24 \beta_{3} + 20 \beta_{2} + 24 \beta_1) q^{95} + (125 \beta_{3} + 372) q^{97}+O(q^{100})$$ q + (-2*b2 - 2) * q^2 + 4*b2 * q^4 + (-6*b2 - b1 - 6) * q^5 + 8 * q^8 + (2*b3 + 12*b2 + 2*b1) * q^10 + (-6*b3 + 2*b2 - 6*b1) * q^11 + (-3*b3 - 24) * q^13 + (-16*b2 - 16) * q^16 + (b3 + 42*b2 + b1) * q^17 + (-36*b2 + 2*b1 - 36) * q^19 + (-4*b3 + 24) * q^20 + (12*b3 + 4) * q^22 + (-154*b2 - 6*b1 - 154) * q^23 + (12*b3 + 9*b2 + 12*b1) * q^25 + (48*b2 - 6*b1 + 48) * q^26 + (-18*b3 + 40) * q^29 + (-6*b3 - 192*b2 - 6*b1) * q^31 + 32*b2 * q^32 + (-2*b3 + 84) * q^34 + (-268*b2 - 12*b1 - 268) * q^37 + (-4*b3 + 72*b2 - 4*b1) * q^38 + (-48*b2 - 8*b1 - 48) * q^40 + (5*b3 + 378) * q^41 + (24*b3 + 200) * q^43 + (-8*b2 + 24*b1 - 8) * q^44 + (12*b3 + 308*b2 + 12*b1) * q^46 + (-156*b2 - 10*b1 - 156) * q^47 + (-24*b3 + 18) * q^50 + (12*b3 - 96*b2 + 12*b1) * q^52 + (-24*b3 + 26*b2 - 24*b1) * q^53 + (34*b3 - 576) * q^55 + (-80*b2 - 36*b1 - 80) * q^58 + (-2*b3 + 432*b2 - 2*b1) * q^59 + (708*b2 + 13*b1 + 708) * q^61 + (12*b3 - 384) * q^62 + 64 * q^64 + (-150*b2 + 6*b1 - 150) * q^65 + (-24*b3 + 72*b2 - 24*b1) * q^67 + (-168*b2 - 4*b1 - 168) * q^68 + (30*b3 + 762) * q^71 + (83*b3 - 372*b2 + 83*b1) * q^73 + (24*b3 + 536*b2 + 24*b1) * q^74 + (8*b3 + 144) * q^76 + (-488*b2 + 84*b1 - 488) * q^79 + (16*b3 + 96*b2 + 16*b1) * q^80 + (-756*b2 + 10*b1 - 756) * q^82 + (136*b3 - 156) * q^83 + (-48*b3 + 350) * q^85 + (-400*b2 + 48*b1 - 400) * q^86 + (-48*b3 + 16*b2 - 48*b1) * q^88 + (-54*b2 - 29*b1 - 54) * q^89 + (-24*b3 + 616) * q^92 + (20*b3 + 312*b2 + 20*b1) * q^94 + (24*b3 + 20*b2 + 24*b1) * q^95 + (125*b3 + 372) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} - 8 q^{4} - 12 q^{5} + 32 q^{8}+O(q^{10})$$ 4 * q - 4 * q^2 - 8 * q^4 - 12 * q^5 + 32 * q^8 $$4 q - 4 q^{2} - 8 q^{4} - 12 q^{5} + 32 q^{8} - 24 q^{10} - 4 q^{11} - 96 q^{13} - 32 q^{16} - 84 q^{17} - 72 q^{19} + 96 q^{20} + 16 q^{22} - 308 q^{23} - 18 q^{25} + 96 q^{26} + 160 q^{29} + 384 q^{31} - 64 q^{32} + 336 q^{34} - 536 q^{37} - 144 q^{38} - 96 q^{40} + 1512 q^{41} + 800 q^{43} - 16 q^{44} - 616 q^{46} - 312 q^{47} + 72 q^{50} + 192 q^{52} - 52 q^{53} - 2304 q^{55} - 160 q^{58} - 864 q^{59} + 1416 q^{61} - 1536 q^{62} + 256 q^{64} - 300 q^{65} - 144 q^{67} - 336 q^{68} + 3048 q^{71} + 744 q^{73} - 1072 q^{74} + 576 q^{76} - 976 q^{79} - 192 q^{80} - 1512 q^{82} - 624 q^{83} + 1400 q^{85} - 800 q^{86} - 32 q^{88} - 108 q^{89} + 2464 q^{92} - 624 q^{94} - 40 q^{95} + 1488 q^{97}+O(q^{100})$$ 4 * q - 4 * q^2 - 8 * q^4 - 12 * q^5 + 32 * q^8 - 24 * q^10 - 4 * q^11 - 96 * q^13 - 32 * q^16 - 84 * q^17 - 72 * q^19 + 96 * q^20 + 16 * q^22 - 308 * q^23 - 18 * q^25 + 96 * q^26 + 160 * q^29 + 384 * q^31 - 64 * q^32 + 336 * q^34 - 536 * q^37 - 144 * q^38 - 96 * q^40 + 1512 * q^41 + 800 * q^43 - 16 * q^44 - 616 * q^46 - 312 * q^47 + 72 * q^50 + 192 * q^52 - 52 * q^53 - 2304 * q^55 - 160 * q^58 - 864 * q^59 + 1416 * q^61 - 1536 * q^62 + 256 * q^64 - 300 * q^65 - 144 * q^67 - 336 * q^68 + 3048 * q^71 + 744 * q^73 - 1072 * q^74 + 576 * q^76 - 976 * q^79 - 192 * q^80 - 1512 * q^82 - 624 * q^83 + 1400 * q^85 - 800 * q^86 - 32 * q^88 - 108 * q^89 + 2464 * q^92 - 624 * q^94 - 40 * q^95 + 1488 * q^97

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

 $$\beta_{1}$$ $$=$$ $$7\nu$$ 7*v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( 7\nu^{3} ) / 2$$ (7*v^3) / 2
 $$\nu$$ $$=$$ $$( \beta_1 ) / 7$$ (b1) / 7 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$( 2\beta_{3} ) / 7$$ (2*b3) / 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 −7.94975 13.7694i 0 0 8.00000 0 −15.8995 + 27.5387i
361.2 −1.00000 1.73205i 0 −2.00000 + 3.46410i 1.94975 + 3.37706i 0 0 8.00000 0 3.89949 6.75412i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i −7.94975 + 13.7694i 0 0 8.00000 0 −15.8995 27.5387i
667.2 −1.00000 + 1.73205i 0 −2.00000 3.46410i 1.94975 3.37706i 0 0 8.00000 0 3.89949 + 6.75412i
 $$n$$: 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.y 4
3.b odd 2 1 294.4.e.n 4
7.b odd 2 1 882.4.g.bd 4
7.c even 3 1 882.4.a.bi 2
7.c even 3 1 inner 882.4.g.y 4
7.d odd 6 1 882.4.a.bc 2
7.d odd 6 1 882.4.g.bd 4
21.c even 2 1 294.4.e.o 4
21.g even 6 1 294.4.a.j 2
21.g even 6 1 294.4.e.o 4
21.h odd 6 1 294.4.a.k yes 2
21.h odd 6 1 294.4.e.n 4
84.j odd 6 1 2352.4.a.cd 2
84.n even 6 1 2352.4.a.bn 2

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

## Hecke kernels

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

 $$T_{5}^{4} + 12T_{5}^{3} + 206T_{5}^{2} - 744T_{5} + 3844$$ T5^4 + 12*T5^3 + 206*T5^2 - 744*T5 + 3844 $$T_{11}^{4} + 4T_{11}^{3} + 3540T_{11}^{2} - 14096T_{11} + 12418576$$ T11^4 + 4*T11^3 + 3540*T11^2 - 14096*T11 + 12418576 $$T_{13}^{2} + 48T_{13} - 306$$ T13^2 + 48*T13 - 306

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2 T + 4)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 12 T^{3} + 206 T^{2} + \cdots + 3844$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 4 T^{3} + 3540 T^{2} + \cdots + 12418576$$
$13$ $$(T^{2} + 48 T - 306)^{2}$$
$17$ $$T^{4} + 84 T^{3} + 5390 T^{2} + \cdots + 2775556$$
$19$ $$T^{4} + 72 T^{3} + 4280 T^{2} + \cdots + 817216$$
$23$ $$T^{4} + 308 T^{3} + \cdots + 407555344$$
$29$ $$(T^{2} - 80 T - 30152)^{2}$$
$31$ $$T^{4} - 384 T^{3} + \cdots + 1111288896$$
$37$ $$T^{4} + 536 T^{3} + \cdots + 3330674944$$
$41$ $$(T^{2} - 756 T + 140434)^{2}$$
$43$ $$(T^{2} - 400 T - 16448)^{2}$$
$47$ $$T^{4} + 312 T^{3} + \cdots + 211295296$$
$53$ $$T^{4} + 52 T^{3} + \cdots + 3110515984$$
$59$ $$T^{4} + 864 T^{3} + \cdots + 34682357824$$
$61$ $$T^{4} - 1416 T^{3} + \cdots + 234936028804$$
$67$ $$T^{4} + 144 T^{3} + \cdots + 2627997696$$
$71$ $$(T^{2} - 1524 T + 492444)^{2}$$
$73$ $$T^{4} - 744 T^{3} + \cdots + 288087680644$$
$79$ $$T^{4} + 976 T^{3} + \cdots + 205520782336$$
$83$ $$(T^{2} + 312 T - 1788272)^{2}$$
$89$ $$T^{4} + 108 T^{3} + \cdots + 6320568004$$
$97$ $$(T^{2} - 744 T - 1392866)^{2}$$
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