# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,4,Mod(361,882)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("882.361");

S:= CuspForms(chi, 4);

N := Newforms(S);

 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

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$52.0396846251$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ 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 294) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q - 2 \beta_{2} q^{2} + ( - 4 \beta_{2} - 4) q^{4} + ( - \beta_{3} + 6 \beta_{2} - \beta_1) q^{5} - 8 q^{8}+O(q^{10})$$ q - 2*b2 * q^2 + (-4*b2 - 4) * q^4 + (-b3 + 6*b2 - b1) * q^5 - 8 * q^8 $$q - 2 \beta_{2} q^{2} + ( - 4 \beta_{2} - 4) q^{4} + ( - \beta_{3} + 6 \beta_{2} - \beta_1) q^{5} - 8 q^{8} + (12 \beta_{2} - 2 \beta_1 + 12) q^{10} + (2 \beta_{2} - 6 \beta_1 + 2) q^{11} + (15 \beta_{3} - 24) q^{13} + 16 \beta_{2} q^{16} + ( - 66 \beta_{2} - 11 \beta_1 - 66) q^{17} + (46 \beta_{3} - 60 \beta_{2} + 46 \beta_1) q^{19} + (4 \beta_{3} + 24) q^{20} + (12 \beta_{3} + 4) q^{22} + ( - 102 \beta_{3} + 38 \beta_{2} - 102 \beta_1) q^{23} + (87 \beta_{2} + 12 \beta_1 + 87) q^{25} + (30 \beta_{3} + 48 \beta_{2} + 30 \beta_1) q^{26} + ( - 150 \beta_{3} + 56) q^{29} + (216 \beta_{2} + 54 \beta_1 + 216) q^{31} + (32 \beta_{2} + 32) q^{32} + (22 \beta_{3} - 132) q^{34} + (180 \beta_{3} - 140 \beta_{2} + 180 \beta_1) q^{37} + ( - 120 \beta_{2} + 92 \beta_1 - 120) q^{38} + (8 \beta_{3} - 48 \beta_{2} + 8 \beta_1) q^{40} + (127 \beta_{3} + 18) q^{41} + (288 \beta_{3} - 64) q^{43} + (24 \beta_{3} - 8 \beta_{2} + 24 \beta_1) q^{44} + (76 \beta_{2} - 204 \beta_1 + 76) q^{46} + (338 \beta_{3} - 132 \beta_{2} + 338 \beta_1) q^{47} + ( - 24 \beta_{3} + 174) q^{50} + (96 \beta_{2} + 60 \beta_1 + 96) q^{52} + (134 \beta_{2} + 192 \beta_1 + 134) q^{53} + ( - 38 \beta_{3} - 24) q^{55} + ( - 300 \beta_{3} - 112 \beta_{2} - 300 \beta_1) q^{58} + ( - 168 \beta_{2} + 298 \beta_1 - 168) q^{59} + (353 \beta_{3} + 252 \beta_{2} + 353 \beta_1) q^{61} + ( - 108 \beta_{3} + 432) q^{62} + 64 q^{64} + ( - 66 \beta_{3} - 114 \beta_{2} - 66 \beta_1) q^{65} + (192 \beta_{2} - 144 \beta_1 + 192) q^{67} + (44 \beta_{3} + 264 \beta_{2} + 44 \beta_1) q^{68} + ( - 342 \beta_{3} + 198) q^{71} + ( - 156 \beta_{2} + 595 \beta_1 - 156) q^{73} + ( - 280 \beta_{2} + 360 \beta_1 - 280) q^{74} + ( - 184 \beta_{3} - 240) q^{76} + (300 \beta_{3} - 424 \beta_{2} + 300 \beta_1) q^{79} + ( - 96 \beta_{2} + 16 \beta_1 - 96) q^{80} + (254 \beta_{3} - 36 \beta_{2} + 254 \beta_1) q^{82} + (80 \beta_{3} - 324) q^{83} + 374 q^{85} + (576 \beta_{3} + 128 \beta_{2} + 576 \beta_1) q^{86} + ( - 16 \beta_{2} + 48 \beta_1 - 16) q^{88} + (175 \beta_{3} - 306 \beta_{2} + 175 \beta_1) q^{89} + (408 \beta_{3} + 152) q^{92} + ( - 264 \beta_{2} + 676 \beta_1 - 264) q^{94} + (452 \beta_{2} - 336 \beta_1 + 452) q^{95} + ( - 133 \beta_{3} + 1092) q^{97}+O(q^{100})$$ q - 2*b2 * q^2 + (-4*b2 - 4) * q^4 + (-b3 + 6*b2 - b1) * q^5 - 8 * q^8 + (12*b2 - 2*b1 + 12) * q^10 + (2*b2 - 6*b1 + 2) * q^11 + (15*b3 - 24) * q^13 + 16*b2 * q^16 + (-66*b2 - 11*b1 - 66) * q^17 + (46*b3 - 60*b2 + 46*b1) * q^19 + (4*b3 + 24) * q^20 + (12*b3 + 4) * q^22 + (-102*b3 + 38*b2 - 102*b1) * q^23 + (87*b2 + 12*b1 + 87) * q^25 + (30*b3 + 48*b2 + 30*b1) * q^26 + (-150*b3 + 56) * q^29 + (216*b2 + 54*b1 + 216) * q^31 + (32*b2 + 32) * q^32 + (22*b3 - 132) * q^34 + (180*b3 - 140*b2 + 180*b1) * q^37 + (-120*b2 + 92*b1 - 120) * q^38 + (8*b3 - 48*b2 + 8*b1) * q^40 + (127*b3 + 18) * q^41 + (288*b3 - 64) * q^43 + (24*b3 - 8*b2 + 24*b1) * q^44 + (76*b2 - 204*b1 + 76) * q^46 + (338*b3 - 132*b2 + 338*b1) * q^47 + (-24*b3 + 174) * q^50 + (96*b2 + 60*b1 + 96) * q^52 + (134*b2 + 192*b1 + 134) * q^53 + (-38*b3 - 24) * q^55 + (-300*b3 - 112*b2 - 300*b1) * q^58 + (-168*b2 + 298*b1 - 168) * q^59 + (353*b3 + 252*b2 + 353*b1) * q^61 + (-108*b3 + 432) * q^62 + 64 * q^64 + (-66*b3 - 114*b2 - 66*b1) * q^65 + (192*b2 - 144*b1 + 192) * q^67 + (44*b3 + 264*b2 + 44*b1) * q^68 + (-342*b3 + 198) * q^71 + (-156*b2 + 595*b1 - 156) * q^73 + (-280*b2 + 360*b1 - 280) * q^74 + (-184*b3 - 240) * q^76 + (300*b3 - 424*b2 + 300*b1) * q^79 + (-96*b2 + 16*b1 - 96) * q^80 + (254*b3 - 36*b2 + 254*b1) * q^82 + (80*b3 - 324) * q^83 + 374 * q^85 + (576*b3 + 128*b2 + 576*b1) * q^86 + (-16*b2 + 48*b1 - 16) * q^88 + (175*b3 - 306*b2 + 175*b1) * q^89 + (408*b3 + 152) * q^92 + (-264*b2 + 676*b1 - 264) * q^94 + (452*b2 - 336*b1 + 452) * q^95 + (-133*b3 + 1092) * 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 $$4 q + 4 q^{2} - 8 q^{4} - 12 q^{5} - 32 q^{8} + 24 q^{10} + 4 q^{11} - 96 q^{13} - 32 q^{16} - 132 q^{17} + 120 q^{19} + 96 q^{20} + 16 q^{22} - 76 q^{23} + 174 q^{25} - 96 q^{26} + 224 q^{29} + 432 q^{31} + 64 q^{32} - 528 q^{34} + 280 q^{37} - 240 q^{38} + 96 q^{40} + 72 q^{41} - 256 q^{43} + 16 q^{44} + 152 q^{46} + 264 q^{47} + 696 q^{50} + 192 q^{52} + 268 q^{53} - 96 q^{55} + 224 q^{58} - 336 q^{59} - 504 q^{61} + 1728 q^{62} + 256 q^{64} + 228 q^{65} + 384 q^{67} - 528 q^{68} + 792 q^{71} - 312 q^{73} - 560 q^{74} - 960 q^{76} + 848 q^{79} - 192 q^{80} + 72 q^{82} - 1296 q^{83} + 1496 q^{85} - 256 q^{86} - 32 q^{88} + 612 q^{89} + 608 q^{92} - 528 q^{94} + 904 q^{95} + 4368 q^{97}+O(q^{100})$$ 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 - 132 * q^17 + 120 * q^19 + 96 * q^20 + 16 * q^22 - 76 * q^23 + 174 * q^25 - 96 * q^26 + 224 * q^29 + 432 * q^31 + 64 * q^32 - 528 * q^34 + 280 * q^37 - 240 * q^38 + 96 * q^40 + 72 * q^41 - 256 * q^43 + 16 * q^44 + 152 * q^46 + 264 * q^47 + 696 * q^50 + 192 * q^52 + 268 * q^53 - 96 * q^55 + 224 * q^58 - 336 * q^59 - 504 * q^61 + 1728 * q^62 + 256 * q^64 + 228 * q^65 + 384 * q^67 - 528 * q^68 + 792 * q^71 - 312 * q^73 - 560 * q^74 - 960 * q^76 + 848 * q^79 - 192 * q^80 + 72 * q^82 - 1296 * q^83 + 1496 * q^85 - 256 * q^86 - 32 * q^88 + 612 * q^89 + 608 * q^92 - 528 * q^94 + 904 * q^95 + 4368 * q^97

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$785$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 −3.70711 6.42090i 0 0 −8.00000 0 7.41421 12.8418i
361.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −2.29289 3.97141i 0 0 −8.00000 0 4.58579 7.94282i
667.1 1.00000 1.73205i 0 −2.00000 3.46410i −3.70711 + 6.42090i 0 0 −8.00000 0 7.41421 + 12.8418i
667.2 1.00000 1.73205i 0 −2.00000 3.46410i −2.29289 + 3.97141i 0 0 −8.00000 0 4.58579 + 7.94282i
 $$n$$: 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.be 4
3.b odd 2 1 294.4.e.m 4
7.b odd 2 1 882.4.g.bk 4
7.c even 3 1 882.4.a.bb 2
7.c even 3 1 inner 882.4.g.be 4
7.d odd 6 1 882.4.a.t 2
7.d odd 6 1 882.4.g.bk 4
21.c even 2 1 294.4.e.k 4
21.g even 6 1 294.4.a.o yes 2
21.g even 6 1 294.4.e.k 4
21.h odd 6 1 294.4.a.l 2
21.h odd 6 1 294.4.e.m 4
84.j odd 6 1 2352.4.a.bu 2
84.n even 6 1 2352.4.a.bw 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.4.a.l 2 21.h odd 6 1
294.4.a.o yes 2 21.g even 6 1
294.4.e.k 4 21.c even 2 1
294.4.e.k 4 21.g even 6 1
294.4.e.m 4 3.b odd 2 1
294.4.e.m 4 21.h odd 6 1
882.4.a.t 2 7.d odd 6 1
882.4.a.bb 2 7.c even 3 1
882.4.g.be 4 1.a even 1 1 trivial
882.4.g.be 4 7.c even 3 1 inner
882.4.g.bk 4 7.b odd 2 1
882.4.g.bk 4 7.d odd 6 1
2352.4.a.bu 2 84.j odd 6 1
2352.4.a.bw 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 $$S_{4}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{4} + 12T_{5}^{3} + 110T_{5}^{2} + 408T_{5} + 1156$$ T5^4 + 12*T5^3 + 110*T5^2 + 408*T5 + 1156 $$T_{11}^{4} - 4T_{11}^{3} + 84T_{11}^{2} + 272T_{11} + 4624$$ T11^4 - 4*T11^3 + 84*T11^2 + 272*T11 + 4624 $$T_{13}^{2} + 48T_{13} + 126$$ T13^2 + 48*T13 + 126

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2 T + 4)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 12 T^{3} + 110 T^{2} + \cdots + 1156$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 4 T^{3} + 84 T^{2} + \cdots + 4624$$
$13$ $$(T^{2} + 48 T + 126)^{2}$$
$17$ $$T^{4} + 132 T^{3} + \cdots + 16924996$$
$19$ $$T^{4} - 120 T^{3} + 15032 T^{2} + \cdots + 399424$$
$23$ $$T^{4} + 76 T^{3} + \cdots + 374964496$$
$29$ $$(T^{2} - 112 T - 41864)^{2}$$
$31$ $$T^{4} - 432 T^{3} + \cdots + 1666598976$$
$37$ $$T^{4} - 280 T^{3} + \cdots + 2043040000$$
$41$ $$(T^{2} - 36 T - 31934)^{2}$$
$43$ $$(T^{2} + 128 T - 161792)^{2}$$
$47$ $$T^{4} - 264 T^{3} + \cdots + 44548012096$$
$53$ $$T^{4} - 268 T^{3} + \cdots + 3110515984$$
$59$ $$T^{4} + 336 T^{3} + \cdots + 22315579456$$
$61$ $$T^{4} + 504 T^{3} + \cdots + 34489689796$$
$67$ $$T^{4} - 384 T^{3} + \cdots + 21233664$$
$71$ $$(T^{2} - 396 T - 194724)^{2}$$
$73$ $$T^{4} + 312 T^{3} + \cdots + 467464833796$$
$79$ $$T^{4} - 848 T^{3} + 719328 T^{2} + \cdots + 50176$$
$83$ $$(T^{2} + 648 T + 92176)^{2}$$
$89$ $$T^{4} - 612 T^{3} + \cdots + 1048852996$$
$97$ $$(T^{2} - 2184 T + 1157086)^{2}$$