# Properties

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

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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: yes 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.

 $$f(q)$$ $$=$$ $$q + (2 \beta_{2} + 2) q^{2} + 4 \beta_{2} q^{4} + 5 \beta_1 q^{5} - 8 q^{8}+O(q^{10})$$ q + (2*b2 + 2) * q^2 + 4*b2 * q^4 + 5*b1 * q^5 - 8 * q^8 $$q + (2 \beta_{2} + 2) q^{2} + 4 \beta_{2} q^{4} + 5 \beta_1 q^{5} - 8 q^{8} + (10 \beta_{3} + 10 \beta_1) q^{10} + 40 \beta_{2} q^{11} + 45 \beta_{3} q^{13} + ( - 16 \beta_{2} - 16) q^{16} + (\beta_{3} + \beta_1) q^{17} - 8 \beta_1 q^{19} + 20 \beta_{3} q^{20} - 80 q^{22} + ( - 68 \beta_{2} - 68) q^{23} - 75 \beta_{2} q^{25} - 90 \beta_1 q^{26} + 110 q^{29} + ( - 84 \beta_{3} - 84 \beta_1) q^{31} - 32 \beta_{2} q^{32} + 2 \beta_{3} q^{34} + (20 \beta_{2} + 20) q^{37} + ( - 16 \beta_{3} - 16 \beta_1) q^{38} - 40 \beta_1 q^{40} + 35 \beta_{3} q^{41} - 340 q^{43} + ( - 160 \beta_{2} - 160) q^{44} - 136 \beta_{2} q^{46} - 64 \beta_1 q^{47} + 150 q^{50} + ( - 180 \beta_{3} - 180 \beta_1) q^{52} + 628 \beta_{2} q^{53} + 200 \beta_{3} q^{55} + (220 \beta_{2} + 220) q^{58} + ( - 620 \beta_{3} - 620 \beta_1) q^{59} - 649 \beta_1 q^{61} - 168 \beta_{3} q^{62} + 64 q^{64} + ( - 450 \beta_{2} - 450) q^{65} + 540 \beta_{2} q^{67} - 4 \beta_1 q^{68} - 420 q^{71} + (205 \beta_{3} + 205 \beta_1) q^{73} + 40 \beta_{2} q^{74} - 32 \beta_{3} q^{76} + (760 \beta_{2} + 760) q^{79} + ( - 80 \beta_{3} - 80 \beta_1) q^{80} - 70 \beta_1 q^{82} - 668 \beta_{3} q^{83} - 10 q^{85} + ( - 680 \beta_{2} - 680) q^{86} - 320 \beta_{2} q^{88} - 815 \beta_1 q^{89} + 272 q^{92} + ( - 128 \beta_{3} - 128 \beta_1) q^{94} - 80 \beta_{2} q^{95} + 355 \beta_{3} q^{97}+O(q^{100})$$ q + (2*b2 + 2) * q^2 + 4*b2 * q^4 + 5*b1 * q^5 - 8 * q^8 + (10*b3 + 10*b1) * q^10 + 40*b2 * q^11 + 45*b3 * q^13 + (-16*b2 - 16) * q^16 + (b3 + b1) * q^17 - 8*b1 * q^19 + 20*b3 * q^20 - 80 * q^22 + (-68*b2 - 68) * q^23 - 75*b2 * q^25 - 90*b1 * q^26 + 110 * q^29 + (-84*b3 - 84*b1) * q^31 - 32*b2 * q^32 + 2*b3 * q^34 + (20*b2 + 20) * q^37 + (-16*b3 - 16*b1) * q^38 - 40*b1 * q^40 + 35*b3 * q^41 - 340 * q^43 + (-160*b2 - 160) * q^44 - 136*b2 * q^46 - 64*b1 * q^47 + 150 * q^50 + (-180*b3 - 180*b1) * q^52 + 628*b2 * q^53 + 200*b3 * q^55 + (220*b2 + 220) * q^58 + (-620*b3 - 620*b1) * q^59 - 649*b1 * q^61 - 168*b3 * q^62 + 64 * q^64 + (-450*b2 - 450) * q^65 + 540*b2 * q^67 - 4*b1 * q^68 - 420 * q^71 + (205*b3 + 205*b1) * q^73 + 40*b2 * q^74 - 32*b3 * q^76 + (760*b2 + 760) * q^79 + (-80*b3 - 80*b1) * q^80 - 70*b1 * q^82 - 668*b3 * q^83 - 10 * q^85 + (-680*b2 - 680) * q^86 - 320*b2 * q^88 - 815*b1 * q^89 + 272 * q^92 + (-128*b3 - 128*b1) * q^94 - 80*b2 * q^95 + 355*b3 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} - 8 q^{4} - 32 q^{8}+O(q^{10})$$ 4 * q + 4 * q^2 - 8 * q^4 - 32 * q^8 $$4 q + 4 q^{2} - 8 q^{4} - 32 q^{8} - 80 q^{11} - 32 q^{16} - 320 q^{22} - 136 q^{23} + 150 q^{25} + 440 q^{29} + 64 q^{32} + 40 q^{37} - 1360 q^{43} - 320 q^{44} + 272 q^{46} + 600 q^{50} - 1256 q^{53} + 440 q^{58} + 256 q^{64} - 900 q^{65} - 1080 q^{67} - 1680 q^{71} - 80 q^{74} + 1520 q^{79} - 40 q^{85} - 1360 q^{86} + 640 q^{88} + 1088 q^{92} + 160 q^{95}+O(q^{100})$$ 4 * q + 4 * q^2 - 8 * q^4 - 32 * q^8 - 80 * q^11 - 32 * q^16 - 320 * q^22 - 136 * q^23 + 150 * q^25 + 440 * q^29 + 64 * q^32 + 40 * q^37 - 1360 * q^43 - 320 * q^44 + 272 * q^46 + 600 * q^50 - 1256 * q^53 + 440 * q^58 + 256 * q^64 - 900 * q^65 - 1080 * q^67 - 1680 * q^71 - 80 * q^74 + 1520 * q^79 - 40 * q^85 - 1360 * q^86 + 640 * q^88 + 1088 * q^92 + 160 * q^95

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 $$\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$$.

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

## 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.

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 −3.53553 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.bg 4
3.b odd 2 1 882.4.g.bc 4
7.b odd 2 1 inner 882.4.g.bg 4
7.c even 3 1 882.4.a.y 2
7.c even 3 1 inner 882.4.g.bg 4
7.d odd 6 1 882.4.a.y 2
7.d odd 6 1 inner 882.4.g.bg 4
21.c even 2 1 882.4.g.bc 4
21.g even 6 1 882.4.a.be yes 2
21.g even 6 1 882.4.g.bc 4
21.h odd 6 1 882.4.a.be yes 2
21.h odd 6 1 882.4.g.bc 4

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

## Hecke kernels

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

 $$T_{5}^{4} + 50T_{5}^{2} + 2500$$ T5^4 + 50*T5^2 + 2500 $$T_{11}^{2} + 40T_{11} + 1600$$ T11^2 + 40*T11 + 1600 $$T_{13}^{2} - 4050$$ T13^2 - 4050

## Hecke characteristic polynomials

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