# Properties

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

# 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: $$\Q(\sqrt{-3}, \sqrt{22})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 22x^{2} + 484$$ x^4 + 22*x^2 + 484 Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 98) 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.

 $$f(q)$$ $$=$$ $$q + (2 \beta_{2} + 2) q^{2} + 4 \beta_{2} q^{4} + \beta_1 q^{5} - 8 q^{8}+O(q^{10})$$ q + (2*b2 + 2) * q^2 + 4*b2 * q^4 + b1 * q^5 - 8 * q^8 $$q + (2 \beta_{2} + 2) q^{2} + 4 \beta_{2} q^{4} + \beta_1 q^{5} - 8 q^{8} + (2 \beta_{3} + 2 \beta_1) q^{10} - 20 \beta_{2} q^{11} - 7 \beta_{3} q^{13} + ( - 16 \beta_{2} - 16) q^{16} + ( - 6 \beta_{3} - 6 \beta_1) q^{17} - \beta_1 q^{19} + 4 \beta_{3} q^{20} + 40 q^{22} + (48 \beta_{2} + 48) q^{23} - 37 \beta_{2} q^{25} + 14 \beta_1 q^{26} + 166 q^{29} + ( - 22 \beta_{3} - 22 \beta_1) q^{31} - 32 \beta_{2} q^{32} - 12 \beta_{3} q^{34} + (78 \beta_{2} + 78) q^{37} + ( - 2 \beta_{3} - 2 \beta_1) q^{38} - 8 \beta_1 q^{40} + 42 \beta_{3} q^{41} + 436 q^{43} + (80 \beta_{2} + 80) q^{44} + 96 \beta_{2} q^{46} + 22 \beta_1 q^{47} + 74 q^{50} + (28 \beta_{3} + 28 \beta_1) q^{52} - 62 \beta_{2} q^{53} - 20 \beta_{3} q^{55} + (332 \beta_{2} + 332) q^{58} + (71 \beta_{3} + 71 \beta_1) q^{59} - 29 \beta_1 q^{61} - 44 \beta_{3} q^{62} + 64 q^{64} + (616 \beta_{2} + 616) q^{65} + 580 \beta_{2} q^{67} + 24 \beta_1 q^{68} + 544 q^{71} + ( - 64 \beta_{3} - 64 \beta_1) q^{73} + 156 \beta_{2} q^{74} - 4 \beta_{3} q^{76} + (680 \beta_{2} + 680) q^{79} + ( - 16 \beta_{3} - 16 \beta_1) q^{80} - 84 \beta_1 q^{82} + 21 \beta_{3} q^{83} + 528 q^{85} + (872 \beta_{2} + 872) q^{86} + 160 \beta_{2} q^{88} - 160 \beta_1 q^{89} - 192 q^{92} + (44 \beta_{3} + 44 \beta_1) q^{94} - 88 \beta_{2} q^{95} + 70 \beta_{3} q^{97}+O(q^{100})$$ q + (2*b2 + 2) * q^2 + 4*b2 * q^4 + b1 * q^5 - 8 * q^8 + (2*b3 + 2*b1) * q^10 - 20*b2 * q^11 - 7*b3 * q^13 + (-16*b2 - 16) * q^16 + (-6*b3 - 6*b1) * q^17 - b1 * q^19 + 4*b3 * q^20 + 40 * q^22 + (48*b2 + 48) * q^23 - 37*b2 * q^25 + 14*b1 * q^26 + 166 * q^29 + (-22*b3 - 22*b1) * q^31 - 32*b2 * q^32 - 12*b3 * q^34 + (78*b2 + 78) * q^37 + (-2*b3 - 2*b1) * q^38 - 8*b1 * q^40 + 42*b3 * q^41 + 436 * q^43 + (80*b2 + 80) * q^44 + 96*b2 * q^46 + 22*b1 * q^47 + 74 * q^50 + (28*b3 + 28*b1) * q^52 - 62*b2 * q^53 - 20*b3 * q^55 + (332*b2 + 332) * q^58 + (71*b3 + 71*b1) * q^59 - 29*b1 * q^61 - 44*b3 * q^62 + 64 * q^64 + (616*b2 + 616) * q^65 + 580*b2 * q^67 + 24*b1 * q^68 + 544 * q^71 + (-64*b3 - 64*b1) * q^73 + 156*b2 * q^74 - 4*b3 * q^76 + (680*b2 + 680) * q^79 + (-16*b3 - 16*b1) * q^80 - 84*b1 * q^82 + 21*b3 * q^83 + 528 * q^85 + (872*b2 + 872) * q^86 + 160*b2 * q^88 - 160*b1 * q^89 - 192 * q^92 + (44*b3 + 44*b1) * q^94 - 88*b2 * q^95 + 70*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 $$4 q + 4 q^{2} - 8 q^{4} - 32 q^{8} + 40 q^{11} - 32 q^{16} + 160 q^{22} + 96 q^{23} + 74 q^{25} + 664 q^{29} + 64 q^{32} + 156 q^{37} + 1744 q^{43} + 160 q^{44} - 192 q^{46} + 296 q^{50} + 124 q^{53} + 664 q^{58} + 256 q^{64} + 1232 q^{65} - 1160 q^{67} + 2176 q^{71} - 312 q^{74} + 1360 q^{79} + 2112 q^{85} + 1744 q^{86} - 320 q^{88} - 768 q^{92} + 176 q^{95}+O(q^{100})$$ 4 * q + 4 * q^2 - 8 * q^4 - 32 * q^8 + 40 * q^11 - 32 * q^16 + 160 * q^22 + 96 * q^23 + 74 * q^25 + 664 * q^29 + 64 * q^32 + 156 * q^37 + 1744 * q^43 + 160 * q^44 - 192 * q^46 + 296 * q^50 + 124 * q^53 + 664 * q^58 + 256 * q^64 + 1232 * q^65 - 1160 * q^67 + 2176 * q^71 - 312 * q^74 + 1360 * q^79 + 2112 * q^85 + 1744 * q^86 - 320 * q^88 - 768 * q^92 + 176 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 22x^{2} + 484$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 22$$ (v^2) / 22 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 11$$ (v^3) / 11
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$22\beta_{2}$$ 22*b2 $$\nu^{3}$$ $$=$$ $$11\beta_{3}$$ 11*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
 −2.34521 − 4.06202i 2.34521 + 4.06202i −2.34521 + 4.06202i 2.34521 − 4.06202i
1.00000 + 1.73205i 0 −2.00000 + 3.46410i −4.69042 8.12404i 0 0 −8.00000 0 9.38083 16.2481i
361.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 4.69042 + 8.12404i 0 0 −8.00000 0 −9.38083 + 16.2481i
667.1 1.00000 1.73205i 0 −2.00000 3.46410i −4.69042 + 8.12404i 0 0 −8.00000 0 9.38083 + 16.2481i
667.2 1.00000 1.73205i 0 −2.00000 3.46410i 4.69042 8.12404i 0 0 −8.00000 0 −9.38083 16.2481i
 $$n$$: 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.bi 4
3.b odd 2 1 98.4.c.g 4
7.b odd 2 1 inner 882.4.g.bi 4
7.c even 3 1 882.4.a.w 2
7.c even 3 1 inner 882.4.g.bi 4
7.d odd 6 1 882.4.a.w 2
7.d odd 6 1 inner 882.4.g.bi 4
21.c even 2 1 98.4.c.g 4
21.g even 6 1 98.4.a.h 2
21.g even 6 1 98.4.c.g 4
21.h odd 6 1 98.4.a.h 2
21.h odd 6 1 98.4.c.g 4
84.j odd 6 1 784.4.a.z 2
84.n even 6 1 784.4.a.z 2
105.o odd 6 1 2450.4.a.bs 2
105.p even 6 1 2450.4.a.bs 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.a.h 2 21.g even 6 1
98.4.a.h 2 21.h odd 6 1
98.4.c.g 4 3.b odd 2 1
98.4.c.g 4 21.c even 2 1
98.4.c.g 4 21.g even 6 1
98.4.c.g 4 21.h odd 6 1
784.4.a.z 2 84.j odd 6 1
784.4.a.z 2 84.n even 6 1
882.4.a.w 2 7.c even 3 1
882.4.a.w 2 7.d odd 6 1
882.4.g.bi 4 1.a even 1 1 trivial
882.4.g.bi 4 7.b odd 2 1 inner
882.4.g.bi 4 7.c even 3 1 inner
882.4.g.bi 4 7.d odd 6 1 inner
2450.4.a.bs 2 105.o odd 6 1
2450.4.a.bs 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 $$S_{4}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{4} + 88T_{5}^{2} + 7744$$ T5^4 + 88*T5^2 + 7744 $$T_{11}^{2} - 20T_{11} + 400$$ T11^2 - 20*T11 + 400 $$T_{13}^{2} - 4312$$ T13^2 - 4312

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2 T + 4)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 88T^{2} + 7744$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - 20 T + 400)^{2}$$
$13$ $$(T^{2} - 4312)^{2}$$
$17$ $$T^{4} + 3168 T^{2} + \cdots + 10036224$$
$19$ $$T^{4} + 88T^{2} + 7744$$
$23$ $$(T^{2} - 48 T + 2304)^{2}$$
$29$ $$(T - 166)^{4}$$
$31$ $$T^{4} + 42592 T^{2} + \cdots + 1814078464$$
$37$ $$(T^{2} - 78 T + 6084)^{2}$$
$41$ $$(T^{2} - 155232)^{2}$$
$43$ $$(T - 436)^{4}$$
$47$ $$T^{4} + 42592 T^{2} + \cdots + 1814078464$$
$53$ $$(T^{2} - 62 T + 3844)^{2}$$
$59$ $$T^{4} + 443608 T^{2} + \cdots + 196788057664$$
$61$ $$T^{4} + 74008 T^{2} + \cdots + 5477184064$$
$67$ $$(T^{2} + 580 T + 336400)^{2}$$
$71$ $$(T - 544)^{4}$$
$73$ $$T^{4} + 360448 T^{2} + \cdots + 129922760704$$
$79$ $$(T^{2} - 680 T + 462400)^{2}$$
$83$ $$(T^{2} - 38808)^{2}$$
$89$ $$T^{4} + 2252800 T^{2} + \cdots + 5075107840000$$
$97$ $$(T^{2} - 431200)^{2}$$