# Properties

 Label 882.4.g.h Level $882$ Weight $4$ Character orbit 882.g Analytic conductor $52.040$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 126) 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} + 6 \zeta_{6} q^{5} + 8 q^{8} +O(q^{10})$$ q - 2*z * q^2 + (4*z - 4) * q^4 + 6*z * q^5 + 8 * q^8 $$q - 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} + 6 \zeta_{6} q^{5} + 8 q^{8} + ( - 12 \zeta_{6} + 12) q^{10} + (30 \zeta_{6} - 30) q^{11} - 2 q^{13} - 16 \zeta_{6} q^{16} + ( - 66 \zeta_{6} + 66) q^{17} - 52 \zeta_{6} q^{19} - 24 q^{20} + 60 q^{22} - 114 \zeta_{6} q^{23} + ( - 89 \zeta_{6} + 89) q^{25} + 4 \zeta_{6} q^{26} + 72 q^{29} + (196 \zeta_{6} - 196) q^{31} + (32 \zeta_{6} - 32) q^{32} - 132 q^{34} + 286 \zeta_{6} q^{37} + (104 \zeta_{6} - 104) q^{38} + 48 \zeta_{6} q^{40} + 378 q^{41} + 164 q^{43} - 120 \zeta_{6} q^{44} + (228 \zeta_{6} - 228) q^{46} - 228 \zeta_{6} q^{47} - 178 q^{50} + ( - 8 \zeta_{6} + 8) q^{52} + ( - 348 \zeta_{6} + 348) q^{53} - 180 q^{55} - 144 \zeta_{6} q^{58} + (348 \zeta_{6} - 348) q^{59} - 106 \zeta_{6} q^{61} + 392 q^{62} + 64 q^{64} - 12 \zeta_{6} q^{65} + (596 \zeta_{6} - 596) q^{67} + 264 \zeta_{6} q^{68} + 630 q^{71} + (1042 \zeta_{6} - 1042) q^{73} + ( - 572 \zeta_{6} + 572) q^{74} + 208 q^{76} + 88 \zeta_{6} q^{79} + ( - 96 \zeta_{6} + 96) q^{80} - 756 \zeta_{6} q^{82} + 1440 q^{83} + 396 q^{85} - 328 \zeta_{6} q^{86} + (240 \zeta_{6} - 240) q^{88} + 1374 \zeta_{6} q^{89} + 456 q^{92} + (456 \zeta_{6} - 456) q^{94} + ( - 312 \zeta_{6} + 312) q^{95} + 34 q^{97} +O(q^{100})$$ q - 2*z * q^2 + (4*z - 4) * q^4 + 6*z * q^5 + 8 * q^8 + (-12*z + 12) * q^10 + (30*z - 30) * q^11 - 2 * q^13 - 16*z * q^16 + (-66*z + 66) * q^17 - 52*z * q^19 - 24 * q^20 + 60 * q^22 - 114*z * q^23 + (-89*z + 89) * q^25 + 4*z * q^26 + 72 * q^29 + (196*z - 196) * q^31 + (32*z - 32) * q^32 - 132 * q^34 + 286*z * q^37 + (104*z - 104) * q^38 + 48*z * q^40 + 378 * q^41 + 164 * q^43 - 120*z * q^44 + (228*z - 228) * q^46 - 228*z * q^47 - 178 * q^50 + (-8*z + 8) * q^52 + (-348*z + 348) * q^53 - 180 * q^55 - 144*z * q^58 + (348*z - 348) * q^59 - 106*z * q^61 + 392 * q^62 + 64 * q^64 - 12*z * q^65 + (596*z - 596) * q^67 + 264*z * q^68 + 630 * q^71 + (1042*z - 1042) * q^73 + (-572*z + 572) * q^74 + 208 * q^76 + 88*z * q^79 + (-96*z + 96) * q^80 - 756*z * q^82 + 1440 * q^83 + 396 * q^85 - 328*z * q^86 + (240*z - 240) * q^88 + 1374*z * q^89 + 456 * q^92 + (456*z - 456) * q^94 + (-312*z + 312) * q^95 + 34 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{4} + 6 q^{5} + 16 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^4 + 6 * q^5 + 16 * q^8 $$2 q - 2 q^{2} - 4 q^{4} + 6 q^{5} + 16 q^{8} + 12 q^{10} - 30 q^{11} - 4 q^{13} - 16 q^{16} + 66 q^{17} - 52 q^{19} - 48 q^{20} + 120 q^{22} - 114 q^{23} + 89 q^{25} + 4 q^{26} + 144 q^{29} - 196 q^{31} - 32 q^{32} - 264 q^{34} + 286 q^{37} - 104 q^{38} + 48 q^{40} + 756 q^{41} + 328 q^{43} - 120 q^{44} - 228 q^{46} - 228 q^{47} - 356 q^{50} + 8 q^{52} + 348 q^{53} - 360 q^{55} - 144 q^{58} - 348 q^{59} - 106 q^{61} + 784 q^{62} + 128 q^{64} - 12 q^{65} - 596 q^{67} + 264 q^{68} + 1260 q^{71} - 1042 q^{73} + 572 q^{74} + 416 q^{76} + 88 q^{79} + 96 q^{80} - 756 q^{82} + 2880 q^{83} + 792 q^{85} - 328 q^{86} - 240 q^{88} + 1374 q^{89} + 912 q^{92} - 456 q^{94} + 312 q^{95} + 68 q^{97}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^4 + 6 * q^5 + 16 * q^8 + 12 * q^10 - 30 * q^11 - 4 * q^13 - 16 * q^16 + 66 * q^17 - 52 * q^19 - 48 * q^20 + 120 * q^22 - 114 * q^23 + 89 * q^25 + 4 * q^26 + 144 * q^29 - 196 * q^31 - 32 * q^32 - 264 * q^34 + 286 * q^37 - 104 * q^38 + 48 * q^40 + 756 * q^41 + 328 * q^43 - 120 * q^44 - 228 * q^46 - 228 * q^47 - 356 * q^50 + 8 * q^52 + 348 * q^53 - 360 * q^55 - 144 * q^58 - 348 * q^59 - 106 * q^61 + 784 * q^62 + 128 * q^64 - 12 * q^65 - 596 * q^67 + 264 * q^68 + 1260 * q^71 - 1042 * q^73 + 572 * q^74 + 416 * q^76 + 88 * q^79 + 96 * q^80 - 756 * q^82 + 2880 * q^83 + 792 * q^85 - 328 * q^86 - 240 * q^88 + 1374 * q^89 + 912 * q^92 - 456 * q^94 + 312 * q^95 + 68 * q^97

## Character values

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

 $$n$$ $$199$$ $$785$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 1.73205i 0 −2.00000 + 3.46410i 3.00000 + 5.19615i 0 0 8.00000 0 6.00000 10.3923i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i 3.00000 5.19615i 0 0 8.00000 0 6.00000 + 10.3923i
 $$n$$: 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.h 2
3.b odd 2 1 882.4.g.q 2
7.b odd 2 1 882.4.g.e 2
7.c even 3 1 882.4.a.m 1
7.c even 3 1 inner 882.4.g.h 2
7.d odd 6 1 126.4.a.g yes 1
7.d odd 6 1 882.4.g.e 2
21.c even 2 1 882.4.g.t 2
21.g even 6 1 126.4.a.b 1
21.g even 6 1 882.4.g.t 2
21.h odd 6 1 882.4.a.e 1
21.h odd 6 1 882.4.g.q 2
28.f even 6 1 1008.4.a.n 1
84.j odd 6 1 1008.4.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.a.b 1 21.g even 6 1
126.4.a.g yes 1 7.d odd 6 1
882.4.a.e 1 21.h odd 6 1
882.4.a.m 1 7.c even 3 1
882.4.g.e 2 7.b odd 2 1
882.4.g.e 2 7.d odd 6 1
882.4.g.h 2 1.a even 1 1 trivial
882.4.g.h 2 7.c even 3 1 inner
882.4.g.q 2 3.b odd 2 1
882.4.g.q 2 21.h odd 6 1
882.4.g.t 2 21.c even 2 1
882.4.g.t 2 21.g even 6 1
1008.4.a.g 1 84.j odd 6 1
1008.4.a.n 1 28.f even 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 6T_{5} + 36$$ T5^2 - 6*T5 + 36 $$T_{11}^{2} + 30T_{11} + 900$$ T11^2 + 30*T11 + 900 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 6T + 36$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 30T + 900$$
$13$ $$(T + 2)^{2}$$
$17$ $$T^{2} - 66T + 4356$$
$19$ $$T^{2} + 52T + 2704$$
$23$ $$T^{2} + 114T + 12996$$
$29$ $$(T - 72)^{2}$$
$31$ $$T^{2} + 196T + 38416$$
$37$ $$T^{2} - 286T + 81796$$
$41$ $$(T - 378)^{2}$$
$43$ $$(T - 164)^{2}$$
$47$ $$T^{2} + 228T + 51984$$
$53$ $$T^{2} - 348T + 121104$$
$59$ $$T^{2} + 348T + 121104$$
$61$ $$T^{2} + 106T + 11236$$
$67$ $$T^{2} + 596T + 355216$$
$71$ $$(T - 630)^{2}$$
$73$ $$T^{2} + 1042 T + 1085764$$
$79$ $$T^{2} - 88T + 7744$$
$83$ $$(T - 1440)^{2}$$
$89$ $$T^{2} - 1374 T + 1887876$$
$97$ $$(T - 34)^{2}$$