# Properties

 Label 882.4.g.o 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 42) 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} - 18 \zeta_{6} q^{5} - 8 q^{8} +O(q^{10})$$ q + 2*z * q^2 + (4*z - 4) * q^4 - 18*z * q^5 - 8 * q^8 $$q + 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} - 18 \zeta_{6} q^{5} - 8 q^{8} + ( - 36 \zeta_{6} + 36) q^{10} + (72 \zeta_{6} - 72) q^{11} + 34 q^{13} - 16 \zeta_{6} q^{16} + (6 \zeta_{6} - 6) q^{17} + 92 \zeta_{6} q^{19} + 72 q^{20} - 144 q^{22} - 180 \zeta_{6} q^{23} + (199 \zeta_{6} - 199) q^{25} + 68 \zeta_{6} q^{26} + 114 q^{29} + ( - 56 \zeta_{6} + 56) q^{31} + ( - 32 \zeta_{6} + 32) q^{32} - 12 q^{34} + 34 \zeta_{6} q^{37} + (184 \zeta_{6} - 184) q^{38} + 144 \zeta_{6} q^{40} + 6 q^{41} + 164 q^{43} - 288 \zeta_{6} q^{44} + ( - 360 \zeta_{6} + 360) q^{46} - 168 \zeta_{6} q^{47} - 398 q^{50} + (136 \zeta_{6} - 136) q^{52} + ( - 654 \zeta_{6} + 654) q^{53} + 1296 q^{55} + 228 \zeta_{6} q^{58} + ( - 492 \zeta_{6} + 492) q^{59} - 250 \zeta_{6} q^{61} + 112 q^{62} + 64 q^{64} - 612 \zeta_{6} q^{65} + ( - 124 \zeta_{6} + 124) q^{67} - 24 \zeta_{6} q^{68} - 36 q^{71} + ( - 1010 \zeta_{6} + 1010) q^{73} + (68 \zeta_{6} - 68) q^{74} - 368 q^{76} - 56 \zeta_{6} q^{79} + (288 \zeta_{6} - 288) q^{80} + 12 \zeta_{6} q^{82} + 228 q^{83} + 108 q^{85} + 328 \zeta_{6} q^{86} + ( - 576 \zeta_{6} + 576) q^{88} - 390 \zeta_{6} q^{89} + 720 q^{92} + ( - 336 \zeta_{6} + 336) q^{94} + ( - 1656 \zeta_{6} + 1656) q^{95} + 70 q^{97} +O(q^{100})$$ q + 2*z * q^2 + (4*z - 4) * q^4 - 18*z * q^5 - 8 * q^8 + (-36*z + 36) * q^10 + (72*z - 72) * q^11 + 34 * q^13 - 16*z * q^16 + (6*z - 6) * q^17 + 92*z * q^19 + 72 * q^20 - 144 * q^22 - 180*z * q^23 + (199*z - 199) * q^25 + 68*z * q^26 + 114 * q^29 + (-56*z + 56) * q^31 + (-32*z + 32) * q^32 - 12 * q^34 + 34*z * q^37 + (184*z - 184) * q^38 + 144*z * q^40 + 6 * q^41 + 164 * q^43 - 288*z * q^44 + (-360*z + 360) * q^46 - 168*z * q^47 - 398 * q^50 + (136*z - 136) * q^52 + (-654*z + 654) * q^53 + 1296 * q^55 + 228*z * q^58 + (-492*z + 492) * q^59 - 250*z * q^61 + 112 * q^62 + 64 * q^64 - 612*z * q^65 + (-124*z + 124) * q^67 - 24*z * q^68 - 36 * q^71 + (-1010*z + 1010) * q^73 + (68*z - 68) * q^74 - 368 * q^76 - 56*z * q^79 + (288*z - 288) * q^80 + 12*z * q^82 + 228 * q^83 + 108 * q^85 + 328*z * q^86 + (-576*z + 576) * q^88 - 390*z * q^89 + 720 * q^92 + (-336*z + 336) * q^94 + (-1656*z + 1656) * q^95 + 70 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 4 q^{4} - 18 q^{5} - 16 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 - 4 * q^4 - 18 * q^5 - 16 * q^8 $$2 q + 2 q^{2} - 4 q^{4} - 18 q^{5} - 16 q^{8} + 36 q^{10} - 72 q^{11} + 68 q^{13} - 16 q^{16} - 6 q^{17} + 92 q^{19} + 144 q^{20} - 288 q^{22} - 180 q^{23} - 199 q^{25} + 68 q^{26} + 228 q^{29} + 56 q^{31} + 32 q^{32} - 24 q^{34} + 34 q^{37} - 184 q^{38} + 144 q^{40} + 12 q^{41} + 328 q^{43} - 288 q^{44} + 360 q^{46} - 168 q^{47} - 796 q^{50} - 136 q^{52} + 654 q^{53} + 2592 q^{55} + 228 q^{58} + 492 q^{59} - 250 q^{61} + 224 q^{62} + 128 q^{64} - 612 q^{65} + 124 q^{67} - 24 q^{68} - 72 q^{71} + 1010 q^{73} - 68 q^{74} - 736 q^{76} - 56 q^{79} - 288 q^{80} + 12 q^{82} + 456 q^{83} + 216 q^{85} + 328 q^{86} + 576 q^{88} - 390 q^{89} + 1440 q^{92} + 336 q^{94} + 1656 q^{95} + 140 q^{97}+O(q^{100})$$ 2 * q + 2 * q^2 - 4 * q^4 - 18 * q^5 - 16 * q^8 + 36 * q^10 - 72 * q^11 + 68 * q^13 - 16 * q^16 - 6 * q^17 + 92 * q^19 + 144 * q^20 - 288 * q^22 - 180 * q^23 - 199 * q^25 + 68 * q^26 + 228 * q^29 + 56 * q^31 + 32 * q^32 - 24 * q^34 + 34 * q^37 - 184 * q^38 + 144 * q^40 + 12 * q^41 + 328 * q^43 - 288 * q^44 + 360 * q^46 - 168 * q^47 - 796 * q^50 - 136 * q^52 + 654 * q^53 + 2592 * q^55 + 228 * q^58 + 492 * q^59 - 250 * q^61 + 224 * q^62 + 128 * q^64 - 612 * q^65 + 124 * q^67 - 24 * q^68 - 72 * q^71 + 1010 * q^73 - 68 * q^74 - 736 * q^76 - 56 * q^79 - 288 * q^80 + 12 * q^82 + 456 * q^83 + 216 * q^85 + 328 * q^86 + 576 * q^88 - 390 * q^89 + 1440 * q^92 + 336 * q^94 + 1656 * q^95 + 140 * 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 −9.00000 15.5885i 0 0 −8.00000 0 18.0000 31.1769i
667.1 1.00000 1.73205i 0 −2.00000 3.46410i −9.00000 + 15.5885i 0 0 −8.00000 0 18.0000 + 31.1769i
 $$n$$: 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.o 2
3.b odd 2 1 294.4.e.b 2
7.b odd 2 1 882.4.g.w 2
7.c even 3 1 882.4.a.g 1
7.c even 3 1 inner 882.4.g.o 2
7.d odd 6 1 126.4.a.a 1
7.d odd 6 1 882.4.g.w 2
21.c even 2 1 294.4.e.c 2
21.g even 6 1 42.4.a.a 1
21.g even 6 1 294.4.e.c 2
21.h odd 6 1 294.4.a.i 1
21.h odd 6 1 294.4.e.b 2
28.f even 6 1 1008.4.a.b 1
84.j odd 6 1 336.4.a.l 1
84.n even 6 1 2352.4.a.a 1
105.p even 6 1 1050.4.a.g 1
105.w odd 12 2 1050.4.g.a 2
168.ba even 6 1 1344.4.a.o 1
168.be odd 6 1 1344.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.a.a 1 21.g even 6 1
126.4.a.a 1 7.d odd 6 1
294.4.a.i 1 21.h odd 6 1
294.4.e.b 2 3.b odd 2 1
294.4.e.b 2 21.h odd 6 1
294.4.e.c 2 21.c even 2 1
294.4.e.c 2 21.g even 6 1
336.4.a.l 1 84.j odd 6 1
882.4.a.g 1 7.c even 3 1
882.4.g.o 2 1.a even 1 1 trivial
882.4.g.o 2 7.c even 3 1 inner
882.4.g.w 2 7.b odd 2 1
882.4.g.w 2 7.d odd 6 1
1008.4.a.b 1 28.f even 6 1
1050.4.a.g 1 105.p even 6 1
1050.4.g.a 2 105.w odd 12 2
1344.4.a.a 1 168.be odd 6 1
1344.4.a.o 1 168.ba even 6 1
2352.4.a.a 1 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 $$S_{4}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{2} + 18T_{5} + 324$$ T5^2 + 18*T5 + 324 $$T_{11}^{2} + 72T_{11} + 5184$$ T11^2 + 72*T11 + 5184 $$T_{13} - 34$$ T13 - 34

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 18T + 324$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 72T + 5184$$
$13$ $$(T - 34)^{2}$$
$17$ $$T^{2} + 6T + 36$$
$19$ $$T^{2} - 92T + 8464$$
$23$ $$T^{2} + 180T + 32400$$
$29$ $$(T - 114)^{2}$$
$31$ $$T^{2} - 56T + 3136$$
$37$ $$T^{2} - 34T + 1156$$
$41$ $$(T - 6)^{2}$$
$43$ $$(T - 164)^{2}$$
$47$ $$T^{2} + 168T + 28224$$
$53$ $$T^{2} - 654T + 427716$$
$59$ $$T^{2} - 492T + 242064$$
$61$ $$T^{2} + 250T + 62500$$
$67$ $$T^{2} - 124T + 15376$$
$71$ $$(T + 36)^{2}$$
$73$ $$T^{2} - 1010 T + 1020100$$
$79$ $$T^{2} + 56T + 3136$$
$83$ $$(T - 228)^{2}$$
$89$ $$T^{2} + 390T + 152100$$
$97$ $$(T - 70)^{2}$$