# Properties

 Label 882.4.g.f 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 6) 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 $$q - 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} - 6 \zeta_{6} q^{5} + 8 q^{8} + (12 \zeta_{6} - 12) q^{10} + ( - 12 \zeta_{6} + 12) q^{11} - 38 q^{13} - 16 \zeta_{6} q^{16} + ( - 126 \zeta_{6} + 126) q^{17} + 20 \zeta_{6} q^{19} + 24 q^{20} - 24 q^{22} + 168 \zeta_{6} q^{23} + ( - 89 \zeta_{6} + 89) q^{25} + 76 \zeta_{6} q^{26} - 30 q^{29} + (88 \zeta_{6} - 88) q^{31} + (32 \zeta_{6} - 32) q^{32} - 252 q^{34} - 254 \zeta_{6} q^{37} + ( - 40 \zeta_{6} + 40) q^{38} - 48 \zeta_{6} q^{40} + 42 q^{41} - 52 q^{43} + 48 \zeta_{6} q^{44} + ( - 336 \zeta_{6} + 336) q^{46} + 96 \zeta_{6} q^{47} - 178 q^{50} + ( - 152 \zeta_{6} + 152) q^{52} + ( - 198 \zeta_{6} + 198) q^{53} - 72 q^{55} + 60 \zeta_{6} q^{58} + ( - 660 \zeta_{6} + 660) q^{59} - 538 \zeta_{6} q^{61} + 176 q^{62} + 64 q^{64} + 228 \zeta_{6} q^{65} + (884 \zeta_{6} - 884) q^{67} + 504 \zeta_{6} q^{68} - 792 q^{71} + ( - 218 \zeta_{6} + 218) q^{73} + (508 \zeta_{6} - 508) q^{74} - 80 q^{76} + 520 \zeta_{6} q^{79} + (96 \zeta_{6} - 96) q^{80} - 84 \zeta_{6} q^{82} - 492 q^{83} - 756 q^{85} + 104 \zeta_{6} q^{86} + ( - 96 \zeta_{6} + 96) q^{88} - 810 \zeta_{6} q^{89} - 672 q^{92} + ( - 192 \zeta_{6} + 192) q^{94} + ( - 120 \zeta_{6} + 120) q^{95} - 1154 q^{97} +O(q^{100})$$ q - 2*z * q^2 + (4*z - 4) * q^4 - 6*z * q^5 + 8 * q^8 + (12*z - 12) * q^10 + (-12*z + 12) * q^11 - 38 * q^13 - 16*z * q^16 + (-126*z + 126) * q^17 + 20*z * q^19 + 24 * q^20 - 24 * q^22 + 168*z * q^23 + (-89*z + 89) * q^25 + 76*z * q^26 - 30 * q^29 + (88*z - 88) * q^31 + (32*z - 32) * q^32 - 252 * q^34 - 254*z * q^37 + (-40*z + 40) * q^38 - 48*z * q^40 + 42 * q^41 - 52 * q^43 + 48*z * q^44 + (-336*z + 336) * q^46 + 96*z * q^47 - 178 * q^50 + (-152*z + 152) * q^52 + (-198*z + 198) * q^53 - 72 * q^55 + 60*z * q^58 + (-660*z + 660) * q^59 - 538*z * q^61 + 176 * q^62 + 64 * q^64 + 228*z * q^65 + (884*z - 884) * q^67 + 504*z * q^68 - 792 * q^71 + (-218*z + 218) * q^73 + (508*z - 508) * q^74 - 80 * q^76 + 520*z * q^79 + (96*z - 96) * q^80 - 84*z * q^82 - 492 * q^83 - 756 * q^85 + 104*z * q^86 + (-96*z + 96) * q^88 - 810*z * q^89 - 672 * q^92 + (-192*z + 192) * q^94 + (-120*z + 120) * q^95 - 1154 * 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 $$2 q - 2 q^{2} - 4 q^{4} - 6 q^{5} + 16 q^{8} - 12 q^{10} + 12 q^{11} - 76 q^{13} - 16 q^{16} + 126 q^{17} + 20 q^{19} + 48 q^{20} - 48 q^{22} + 168 q^{23} + 89 q^{25} + 76 q^{26} - 60 q^{29} - 88 q^{31} - 32 q^{32} - 504 q^{34} - 254 q^{37} + 40 q^{38} - 48 q^{40} + 84 q^{41} - 104 q^{43} + 48 q^{44} + 336 q^{46} + 96 q^{47} - 356 q^{50} + 152 q^{52} + 198 q^{53} - 144 q^{55} + 60 q^{58} + 660 q^{59} - 538 q^{61} + 352 q^{62} + 128 q^{64} + 228 q^{65} - 884 q^{67} + 504 q^{68} - 1584 q^{71} + 218 q^{73} - 508 q^{74} - 160 q^{76} + 520 q^{79} - 96 q^{80} - 84 q^{82} - 984 q^{83} - 1512 q^{85} + 104 q^{86} + 96 q^{88} - 810 q^{89} - 1344 q^{92} + 192 q^{94} + 120 q^{95} - 2308 q^{97}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^4 - 6 * q^5 + 16 * q^8 - 12 * q^10 + 12 * q^11 - 76 * q^13 - 16 * q^16 + 126 * q^17 + 20 * q^19 + 48 * q^20 - 48 * q^22 + 168 * q^23 + 89 * q^25 + 76 * q^26 - 60 * q^29 - 88 * q^31 - 32 * q^32 - 504 * q^34 - 254 * q^37 + 40 * q^38 - 48 * q^40 + 84 * q^41 - 104 * q^43 + 48 * q^44 + 336 * q^46 + 96 * q^47 - 356 * q^50 + 152 * q^52 + 198 * q^53 - 144 * q^55 + 60 * q^58 + 660 * q^59 - 538 * q^61 + 352 * q^62 + 128 * q^64 + 228 * q^65 - 884 * q^67 + 504 * q^68 - 1584 * q^71 + 218 * q^73 - 508 * q^74 - 160 * q^76 + 520 * q^79 - 96 * q^80 - 84 * q^82 - 984 * q^83 - 1512 * q^85 + 104 * q^86 + 96 * q^88 - 810 * q^89 - 1344 * q^92 + 192 * q^94 + 120 * q^95 - 2308 * 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.f 2
3.b odd 2 1 294.4.e.g 2
7.b odd 2 1 882.4.g.i 2
7.c even 3 1 882.4.a.n 1
7.c even 3 1 inner 882.4.g.f 2
7.d odd 6 1 18.4.a.a 1
7.d odd 6 1 882.4.g.i 2
21.c even 2 1 294.4.e.h 2
21.g even 6 1 6.4.a.a 1
21.g even 6 1 294.4.e.h 2
21.h odd 6 1 294.4.a.e 1
21.h odd 6 1 294.4.e.g 2
28.f even 6 1 144.4.a.c 1
35.i odd 6 1 450.4.a.h 1
35.k even 12 2 450.4.c.e 2
56.j odd 6 1 576.4.a.q 1
56.m even 6 1 576.4.a.r 1
63.i even 6 1 162.4.c.f 2
63.k odd 6 1 162.4.c.c 2
63.s even 6 1 162.4.c.f 2
63.t odd 6 1 162.4.c.c 2
77.i even 6 1 2178.4.a.e 1
84.j odd 6 1 48.4.a.c 1
84.n even 6 1 2352.4.a.e 1
105.p even 6 1 150.4.a.i 1
105.w odd 12 2 150.4.c.d 2
168.ba even 6 1 192.4.a.i 1
168.be odd 6 1 192.4.a.c 1
231.k odd 6 1 726.4.a.f 1
273.ba even 6 1 1014.4.a.g 1
273.cb odd 12 2 1014.4.b.d 2
336.bo even 12 2 768.4.d.n 2
336.br odd 12 2 768.4.d.c 2
357.s even 6 1 1734.4.a.d 1
399.s odd 6 1 2166.4.a.i 1
420.be odd 6 1 1200.4.a.b 1
420.br even 12 2 1200.4.f.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 21.g even 6 1
18.4.a.a 1 7.d odd 6 1
48.4.a.c 1 84.j odd 6 1
144.4.a.c 1 28.f even 6 1
150.4.a.i 1 105.p even 6 1
150.4.c.d 2 105.w odd 12 2
162.4.c.c 2 63.k odd 6 1
162.4.c.c 2 63.t odd 6 1
162.4.c.f 2 63.i even 6 1
162.4.c.f 2 63.s even 6 1
192.4.a.c 1 168.be odd 6 1
192.4.a.i 1 168.ba even 6 1
294.4.a.e 1 21.h odd 6 1
294.4.e.g 2 3.b odd 2 1
294.4.e.g 2 21.h odd 6 1
294.4.e.h 2 21.c even 2 1
294.4.e.h 2 21.g even 6 1
450.4.a.h 1 35.i odd 6 1
450.4.c.e 2 35.k even 12 2
576.4.a.q 1 56.j odd 6 1
576.4.a.r 1 56.m even 6 1
726.4.a.f 1 231.k odd 6 1
768.4.d.c 2 336.br odd 12 2
768.4.d.n 2 336.bo even 12 2
882.4.a.n 1 7.c even 3 1
882.4.g.f 2 1.a even 1 1 trivial
882.4.g.f 2 7.c even 3 1 inner
882.4.g.i 2 7.b odd 2 1
882.4.g.i 2 7.d odd 6 1
1014.4.a.g 1 273.ba even 6 1
1014.4.b.d 2 273.cb odd 12 2
1200.4.a.b 1 420.be odd 6 1
1200.4.f.j 2 420.br even 12 2
1734.4.a.d 1 357.s even 6 1
2166.4.a.i 1 399.s odd 6 1
2178.4.a.e 1 77.i even 6 1
2352.4.a.e 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 $$S_{4}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{2} + 6T_{5} + 36$$ T5^2 + 6*T5 + 36 $$T_{11}^{2} - 12T_{11} + 144$$ T11^2 - 12*T11 + 144 $$T_{13} + 38$$ T13 + 38

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 6T + 36$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 12T + 144$$
$13$ $$(T + 38)^{2}$$
$17$ $$T^{2} - 126T + 15876$$
$19$ $$T^{2} - 20T + 400$$
$23$ $$T^{2} - 168T + 28224$$
$29$ $$(T + 30)^{2}$$
$31$ $$T^{2} + 88T + 7744$$
$37$ $$T^{2} + 254T + 64516$$
$41$ $$(T - 42)^{2}$$
$43$ $$(T + 52)^{2}$$
$47$ $$T^{2} - 96T + 9216$$
$53$ $$T^{2} - 198T + 39204$$
$59$ $$T^{2} - 660T + 435600$$
$61$ $$T^{2} + 538T + 289444$$
$67$ $$T^{2} + 884T + 781456$$
$71$ $$(T + 792)^{2}$$
$73$ $$T^{2} - 218T + 47524$$
$79$ $$T^{2} - 520T + 270400$$
$83$ $$(T + 492)^{2}$$
$89$ $$T^{2} + 810T + 656100$$
$97$ $$(T + 1154)^{2}$$