# Properties

 Label 882.4.g.a 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.

 $$f(q)$$ $$=$$ $$q - 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} - 22 \zeta_{6} q^{5} + 8 q^{8} +O(q^{10})$$ q - 2*z * q^2 + (4*z - 4) * q^4 - 22*z * q^5 + 8 * q^8 $$q - 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} - 22 \zeta_{6} q^{5} + 8 q^{8} + (44 \zeta_{6} - 44) q^{10} + ( - 26 \zeta_{6} + 26) q^{11} + 54 q^{13} - 16 \zeta_{6} q^{16} + (74 \zeta_{6} - 74) q^{17} + 116 \zeta_{6} q^{19} + 88 q^{20} - 52 q^{22} - 58 \zeta_{6} q^{23} + (359 \zeta_{6} - 359) q^{25} - 108 \zeta_{6} q^{26} - 208 q^{29} + (252 \zeta_{6} - 252) q^{31} + (32 \zeta_{6} - 32) q^{32} + 148 q^{34} - 50 \zeta_{6} q^{37} + ( - 232 \zeta_{6} + 232) q^{38} - 176 \zeta_{6} q^{40} + 126 q^{41} + 164 q^{43} + 104 \zeta_{6} q^{44} + (116 \zeta_{6} - 116) q^{46} + 444 \zeta_{6} q^{47} + 718 q^{50} + (216 \zeta_{6} - 216) q^{52} + ( - 12 \zeta_{6} + 12) q^{53} - 572 q^{55} + 416 \zeta_{6} q^{58} + (124 \zeta_{6} - 124) q^{59} - 162 \zeta_{6} q^{61} + 504 q^{62} + 64 q^{64} - 1188 \zeta_{6} q^{65} + ( - 860 \zeta_{6} + 860) q^{67} - 296 \zeta_{6} q^{68} + 238 q^{71} + (146 \zeta_{6} - 146) q^{73} + (100 \zeta_{6} - 100) q^{74} - 464 q^{76} + 984 \zeta_{6} q^{79} + (352 \zeta_{6} - 352) q^{80} - 252 \zeta_{6} q^{82} + 656 q^{83} + 1628 q^{85} - 328 \zeta_{6} q^{86} + ( - 208 \zeta_{6} + 208) q^{88} + 954 \zeta_{6} q^{89} + 232 q^{92} + ( - 888 \zeta_{6} + 888) q^{94} + ( - 2552 \zeta_{6} + 2552) q^{95} - 526 q^{97} +O(q^{100})$$ q - 2*z * q^2 + (4*z - 4) * q^4 - 22*z * q^5 + 8 * q^8 + (44*z - 44) * q^10 + (-26*z + 26) * q^11 + 54 * q^13 - 16*z * q^16 + (74*z - 74) * q^17 + 116*z * q^19 + 88 * q^20 - 52 * q^22 - 58*z * q^23 + (359*z - 359) * q^25 - 108*z * q^26 - 208 * q^29 + (252*z - 252) * q^31 + (32*z - 32) * q^32 + 148 * q^34 - 50*z * q^37 + (-232*z + 232) * q^38 - 176*z * q^40 + 126 * q^41 + 164 * q^43 + 104*z * q^44 + (116*z - 116) * q^46 + 444*z * q^47 + 718 * q^50 + (216*z - 216) * q^52 + (-12*z + 12) * q^53 - 572 * q^55 + 416*z * q^58 + (124*z - 124) * q^59 - 162*z * q^61 + 504 * q^62 + 64 * q^64 - 1188*z * q^65 + (-860*z + 860) * q^67 - 296*z * q^68 + 238 * q^71 + (146*z - 146) * q^73 + (100*z - 100) * q^74 - 464 * q^76 + 984*z * q^79 + (352*z - 352) * q^80 - 252*z * q^82 + 656 * q^83 + 1628 * q^85 - 328*z * q^86 + (-208*z + 208) * q^88 + 954*z * q^89 + 232 * q^92 + (-888*z + 888) * q^94 + (-2552*z + 2552) * q^95 - 526 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{4} - 22 q^{5} + 16 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^4 - 22 * q^5 + 16 * q^8 $$2 q - 2 q^{2} - 4 q^{4} - 22 q^{5} + 16 q^{8} - 44 q^{10} + 26 q^{11} + 108 q^{13} - 16 q^{16} - 74 q^{17} + 116 q^{19} + 176 q^{20} - 104 q^{22} - 58 q^{23} - 359 q^{25} - 108 q^{26} - 416 q^{29} - 252 q^{31} - 32 q^{32} + 296 q^{34} - 50 q^{37} + 232 q^{38} - 176 q^{40} + 252 q^{41} + 328 q^{43} + 104 q^{44} - 116 q^{46} + 444 q^{47} + 1436 q^{50} - 216 q^{52} + 12 q^{53} - 1144 q^{55} + 416 q^{58} - 124 q^{59} - 162 q^{61} + 1008 q^{62} + 128 q^{64} - 1188 q^{65} + 860 q^{67} - 296 q^{68} + 476 q^{71} - 146 q^{73} - 100 q^{74} - 928 q^{76} + 984 q^{79} - 352 q^{80} - 252 q^{82} + 1312 q^{83} + 3256 q^{85} - 328 q^{86} + 208 q^{88} + 954 q^{89} + 464 q^{92} + 888 q^{94} + 2552 q^{95} - 1052 q^{97}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^4 - 22 * q^5 + 16 * q^8 - 44 * q^10 + 26 * q^11 + 108 * q^13 - 16 * q^16 - 74 * q^17 + 116 * q^19 + 176 * q^20 - 104 * q^22 - 58 * q^23 - 359 * q^25 - 108 * q^26 - 416 * q^29 - 252 * q^31 - 32 * q^32 + 296 * q^34 - 50 * q^37 + 232 * q^38 - 176 * q^40 + 252 * q^41 + 328 * q^43 + 104 * q^44 - 116 * q^46 + 444 * q^47 + 1436 * q^50 - 216 * q^52 + 12 * q^53 - 1144 * q^55 + 416 * q^58 - 124 * q^59 - 162 * q^61 + 1008 * q^62 + 128 * q^64 - 1188 * q^65 + 860 * q^67 - 296 * q^68 + 476 * q^71 - 146 * q^73 - 100 * q^74 - 928 * q^76 + 984 * q^79 - 352 * q^80 - 252 * q^82 + 1312 * q^83 + 3256 * q^85 - 328 * q^86 + 208 * q^88 + 954 * q^89 + 464 * q^92 + 888 * q^94 + 2552 * q^95 - 1052 * 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 −11.0000 19.0526i 0 0 8.00000 0 −22.0000 + 38.1051i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i −11.0000 + 19.0526i 0 0 8.00000 0 −22.0000 38.1051i
 $$n$$: 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.a 2
3.b odd 2 1 882.4.g.x 2
7.b odd 2 1 882.4.g.m 2
7.c even 3 1 882.4.a.s 1
7.c even 3 1 inner 882.4.g.a 2
7.d odd 6 1 126.4.a.f yes 1
7.d odd 6 1 882.4.g.m 2
21.c even 2 1 882.4.g.n 2
21.g even 6 1 126.4.a.e 1
21.g even 6 1 882.4.g.n 2
21.h odd 6 1 882.4.a.a 1
21.h odd 6 1 882.4.g.x 2
28.f even 6 1 1008.4.a.a 1
84.j odd 6 1 1008.4.a.w 1

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

## Hecke kernels

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

 $$T_{5}^{2} + 22T_{5} + 484$$ T5^2 + 22*T5 + 484 $$T_{11}^{2} - 26T_{11} + 676$$ T11^2 - 26*T11 + 676 $$T_{13} - 54$$ T13 - 54

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 22T + 484$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 26T + 676$$
$13$ $$(T - 54)^{2}$$
$17$ $$T^{2} + 74T + 5476$$
$19$ $$T^{2} - 116T + 13456$$
$23$ $$T^{2} + 58T + 3364$$
$29$ $$(T + 208)^{2}$$
$31$ $$T^{2} + 252T + 63504$$
$37$ $$T^{2} + 50T + 2500$$
$41$ $$(T - 126)^{2}$$
$43$ $$(T - 164)^{2}$$
$47$ $$T^{2} - 444T + 197136$$
$53$ $$T^{2} - 12T + 144$$
$59$ $$T^{2} + 124T + 15376$$
$61$ $$T^{2} + 162T + 26244$$
$67$ $$T^{2} - 860T + 739600$$
$71$ $$(T - 238)^{2}$$
$73$ $$T^{2} + 146T + 21316$$
$79$ $$T^{2} - 984T + 968256$$
$83$ $$(T - 656)^{2}$$
$89$ $$T^{2} - 954T + 910116$$
$97$ $$(T + 526)^{2}$$