# Properties

 Label 882.4.g.l 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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q - 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} + 15 \zeta_{6} q^{5} + 8 q^{8} +O(q^{10})$$ q - 2*z * q^2 + (4*z - 4) * q^4 + 15*z * q^5 + 8 * q^8 $$q - 2 \zeta_{6} q^{2} + (4 \zeta_{6} - 4) q^{4} + 15 \zeta_{6} q^{5} + 8 q^{8} + ( - 30 \zeta_{6} + 30) q^{10} + (9 \zeta_{6} - 9) q^{11} + 88 q^{13} - 16 \zeta_{6} q^{16} + ( - 84 \zeta_{6} + 84) q^{17} + 104 \zeta_{6} q^{19} - 60 q^{20} + 18 q^{22} - 84 \zeta_{6} q^{23} + (100 \zeta_{6} - 100) q^{25} - 176 \zeta_{6} q^{26} - 51 q^{29} + ( - 185 \zeta_{6} + 185) q^{31} + (32 \zeta_{6} - 32) q^{32} - 168 q^{34} - 44 \zeta_{6} q^{37} + ( - 208 \zeta_{6} + 208) q^{38} + 120 \zeta_{6} q^{40} - 168 q^{41} + 326 q^{43} - 36 \zeta_{6} q^{44} + (168 \zeta_{6} - 168) q^{46} + 138 \zeta_{6} q^{47} + 200 q^{50} + (352 \zeta_{6} - 352) q^{52} + ( - 639 \zeta_{6} + 639) q^{53} - 135 q^{55} + 102 \zeta_{6} q^{58} + (159 \zeta_{6} - 159) q^{59} + 722 \zeta_{6} q^{61} - 370 q^{62} + 64 q^{64} + 1320 \zeta_{6} q^{65} + ( - 166 \zeta_{6} + 166) q^{67} + 336 \zeta_{6} q^{68} - 1086 q^{71} + ( - 218 \zeta_{6} + 218) q^{73} + (88 \zeta_{6} - 88) q^{74} - 416 q^{76} + 583 \zeta_{6} q^{79} + ( - 240 \zeta_{6} + 240) q^{80} + 336 \zeta_{6} q^{82} - 597 q^{83} + 1260 q^{85} - 652 \zeta_{6} q^{86} + (72 \zeta_{6} - 72) q^{88} + 1038 \zeta_{6} q^{89} + 336 q^{92} + ( - 276 \zeta_{6} + 276) q^{94} + (1560 \zeta_{6} - 1560) q^{95} + 169 q^{97} +O(q^{100})$$ q - 2*z * q^2 + (4*z - 4) * q^4 + 15*z * q^5 + 8 * q^8 + (-30*z + 30) * q^10 + (9*z - 9) * q^11 + 88 * q^13 - 16*z * q^16 + (-84*z + 84) * q^17 + 104*z * q^19 - 60 * q^20 + 18 * q^22 - 84*z * q^23 + (100*z - 100) * q^25 - 176*z * q^26 - 51 * q^29 + (-185*z + 185) * q^31 + (32*z - 32) * q^32 - 168 * q^34 - 44*z * q^37 + (-208*z + 208) * q^38 + 120*z * q^40 - 168 * q^41 + 326 * q^43 - 36*z * q^44 + (168*z - 168) * q^46 + 138*z * q^47 + 200 * q^50 + (352*z - 352) * q^52 + (-639*z + 639) * q^53 - 135 * q^55 + 102*z * q^58 + (159*z - 159) * q^59 + 722*z * q^61 - 370 * q^62 + 64 * q^64 + 1320*z * q^65 + (-166*z + 166) * q^67 + 336*z * q^68 - 1086 * q^71 + (-218*z + 218) * q^73 + (88*z - 88) * q^74 - 416 * q^76 + 583*z * q^79 + (-240*z + 240) * q^80 + 336*z * q^82 - 597 * q^83 + 1260 * q^85 - 652*z * q^86 + (72*z - 72) * q^88 + 1038*z * q^89 + 336 * q^92 + (-276*z + 276) * q^94 + (1560*z - 1560) * q^95 + 169 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{4} + 15 q^{5} + 16 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^4 + 15 * q^5 + 16 * q^8 $$2 q - 2 q^{2} - 4 q^{4} + 15 q^{5} + 16 q^{8} + 30 q^{10} - 9 q^{11} + 176 q^{13} - 16 q^{16} + 84 q^{17} + 104 q^{19} - 120 q^{20} + 36 q^{22} - 84 q^{23} - 100 q^{25} - 176 q^{26} - 102 q^{29} + 185 q^{31} - 32 q^{32} - 336 q^{34} - 44 q^{37} + 208 q^{38} + 120 q^{40} - 336 q^{41} + 652 q^{43} - 36 q^{44} - 168 q^{46} + 138 q^{47} + 400 q^{50} - 352 q^{52} + 639 q^{53} - 270 q^{55} + 102 q^{58} - 159 q^{59} + 722 q^{61} - 740 q^{62} + 128 q^{64} + 1320 q^{65} + 166 q^{67} + 336 q^{68} - 2172 q^{71} + 218 q^{73} - 88 q^{74} - 832 q^{76} + 583 q^{79} + 240 q^{80} + 336 q^{82} - 1194 q^{83} + 2520 q^{85} - 652 q^{86} - 72 q^{88} + 1038 q^{89} + 672 q^{92} + 276 q^{94} - 1560 q^{95} + 338 q^{97}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^4 + 15 * q^5 + 16 * q^8 + 30 * q^10 - 9 * q^11 + 176 * q^13 - 16 * q^16 + 84 * q^17 + 104 * q^19 - 120 * q^20 + 36 * q^22 - 84 * q^23 - 100 * q^25 - 176 * q^26 - 102 * q^29 + 185 * q^31 - 32 * q^32 - 336 * q^34 - 44 * q^37 + 208 * q^38 + 120 * q^40 - 336 * q^41 + 652 * q^43 - 36 * q^44 - 168 * q^46 + 138 * q^47 + 400 * q^50 - 352 * q^52 + 639 * q^53 - 270 * q^55 + 102 * q^58 - 159 * q^59 + 722 * q^61 - 740 * q^62 + 128 * q^64 + 1320 * q^65 + 166 * q^67 + 336 * q^68 - 2172 * q^71 + 218 * q^73 - 88 * q^74 - 832 * q^76 + 583 * q^79 + 240 * q^80 + 336 * q^82 - 1194 * q^83 + 2520 * q^85 - 652 * q^86 - 72 * q^88 + 1038 * q^89 + 672 * q^92 + 276 * q^94 - 1560 * q^95 + 338 * 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 7.50000 + 12.9904i 0 0 8.00000 0 15.0000 25.9808i
667.1 −1.00000 + 1.73205i 0 −2.00000 3.46410i 7.50000 12.9904i 0 0 8.00000 0 15.0000 + 25.9808i
 $$n$$: 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.l 2
3.b odd 2 1 294.4.e.e 2
7.b odd 2 1 126.4.g.a 2
7.c even 3 1 882.4.a.h 1
7.c even 3 1 inner 882.4.g.l 2
7.d odd 6 1 126.4.g.a 2
7.d odd 6 1 882.4.a.r 1
21.c even 2 1 42.4.e.b 2
21.g even 6 1 42.4.e.b 2
21.g even 6 1 294.4.a.a 1
21.h odd 6 1 294.4.a.g 1
21.h odd 6 1 294.4.e.e 2
84.h odd 2 1 336.4.q.d 2
84.j odd 6 1 336.4.q.d 2
84.j odd 6 1 2352.4.a.u 1
84.n even 6 1 2352.4.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.b 2 21.c even 2 1
42.4.e.b 2 21.g even 6 1
126.4.g.a 2 7.b odd 2 1
126.4.g.a 2 7.d odd 6 1
294.4.a.a 1 21.g even 6 1
294.4.a.g 1 21.h odd 6 1
294.4.e.e 2 3.b odd 2 1
294.4.e.e 2 21.h odd 6 1
336.4.q.d 2 84.h odd 2 1
336.4.q.d 2 84.j odd 6 1
882.4.a.h 1 7.c even 3 1
882.4.a.r 1 7.d odd 6 1
882.4.g.l 2 1.a even 1 1 trivial
882.4.g.l 2 7.c even 3 1 inner
2352.4.a.q 1 84.n even 6 1
2352.4.a.u 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 $$S_{4}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{2} - 15T_{5} + 225$$ T5^2 - 15*T5 + 225 $$T_{11}^{2} + 9T_{11} + 81$$ T11^2 + 9*T11 + 81 $$T_{13} - 88$$ T13 - 88

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 15T + 225$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 9T + 81$$
$13$ $$(T - 88)^{2}$$
$17$ $$T^{2} - 84T + 7056$$
$19$ $$T^{2} - 104T + 10816$$
$23$ $$T^{2} + 84T + 7056$$
$29$ $$(T + 51)^{2}$$
$31$ $$T^{2} - 185T + 34225$$
$37$ $$T^{2} + 44T + 1936$$
$41$ $$(T + 168)^{2}$$
$43$ $$(T - 326)^{2}$$
$47$ $$T^{2} - 138T + 19044$$
$53$ $$T^{2} - 639T + 408321$$
$59$ $$T^{2} + 159T + 25281$$
$61$ $$T^{2} - 722T + 521284$$
$67$ $$T^{2} - 166T + 27556$$
$71$ $$(T + 1086)^{2}$$
$73$ $$T^{2} - 218T + 47524$$
$79$ $$T^{2} - 583T + 339889$$
$83$ $$(T + 597)^{2}$$
$89$ $$T^{2} - 1038 T + 1077444$$
$97$ $$(T - 169)^{2}$$