# Properties

 Label 882.2.f.g Level $882$ Weight $2$ Character orbit 882.f Analytic conductor $7.043$ 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] = [882,2,Mod(295,882)]

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([4, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("882.295");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 882.f (of order $$3$$, degree $$2$$, 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: $$7.04280545828$$ 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{2} + (\zeta_{6} - 2) q^{3} - \zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + (2 \zeta_{6} - 1) q^{6} - q^{8} + ( - 3 \zeta_{6} + 3) q^{9} +O(q^{10})$$ q + (-z + 1) * q^2 + (z - 2) * q^3 - z * q^4 + 3*z * q^5 + (2*z - 1) * q^6 - q^8 + (-3*z + 3) * q^9 $$q + ( - \zeta_{6} + 1) q^{2} + (\zeta_{6} - 2) q^{3} - \zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + (2 \zeta_{6} - 1) q^{6} - q^{8} + ( - 3 \zeta_{6} + 3) q^{9} + 3 q^{10} + ( - 3 \zeta_{6} + 3) q^{11} + (\zeta_{6} + 1) q^{12} - \zeta_{6} q^{13} + ( - 3 \zeta_{6} - 3) q^{15} + (\zeta_{6} - 1) q^{16} - 3 q^{17} - 3 \zeta_{6} q^{18} + 7 q^{19} + ( - 3 \zeta_{6} + 3) q^{20} - 3 \zeta_{6} q^{22} + 9 \zeta_{6} q^{23} + ( - \zeta_{6} + 2) q^{24} + (4 \zeta_{6} - 4) q^{25} - q^{26} + (6 \zeta_{6} - 3) q^{27} + (3 \zeta_{6} - 3) q^{29} + (3 \zeta_{6} - 6) q^{30} + 8 \zeta_{6} q^{31} + \zeta_{6} q^{32} + (6 \zeta_{6} - 3) q^{33} + (3 \zeta_{6} - 3) q^{34} - 3 q^{36} - q^{37} + ( - 7 \zeta_{6} + 7) q^{38} + (\zeta_{6} + 1) q^{39} - 3 \zeta_{6} q^{40} + 3 \zeta_{6} q^{41} + ( - \zeta_{6} + 1) q^{43} - 3 q^{44} + 9 q^{45} + 9 q^{46} + ( - 2 \zeta_{6} + 1) q^{48} + 4 \zeta_{6} q^{50} + ( - 3 \zeta_{6} + 6) q^{51} + (\zeta_{6} - 1) q^{52} + 3 q^{53} + (3 \zeta_{6} + 3) q^{54} + 9 q^{55} + (7 \zeta_{6} - 14) q^{57} + 3 \zeta_{6} q^{58} + (6 \zeta_{6} - 3) q^{60} + ( - 2 \zeta_{6} + 2) q^{61} + 8 q^{62} + q^{64} + ( - 3 \zeta_{6} + 3) q^{65} + (3 \zeta_{6} + 3) q^{66} + 4 \zeta_{6} q^{67} + 3 \zeta_{6} q^{68} + ( - 9 \zeta_{6} - 9) q^{69} + 12 q^{71} + (3 \zeta_{6} - 3) q^{72} - 11 q^{73} + (\zeta_{6} - 1) q^{74} + ( - 8 \zeta_{6} + 4) q^{75} - 7 \zeta_{6} q^{76} + ( - \zeta_{6} + 2) q^{78} + ( - 16 \zeta_{6} + 16) q^{79} - 3 q^{80} - 9 \zeta_{6} q^{81} + 3 q^{82} + (9 \zeta_{6} - 9) q^{83} - 9 \zeta_{6} q^{85} - \zeta_{6} q^{86} + ( - 6 \zeta_{6} + 3) q^{87} + (3 \zeta_{6} - 3) q^{88} - 3 q^{89} + ( - 9 \zeta_{6} + 9) q^{90} + ( - 9 \zeta_{6} + 9) q^{92} + ( - 8 \zeta_{6} - 8) q^{93} + 21 \zeta_{6} q^{95} + ( - \zeta_{6} - 1) q^{96} + (\zeta_{6} - 1) q^{97} - 9 \zeta_{6} q^{99} +O(q^{100})$$ q + (-z + 1) * q^2 + (z - 2) * q^3 - z * q^4 + 3*z * q^5 + (2*z - 1) * q^6 - q^8 + (-3*z + 3) * q^9 + 3 * q^10 + (-3*z + 3) * q^11 + (z + 1) * q^12 - z * q^13 + (-3*z - 3) * q^15 + (z - 1) * q^16 - 3 * q^17 - 3*z * q^18 + 7 * q^19 + (-3*z + 3) * q^20 - 3*z * q^22 + 9*z * q^23 + (-z + 2) * q^24 + (4*z - 4) * q^25 - q^26 + (6*z - 3) * q^27 + (3*z - 3) * q^29 + (3*z - 6) * q^30 + 8*z * q^31 + z * q^32 + (6*z - 3) * q^33 + (3*z - 3) * q^34 - 3 * q^36 - q^37 + (-7*z + 7) * q^38 + (z + 1) * q^39 - 3*z * q^40 + 3*z * q^41 + (-z + 1) * q^43 - 3 * q^44 + 9 * q^45 + 9 * q^46 + (-2*z + 1) * q^48 + 4*z * q^50 + (-3*z + 6) * q^51 + (z - 1) * q^52 + 3 * q^53 + (3*z + 3) * q^54 + 9 * q^55 + (7*z - 14) * q^57 + 3*z * q^58 + (6*z - 3) * q^60 + (-2*z + 2) * q^61 + 8 * q^62 + q^64 + (-3*z + 3) * q^65 + (3*z + 3) * q^66 + 4*z * q^67 + 3*z * q^68 + (-9*z - 9) * q^69 + 12 * q^71 + (3*z - 3) * q^72 - 11 * q^73 + (z - 1) * q^74 + (-8*z + 4) * q^75 - 7*z * q^76 + (-z + 2) * q^78 + (-16*z + 16) * q^79 - 3 * q^80 - 9*z * q^81 + 3 * q^82 + (9*z - 9) * q^83 - 9*z * q^85 - z * q^86 + (-6*z + 3) * q^87 + (3*z - 3) * q^88 - 3 * q^89 + (-9*z + 9) * q^90 + (-9*z + 9) * q^92 + (-8*z - 8) * q^93 + 21*z * q^95 + (-z - 1) * q^96 + (z - 1) * q^97 - 9*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - 3 q^{3} - q^{4} + 3 q^{5} - 2 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q + q^2 - 3 * q^3 - q^4 + 3 * q^5 - 2 * q^8 + 3 * q^9 $$2 q + q^{2} - 3 q^{3} - q^{4} + 3 q^{5} - 2 q^{8} + 3 q^{9} + 6 q^{10} + 3 q^{11} + 3 q^{12} - q^{13} - 9 q^{15} - q^{16} - 6 q^{17} - 3 q^{18} + 14 q^{19} + 3 q^{20} - 3 q^{22} + 9 q^{23} + 3 q^{24} - 4 q^{25} - 2 q^{26} - 3 q^{29} - 9 q^{30} + 8 q^{31} + q^{32} - 3 q^{34} - 6 q^{36} - 2 q^{37} + 7 q^{38} + 3 q^{39} - 3 q^{40} + 3 q^{41} + q^{43} - 6 q^{44} + 18 q^{45} + 18 q^{46} + 4 q^{50} + 9 q^{51} - q^{52} + 6 q^{53} + 9 q^{54} + 18 q^{55} - 21 q^{57} + 3 q^{58} + 2 q^{61} + 16 q^{62} + 2 q^{64} + 3 q^{65} + 9 q^{66} + 4 q^{67} + 3 q^{68} - 27 q^{69} + 24 q^{71} - 3 q^{72} - 22 q^{73} - q^{74} - 7 q^{76} + 3 q^{78} + 16 q^{79} - 6 q^{80} - 9 q^{81} + 6 q^{82} - 9 q^{83} - 9 q^{85} - q^{86} - 3 q^{88} - 6 q^{89} + 9 q^{90} + 9 q^{92} - 24 q^{93} + 21 q^{95} - 3 q^{96} - q^{97} - 9 q^{99}+O(q^{100})$$ 2 * q + q^2 - 3 * q^3 - q^4 + 3 * q^5 - 2 * q^8 + 3 * q^9 + 6 * q^10 + 3 * q^11 + 3 * q^12 - q^13 - 9 * q^15 - q^16 - 6 * q^17 - 3 * q^18 + 14 * q^19 + 3 * q^20 - 3 * q^22 + 9 * q^23 + 3 * q^24 - 4 * q^25 - 2 * q^26 - 3 * q^29 - 9 * q^30 + 8 * q^31 + q^32 - 3 * q^34 - 6 * q^36 - 2 * q^37 + 7 * q^38 + 3 * q^39 - 3 * q^40 + 3 * q^41 + q^43 - 6 * q^44 + 18 * q^45 + 18 * q^46 + 4 * q^50 + 9 * q^51 - q^52 + 6 * q^53 + 9 * q^54 + 18 * q^55 - 21 * q^57 + 3 * q^58 + 2 * q^61 + 16 * q^62 + 2 * q^64 + 3 * q^65 + 9 * q^66 + 4 * q^67 + 3 * q^68 - 27 * q^69 + 24 * q^71 - 3 * q^72 - 22 * q^73 - q^74 - 7 * q^76 + 3 * q^78 + 16 * q^79 - 6 * q^80 - 9 * q^81 + 6 * q^82 - 9 * q^83 - 9 * q^85 - q^86 - 3 * q^88 - 6 * q^89 + 9 * q^90 + 9 * q^92 - 24 * q^93 + 21 * q^95 - 3 * q^96 - q^97 - 9 * q^99

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
295.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i −1.50000 + 0.866025i −0.500000 0.866025i 1.50000 + 2.59808i 1.73205i 0 −1.00000 1.50000 2.59808i 3.00000
589.1 0.500000 + 0.866025i −1.50000 0.866025i −0.500000 + 0.866025i 1.50000 2.59808i 1.73205i 0 −1.00000 1.50000 + 2.59808i 3.00000
 $$n$$: 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.2.f.g 2
3.b odd 2 1 2646.2.f.a 2
7.b odd 2 1 882.2.f.i 2
7.c even 3 1 882.2.e.c 2
7.c even 3 1 882.2.h.i 2
7.d odd 6 1 126.2.e.a 2
7.d odd 6 1 126.2.h.b yes 2
9.c even 3 1 inner 882.2.f.g 2
9.c even 3 1 7938.2.a.b 1
9.d odd 6 1 2646.2.f.a 2
9.d odd 6 1 7938.2.a.be 1
21.c even 2 1 2646.2.f.d 2
21.g even 6 1 378.2.e.b 2
21.g even 6 1 378.2.h.a 2
21.h odd 6 1 2646.2.e.g 2
21.h odd 6 1 2646.2.h.d 2
28.f even 6 1 1008.2.q.a 2
28.f even 6 1 1008.2.t.f 2
63.g even 3 1 882.2.e.c 2
63.h even 3 1 882.2.h.i 2
63.i even 6 1 378.2.h.a 2
63.i even 6 1 1134.2.g.c 2
63.j odd 6 1 2646.2.h.d 2
63.k odd 6 1 126.2.e.a 2
63.k odd 6 1 1134.2.g.e 2
63.l odd 6 1 882.2.f.i 2
63.l odd 6 1 7938.2.a.m 1
63.n odd 6 1 2646.2.e.g 2
63.o even 6 1 2646.2.f.d 2
63.o even 6 1 7938.2.a.t 1
63.s even 6 1 378.2.e.b 2
63.s even 6 1 1134.2.g.c 2
63.t odd 6 1 126.2.h.b yes 2
63.t odd 6 1 1134.2.g.e 2
84.j odd 6 1 3024.2.q.f 2
84.j odd 6 1 3024.2.t.a 2
252.n even 6 1 1008.2.q.a 2
252.r odd 6 1 3024.2.t.a 2
252.bj even 6 1 1008.2.t.f 2
252.bn odd 6 1 3024.2.q.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.a 2 7.d odd 6 1
126.2.e.a 2 63.k odd 6 1
126.2.h.b yes 2 7.d odd 6 1
126.2.h.b yes 2 63.t odd 6 1
378.2.e.b 2 21.g even 6 1
378.2.e.b 2 63.s even 6 1
378.2.h.a 2 21.g even 6 1
378.2.h.a 2 63.i even 6 1
882.2.e.c 2 7.c even 3 1
882.2.e.c 2 63.g even 3 1
882.2.f.g 2 1.a even 1 1 trivial
882.2.f.g 2 9.c even 3 1 inner
882.2.f.i 2 7.b odd 2 1
882.2.f.i 2 63.l odd 6 1
882.2.h.i 2 7.c even 3 1
882.2.h.i 2 63.h even 3 1
1008.2.q.a 2 28.f even 6 1
1008.2.q.a 2 252.n even 6 1
1008.2.t.f 2 28.f even 6 1
1008.2.t.f 2 252.bj even 6 1
1134.2.g.c 2 63.i even 6 1
1134.2.g.c 2 63.s even 6 1
1134.2.g.e 2 63.k odd 6 1
1134.2.g.e 2 63.t odd 6 1
2646.2.e.g 2 21.h odd 6 1
2646.2.e.g 2 63.n odd 6 1
2646.2.f.a 2 3.b odd 2 1
2646.2.f.a 2 9.d odd 6 1
2646.2.f.d 2 21.c even 2 1
2646.2.f.d 2 63.o even 6 1
2646.2.h.d 2 21.h odd 6 1
2646.2.h.d 2 63.j odd 6 1
3024.2.q.f 2 84.j odd 6 1
3024.2.q.f 2 252.bn odd 6 1
3024.2.t.a 2 84.j odd 6 1
3024.2.t.a 2 252.r odd 6 1
7938.2.a.b 1 9.c even 3 1
7938.2.a.m 1 63.l odd 6 1
7938.2.a.t 1 63.o even 6 1
7938.2.a.be 1 9.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(882, [\chi])$$:

 $$T_{5}^{2} - 3T_{5} + 9$$ T5^2 - 3*T5 + 9 $$T_{11}^{2} - 3T_{11} + 9$$ T11^2 - 3*T11 + 9 $$T_{13}^{2} + T_{13} + 1$$ T13^2 + T13 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} + 3T + 3$$
$5$ $$T^{2} - 3T + 9$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$T^{2} + T + 1$$
$17$ $$(T + 3)^{2}$$
$19$ $$(T - 7)^{2}$$
$23$ $$T^{2} - 9T + 81$$
$29$ $$T^{2} + 3T + 9$$
$31$ $$T^{2} - 8T + 64$$
$37$ $$(T + 1)^{2}$$
$41$ $$T^{2} - 3T + 9$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2}$$
$53$ $$(T - 3)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 2T + 4$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$(T - 12)^{2}$$
$73$ $$(T + 11)^{2}$$
$79$ $$T^{2} - 16T + 256$$
$83$ $$T^{2} + 9T + 81$$
$89$ $$(T + 3)^{2}$$
$97$ $$T^{2} + T + 1$$