# Properties

 Label 882.2.e.d Level $882$ Weight $2$ Character orbit 882.e 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(373,882)]

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([2, 2]))

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

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

chi := DirichletCharacter("882.373");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$882 = 2 \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 882.e (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: $$7.04280545828$$ 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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + (\zeta_{6} + 1) q^{3} + q^{4} + (3 \zeta_{6} - 3) q^{5} + ( - \zeta_{6} - 1) q^{6} - q^{8} + 3 \zeta_{6} q^{9}+O(q^{10})$$ q - q^2 + (z + 1) * q^3 + q^4 + (3*z - 3) * q^5 + (-z - 1) * q^6 - q^8 + 3*z * q^9 $$q - q^{2} + (\zeta_{6} + 1) q^{3} + q^{4} + (3 \zeta_{6} - 3) q^{5} + ( - \zeta_{6} - 1) q^{6} - q^{8} + 3 \zeta_{6} q^{9} + ( - 3 \zeta_{6} + 3) q^{10} + 6 \zeta_{6} q^{11} + (\zeta_{6} + 1) q^{12} + 2 \zeta_{6} q^{13} + (3 \zeta_{6} - 6) q^{15} + q^{16} + ( - 6 \zeta_{6} + 6) q^{17} - 3 \zeta_{6} q^{18} - 7 \zeta_{6} q^{19} + (3 \zeta_{6} - 3) q^{20} - 6 \zeta_{6} q^{22} + (3 \zeta_{6} - 3) q^{23} + ( - \zeta_{6} - 1) q^{24} - 4 \zeta_{6} q^{25} - 2 \zeta_{6} q^{26} + (6 \zeta_{6} - 3) q^{27} + (6 \zeta_{6} - 6) q^{29} + ( - 3 \zeta_{6} + 6) q^{30} - 2 q^{31} - q^{32} + (12 \zeta_{6} - 6) q^{33} + (6 \zeta_{6} - 6) q^{34} + 3 \zeta_{6} q^{36} - 2 \zeta_{6} q^{37} + 7 \zeta_{6} q^{38} + (4 \zeta_{6} - 2) q^{39} + ( - 3 \zeta_{6} + 3) q^{40} + (2 \zeta_{6} - 2) q^{43} + 6 \zeta_{6} q^{44} - 9 q^{45} + ( - 3 \zeta_{6} + 3) q^{46} + (\zeta_{6} + 1) q^{48} + 4 \zeta_{6} q^{50} + ( - 6 \zeta_{6} + 12) q^{51} + 2 \zeta_{6} q^{52} + (6 \zeta_{6} - 6) q^{53} + ( - 6 \zeta_{6} + 3) q^{54} - 18 q^{55} + ( - 14 \zeta_{6} + 7) q^{57} + ( - 6 \zeta_{6} + 6) q^{58} + (3 \zeta_{6} - 6) q^{60} - 5 q^{61} + 2 q^{62} + q^{64} - 6 q^{65} + ( - 12 \zeta_{6} + 6) q^{66} + 8 q^{67} + ( - 6 \zeta_{6} + 6) q^{68} + (3 \zeta_{6} - 6) q^{69} + 3 q^{71} - 3 \zeta_{6} q^{72} + ( - 2 \zeta_{6} + 2) q^{73} + 2 \zeta_{6} q^{74} + ( - 8 \zeta_{6} + 4) q^{75} - 7 \zeta_{6} q^{76} + ( - 4 \zeta_{6} + 2) q^{78} + 5 q^{79} + (3 \zeta_{6} - 3) q^{80} + (9 \zeta_{6} - 9) q^{81} + ( - 12 \zeta_{6} + 12) q^{83} + 18 \zeta_{6} q^{85} + ( - 2 \zeta_{6} + 2) q^{86} + (6 \zeta_{6} - 12) q^{87} - 6 \zeta_{6} q^{88} + 9 q^{90} + (3 \zeta_{6} - 3) q^{92} + ( - 2 \zeta_{6} - 2) q^{93} + 21 q^{95} + ( - \zeta_{6} - 1) q^{96} + ( - 2 \zeta_{6} + 2) q^{97} + (18 \zeta_{6} - 18) q^{99} +O(q^{100})$$ q - q^2 + (z + 1) * q^3 + q^4 + (3*z - 3) * q^5 + (-z - 1) * q^6 - q^8 + 3*z * q^9 + (-3*z + 3) * q^10 + 6*z * q^11 + (z + 1) * q^12 + 2*z * q^13 + (3*z - 6) * q^15 + q^16 + (-6*z + 6) * q^17 - 3*z * q^18 - 7*z * q^19 + (3*z - 3) * q^20 - 6*z * q^22 + (3*z - 3) * q^23 + (-z - 1) * q^24 - 4*z * q^25 - 2*z * q^26 + (6*z - 3) * q^27 + (6*z - 6) * q^29 + (-3*z + 6) * q^30 - 2 * q^31 - q^32 + (12*z - 6) * q^33 + (6*z - 6) * q^34 + 3*z * q^36 - 2*z * q^37 + 7*z * q^38 + (4*z - 2) * q^39 + (-3*z + 3) * q^40 + (2*z - 2) * q^43 + 6*z * q^44 - 9 * q^45 + (-3*z + 3) * q^46 + (z + 1) * q^48 + 4*z * q^50 + (-6*z + 12) * q^51 + 2*z * q^52 + (6*z - 6) * q^53 + (-6*z + 3) * q^54 - 18 * q^55 + (-14*z + 7) * q^57 + (-6*z + 6) * q^58 + (3*z - 6) * q^60 - 5 * q^61 + 2 * q^62 + q^64 - 6 * q^65 + (-12*z + 6) * q^66 + 8 * q^67 + (-6*z + 6) * q^68 + (3*z - 6) * q^69 + 3 * q^71 - 3*z * q^72 + (-2*z + 2) * q^73 + 2*z * q^74 + (-8*z + 4) * q^75 - 7*z * q^76 + (-4*z + 2) * q^78 + 5 * q^79 + (3*z - 3) * q^80 + (9*z - 9) * q^81 + (-12*z + 12) * q^83 + 18*z * q^85 + (-2*z + 2) * q^86 + (6*z - 12) * q^87 - 6*z * q^88 + 9 * q^90 + (3*z - 3) * q^92 + (-2*z - 2) * q^93 + 21 * q^95 + (-z - 1) * q^96 + (-2*z + 2) * q^97 + (18*z - 18) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} - 3 q^{5} - 3 q^{6} - 2 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 3 * q^3 + 2 * q^4 - 3 * q^5 - 3 * q^6 - 2 * q^8 + 3 * q^9 $$2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} - 3 q^{5} - 3 q^{6} - 2 q^{8} + 3 q^{9} + 3 q^{10} + 6 q^{11} + 3 q^{12} + 2 q^{13} - 9 q^{15} + 2 q^{16} + 6 q^{17} - 3 q^{18} - 7 q^{19} - 3 q^{20} - 6 q^{22} - 3 q^{23} - 3 q^{24} - 4 q^{25} - 2 q^{26} - 6 q^{29} + 9 q^{30} - 4 q^{31} - 2 q^{32} - 6 q^{34} + 3 q^{36} - 2 q^{37} + 7 q^{38} + 3 q^{40} - 2 q^{43} + 6 q^{44} - 18 q^{45} + 3 q^{46} + 3 q^{48} + 4 q^{50} + 18 q^{51} + 2 q^{52} - 6 q^{53} - 36 q^{55} + 6 q^{58} - 9 q^{60} - 10 q^{61} + 4 q^{62} + 2 q^{64} - 12 q^{65} + 16 q^{67} + 6 q^{68} - 9 q^{69} + 6 q^{71} - 3 q^{72} + 2 q^{73} + 2 q^{74} - 7 q^{76} + 10 q^{79} - 3 q^{80} - 9 q^{81} + 12 q^{83} + 18 q^{85} + 2 q^{86} - 18 q^{87} - 6 q^{88} + 18 q^{90} - 3 q^{92} - 6 q^{93} + 42 q^{95} - 3 q^{96} + 2 q^{97} - 18 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 3 * q^3 + 2 * q^4 - 3 * q^5 - 3 * q^6 - 2 * q^8 + 3 * q^9 + 3 * q^10 + 6 * q^11 + 3 * q^12 + 2 * q^13 - 9 * q^15 + 2 * q^16 + 6 * q^17 - 3 * q^18 - 7 * q^19 - 3 * q^20 - 6 * q^22 - 3 * q^23 - 3 * q^24 - 4 * q^25 - 2 * q^26 - 6 * q^29 + 9 * q^30 - 4 * q^31 - 2 * q^32 - 6 * q^34 + 3 * q^36 - 2 * q^37 + 7 * q^38 + 3 * q^40 - 2 * q^43 + 6 * q^44 - 18 * q^45 + 3 * q^46 + 3 * q^48 + 4 * q^50 + 18 * q^51 + 2 * q^52 - 6 * q^53 - 36 * q^55 + 6 * q^58 - 9 * q^60 - 10 * q^61 + 4 * q^62 + 2 * q^64 - 12 * q^65 + 16 * q^67 + 6 * q^68 - 9 * q^69 + 6 * q^71 - 3 * q^72 + 2 * q^73 + 2 * q^74 - 7 * q^76 + 10 * q^79 - 3 * q^80 - 9 * q^81 + 12 * q^83 + 18 * q^85 + 2 * q^86 - 18 * q^87 - 6 * q^88 + 18 * q^90 - 3 * q^92 - 6 * q^93 + 42 * q^95 - 3 * q^96 + 2 * q^97 - 18 * q^99

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
373.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.00000 1.50000 0.866025i 1.00000 −1.50000 2.59808i −1.50000 + 0.866025i 0 −1.00000 1.50000 2.59808i 1.50000 + 2.59808i
655.1 −1.00000 1.50000 + 0.866025i 1.00000 −1.50000 + 2.59808i −1.50000 0.866025i 0 −1.00000 1.50000 + 2.59808i 1.50000 2.59808i
 $$n$$: 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
63.h 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.e.d 2
3.b odd 2 1 2646.2.e.j 2
7.b odd 2 1 882.2.e.b 2
7.c even 3 1 882.2.f.h 2
7.c even 3 1 882.2.h.f 2
7.d odd 6 1 126.2.f.a 2
7.d odd 6 1 882.2.h.j 2
9.c even 3 1 882.2.h.f 2
9.d odd 6 1 2646.2.h.a 2
21.c even 2 1 2646.2.e.f 2
21.g even 6 1 378.2.f.a 2
21.g even 6 1 2646.2.h.e 2
21.h odd 6 1 2646.2.f.c 2
21.h odd 6 1 2646.2.h.a 2
28.f even 6 1 1008.2.r.d 2
63.g even 3 1 882.2.f.h 2
63.h even 3 1 inner 882.2.e.d 2
63.h even 3 1 7938.2.a.l 1
63.i even 6 1 1134.2.a.h 1
63.i even 6 1 2646.2.e.f 2
63.j odd 6 1 2646.2.e.j 2
63.j odd 6 1 7938.2.a.u 1
63.k odd 6 1 126.2.f.a 2
63.l odd 6 1 882.2.h.j 2
63.n odd 6 1 2646.2.f.c 2
63.o even 6 1 2646.2.h.e 2
63.s even 6 1 378.2.f.a 2
63.t odd 6 1 882.2.e.b 2
63.t odd 6 1 1134.2.a.a 1
84.j odd 6 1 3024.2.r.a 2
252.n even 6 1 1008.2.r.d 2
252.r odd 6 1 9072.2.a.w 1
252.bj even 6 1 9072.2.a.c 1
252.bn odd 6 1 3024.2.r.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.a 2 7.d odd 6 1
126.2.f.a 2 63.k odd 6 1
378.2.f.a 2 21.g even 6 1
378.2.f.a 2 63.s even 6 1
882.2.e.b 2 7.b odd 2 1
882.2.e.b 2 63.t odd 6 1
882.2.e.d 2 1.a even 1 1 trivial
882.2.e.d 2 63.h even 3 1 inner
882.2.f.h 2 7.c even 3 1
882.2.f.h 2 63.g even 3 1
882.2.h.f 2 7.c even 3 1
882.2.h.f 2 9.c even 3 1
882.2.h.j 2 7.d odd 6 1
882.2.h.j 2 63.l odd 6 1
1008.2.r.d 2 28.f even 6 1
1008.2.r.d 2 252.n even 6 1
1134.2.a.a 1 63.t odd 6 1
1134.2.a.h 1 63.i even 6 1
2646.2.e.f 2 21.c even 2 1
2646.2.e.f 2 63.i even 6 1
2646.2.e.j 2 3.b odd 2 1
2646.2.e.j 2 63.j odd 6 1
2646.2.f.c 2 21.h odd 6 1
2646.2.f.c 2 63.n odd 6 1
2646.2.h.a 2 9.d odd 6 1
2646.2.h.a 2 21.h odd 6 1
2646.2.h.e 2 21.g even 6 1
2646.2.h.e 2 63.o even 6 1
3024.2.r.a 2 84.j odd 6 1
3024.2.r.a 2 252.bn odd 6 1
7938.2.a.l 1 63.h even 3 1
7938.2.a.u 1 63.j odd 6 1
9072.2.a.c 1 252.bj even 6 1
9072.2.a.w 1 252.r 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 $$T_{11}^{2} - 6T_{11} + 36$$ T11^2 - 6*T11 + 36 $$T_{13}^{2} - 2T_{13} + 4$$ T13^2 - 2*T13 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} - 3T + 3$$
$5$ $$T^{2} + 3T + 9$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 6T + 36$$
$13$ $$T^{2} - 2T + 4$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} + 7T + 49$$
$23$ $$T^{2} + 3T + 9$$
$29$ $$T^{2} + 6T + 36$$
$31$ $$(T + 2)^{2}$$
$37$ $$T^{2} + 2T + 4$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 2T + 4$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 6T + 36$$
$59$ $$T^{2}$$
$61$ $$(T + 5)^{2}$$
$67$ $$(T - 8)^{2}$$
$71$ $$(T - 3)^{2}$$
$73$ $$T^{2} - 2T + 4$$
$79$ $$(T - 5)^{2}$$
$83$ $$T^{2} - 12T + 144$$
$89$ $$T^{2}$$
$97$ $$T^{2} - 2T + 4$$