# Properties

 Label 2646.2.h.g Level $2646$ Weight $2$ Character orbit 2646.h Analytic conductor $21.128$ 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] = [2646,2,Mod(361,2646)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2646, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 4]))

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

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

chi := DirichletCharacter("2646.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2646 = 2 \cdot 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2646.h (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: $$21.1284163748$$ 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 882) 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} - q^{5} - q^{8} +O(q^{10})$$ q + (-z + 1) * q^2 - z * q^4 - q^5 - q^8 $$q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} - q^{5} - q^{8} + (\zeta_{6} - 1) q^{10} + 2 q^{11} + (2 \zeta_{6} - 2) q^{13} + (\zeta_{6} - 1) q^{16} + 7 \zeta_{6} q^{19} + \zeta_{6} q^{20} + ( - 2 \zeta_{6} + 2) q^{22} - 3 q^{23} - 4 q^{25} + 2 \zeta_{6} q^{26} - 8 \zeta_{6} q^{29} - 4 \zeta_{6} q^{31} + \zeta_{6} q^{32} + 6 \zeta_{6} q^{37} + 7 q^{38} + q^{40} + (12 \zeta_{6} - 12) q^{41} + 8 \zeta_{6} q^{43} - 2 \zeta_{6} q^{44} + (3 \zeta_{6} - 3) q^{46} + (8 \zeta_{6} - 8) q^{47} + (4 \zeta_{6} - 4) q^{50} + 2 q^{52} + ( - 4 \zeta_{6} + 4) q^{53} - 2 q^{55} - 8 q^{58} + 4 \zeta_{6} q^{59} + (13 \zeta_{6} - 13) q^{61} - 4 q^{62} + q^{64} + ( - 2 \zeta_{6} + 2) q^{65} + 2 \zeta_{6} q^{67} + 5 q^{71} + ( - 14 \zeta_{6} + 14) q^{73} + 6 q^{74} + ( - 7 \zeta_{6} + 7) q^{76} + ( - 11 \zeta_{6} + 11) q^{79} + ( - \zeta_{6} + 1) q^{80} + 12 \zeta_{6} q^{82} + 12 \zeta_{6} q^{83} + 8 q^{86} - 2 q^{88} + 14 \zeta_{6} q^{89} + 3 \zeta_{6} q^{92} + 8 \zeta_{6} q^{94} - 7 \zeta_{6} q^{95} + 2 \zeta_{6} q^{97} +O(q^{100})$$ q + (-z + 1) * q^2 - z * q^4 - q^5 - q^8 + (z - 1) * q^10 + 2 * q^11 + (2*z - 2) * q^13 + (z - 1) * q^16 + 7*z * q^19 + z * q^20 + (-2*z + 2) * q^22 - 3 * q^23 - 4 * q^25 + 2*z * q^26 - 8*z * q^29 - 4*z * q^31 + z * q^32 + 6*z * q^37 + 7 * q^38 + q^40 + (12*z - 12) * q^41 + 8*z * q^43 - 2*z * q^44 + (3*z - 3) * q^46 + (8*z - 8) * q^47 + (4*z - 4) * q^50 + 2 * q^52 + (-4*z + 4) * q^53 - 2 * q^55 - 8 * q^58 + 4*z * q^59 + (13*z - 13) * q^61 - 4 * q^62 + q^64 + (-2*z + 2) * q^65 + 2*z * q^67 + 5 * q^71 + (-14*z + 14) * q^73 + 6 * q^74 + (-7*z + 7) * q^76 + (-11*z + 11) * q^79 + (-z + 1) * q^80 + 12*z * q^82 + 12*z * q^83 + 8 * q^86 - 2 * q^88 + 14*z * q^89 + 3*z * q^92 + 8*z * q^94 - 7*z * q^95 + 2*z * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} - 2 q^{5} - 2 q^{8}+O(q^{10})$$ 2 * q + q^2 - q^4 - 2 * q^5 - 2 * q^8 $$2 q + q^{2} - q^{4} - 2 q^{5} - 2 q^{8} - q^{10} + 4 q^{11} - 2 q^{13} - q^{16} + 7 q^{19} + q^{20} + 2 q^{22} - 6 q^{23} - 8 q^{25} + 2 q^{26} - 8 q^{29} - 4 q^{31} + q^{32} + 6 q^{37} + 14 q^{38} + 2 q^{40} - 12 q^{41} + 8 q^{43} - 2 q^{44} - 3 q^{46} - 8 q^{47} - 4 q^{50} + 4 q^{52} + 4 q^{53} - 4 q^{55} - 16 q^{58} + 4 q^{59} - 13 q^{61} - 8 q^{62} + 2 q^{64} + 2 q^{65} + 2 q^{67} + 10 q^{71} + 14 q^{73} + 12 q^{74} + 7 q^{76} + 11 q^{79} + q^{80} + 12 q^{82} + 12 q^{83} + 16 q^{86} - 4 q^{88} + 14 q^{89} + 3 q^{92} + 8 q^{94} - 7 q^{95} + 2 q^{97}+O(q^{100})$$ 2 * q + q^2 - q^4 - 2 * q^5 - 2 * q^8 - q^10 + 4 * q^11 - 2 * q^13 - q^16 + 7 * q^19 + q^20 + 2 * q^22 - 6 * q^23 - 8 * q^25 + 2 * q^26 - 8 * q^29 - 4 * q^31 + q^32 + 6 * q^37 + 14 * q^38 + 2 * q^40 - 12 * q^41 + 8 * q^43 - 2 * q^44 - 3 * q^46 - 8 * q^47 - 4 * q^50 + 4 * q^52 + 4 * q^53 - 4 * q^55 - 16 * q^58 + 4 * q^59 - 13 * q^61 - 8 * q^62 + 2 * q^64 + 2 * q^65 + 2 * q^67 + 10 * q^71 + 14 * q^73 + 12 * q^74 + 7 * q^76 + 11 * q^79 + q^80 + 12 * q^82 + 12 * q^83 + 16 * q^86 - 4 * q^88 + 14 * q^89 + 3 * q^92 + 8 * q^94 - 7 * q^95 + 2 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1081$$ $$\chi(n)$$ $$-1 + \zeta_{6}$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i 0 −0.500000 0.866025i −1.00000 0 0 −1.00000 0 −0.500000 + 0.866025i
667.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.00000 0 0 −1.00000 0 −0.500000 0.866025i
 $$n$$: 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.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2646.2.h.g 2
3.b odd 2 1 882.2.h.d 2
7.b odd 2 1 2646.2.h.j 2
7.c even 3 1 2646.2.e.d 2
7.c even 3 1 2646.2.f.h 2
7.d odd 6 1 2646.2.e.a 2
7.d odd 6 1 2646.2.f.f 2
9.c even 3 1 2646.2.e.d 2
9.d odd 6 1 882.2.e.f 2
21.c even 2 1 882.2.h.a 2
21.g even 6 1 882.2.e.j 2
21.g even 6 1 882.2.f.c yes 2
21.h odd 6 1 882.2.e.f 2
21.h odd 6 1 882.2.f.b 2
63.g even 3 1 inner 2646.2.h.g 2
63.g even 3 1 7938.2.a.f 1
63.h even 3 1 2646.2.f.h 2
63.i even 6 1 882.2.f.c yes 2
63.j odd 6 1 882.2.f.b 2
63.k odd 6 1 2646.2.h.j 2
63.k odd 6 1 7938.2.a.k 1
63.l odd 6 1 2646.2.e.a 2
63.n odd 6 1 882.2.h.d 2
63.n odd 6 1 7938.2.a.ba 1
63.o even 6 1 882.2.e.j 2
63.s even 6 1 882.2.h.a 2
63.s even 6 1 7938.2.a.v 1
63.t odd 6 1 2646.2.f.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.2.e.f 2 9.d odd 6 1
882.2.e.f 2 21.h odd 6 1
882.2.e.j 2 21.g even 6 1
882.2.e.j 2 63.o even 6 1
882.2.f.b 2 21.h odd 6 1
882.2.f.b 2 63.j odd 6 1
882.2.f.c yes 2 21.g even 6 1
882.2.f.c yes 2 63.i even 6 1
882.2.h.a 2 21.c even 2 1
882.2.h.a 2 63.s even 6 1
882.2.h.d 2 3.b odd 2 1
882.2.h.d 2 63.n odd 6 1
2646.2.e.a 2 7.d odd 6 1
2646.2.e.a 2 63.l odd 6 1
2646.2.e.d 2 7.c even 3 1
2646.2.e.d 2 9.c even 3 1
2646.2.f.f 2 7.d odd 6 1
2646.2.f.f 2 63.t odd 6 1
2646.2.f.h 2 7.c even 3 1
2646.2.f.h 2 63.h even 3 1
2646.2.h.g 2 1.a even 1 1 trivial
2646.2.h.g 2 63.g even 3 1 inner
2646.2.h.j 2 7.b odd 2 1
2646.2.h.j 2 63.k odd 6 1
7938.2.a.f 1 63.g even 3 1
7938.2.a.k 1 63.k odd 6 1
7938.2.a.v 1 63.s even 6 1
7938.2.a.ba 1 63.n odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2646, [\chi])$$:

 $$T_{5} + 1$$ T5 + 1 $$T_{11} - 2$$ T11 - 2 $$T_{13}^{2} + 2T_{13} + 4$$ T13^2 + 2*T13 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2}$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} + 2T + 4$$
$17$ $$T^{2}$$
$19$ $$T^{2} - 7T + 49$$
$23$ $$(T + 3)^{2}$$
$29$ $$T^{2} + 8T + 64$$
$31$ $$T^{2} + 4T + 16$$
$37$ $$T^{2} - 6T + 36$$
$41$ $$T^{2} + 12T + 144$$
$43$ $$T^{2} - 8T + 64$$
$47$ $$T^{2} + 8T + 64$$
$53$ $$T^{2} - 4T + 16$$
$59$ $$T^{2} - 4T + 16$$
$61$ $$T^{2} + 13T + 169$$
$67$ $$T^{2} - 2T + 4$$
$71$ $$(T - 5)^{2}$$
$73$ $$T^{2} - 14T + 196$$
$79$ $$T^{2} - 11T + 121$$
$83$ $$T^{2} - 12T + 144$$
$89$ $$T^{2} - 14T + 196$$
$97$ $$T^{2} - 2T + 4$$