# Properties

 Label 2646.2.h.h 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 # Warning: the index may be different

gp: f = lf \\ 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 18) 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} - q^{8} +O(q^{10})$$ q + (-z + 1) * q^2 - z * q^4 - q^8 $$q + ( - \zeta_{6} + 1) q^{2} - \zeta_{6} q^{4} - q^{8} + 3 q^{11} + (2 \zeta_{6} - 2) q^{13} + (\zeta_{6} - 1) q^{16} + (3 \zeta_{6} - 3) q^{17} + \zeta_{6} q^{19} + ( - 3 \zeta_{6} + 3) q^{22} + 6 q^{23} - 5 q^{25} + 2 \zeta_{6} q^{26} + 6 \zeta_{6} q^{29} + 4 \zeta_{6} q^{31} + \zeta_{6} q^{32} + 3 \zeta_{6} q^{34} + 4 \zeta_{6} q^{37} + q^{38} + ( - 9 \zeta_{6} + 9) q^{41} + \zeta_{6} q^{43} - 3 \zeta_{6} q^{44} + ( - 6 \zeta_{6} + 6) q^{46} + (6 \zeta_{6} - 6) q^{47} + (5 \zeta_{6} - 5) q^{50} + 2 q^{52} + ( - 12 \zeta_{6} + 12) q^{53} + 6 q^{58} + 3 \zeta_{6} q^{59} + (8 \zeta_{6} - 8) q^{61} + 4 q^{62} + q^{64} - 5 \zeta_{6} q^{67} + 3 q^{68} + 12 q^{71} + (11 \zeta_{6} - 11) q^{73} + 4 q^{74} + ( - \zeta_{6} + 1) q^{76} + ( - 4 \zeta_{6} + 4) q^{79} - 9 \zeta_{6} q^{82} + 12 \zeta_{6} q^{83} + q^{86} - 3 q^{88} + 6 \zeta_{6} q^{89} - 6 \zeta_{6} q^{92} + 6 \zeta_{6} q^{94} - 5 \zeta_{6} q^{97} +O(q^{100})$$ q + (-z + 1) * q^2 - z * q^4 - q^8 + 3 * q^11 + (2*z - 2) * q^13 + (z - 1) * q^16 + (3*z - 3) * q^17 + z * q^19 + (-3*z + 3) * q^22 + 6 * q^23 - 5 * q^25 + 2*z * q^26 + 6*z * q^29 + 4*z * q^31 + z * q^32 + 3*z * q^34 + 4*z * q^37 + q^38 + (-9*z + 9) * q^41 + z * q^43 - 3*z * q^44 + (-6*z + 6) * q^46 + (6*z - 6) * q^47 + (5*z - 5) * q^50 + 2 * q^52 + (-12*z + 12) * q^53 + 6 * q^58 + 3*z * q^59 + (8*z - 8) * q^61 + 4 * q^62 + q^64 - 5*z * q^67 + 3 * q^68 + 12 * q^71 + (11*z - 11) * q^73 + 4 * q^74 + (-z + 1) * q^76 + (-4*z + 4) * q^79 - 9*z * q^82 + 12*z * q^83 + q^86 - 3 * q^88 + 6*z * q^89 - 6*z * q^92 + 6*z * q^94 - 5*z * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} - 2 q^{8}+O(q^{10})$$ 2 * q + q^2 - q^4 - 2 * q^8 $$2 q + q^{2} - q^{4} - 2 q^{8} + 6 q^{11} - 2 q^{13} - q^{16} - 3 q^{17} + q^{19} + 3 q^{22} + 12 q^{23} - 10 q^{25} + 2 q^{26} + 6 q^{29} + 4 q^{31} + q^{32} + 3 q^{34} + 4 q^{37} + 2 q^{38} + 9 q^{41} + q^{43} - 3 q^{44} + 6 q^{46} - 6 q^{47} - 5 q^{50} + 4 q^{52} + 12 q^{53} + 12 q^{58} + 3 q^{59} - 8 q^{61} + 8 q^{62} + 2 q^{64} - 5 q^{67} + 6 q^{68} + 24 q^{71} - 11 q^{73} + 8 q^{74} + q^{76} + 4 q^{79} - 9 q^{82} + 12 q^{83} + 2 q^{86} - 6 q^{88} + 6 q^{89} - 6 q^{92} + 6 q^{94} - 5 q^{97}+O(q^{100})$$ 2 * q + q^2 - q^4 - 2 * q^8 + 6 * q^11 - 2 * q^13 - q^16 - 3 * q^17 + q^19 + 3 * q^22 + 12 * q^23 - 10 * q^25 + 2 * q^26 + 6 * q^29 + 4 * q^31 + q^32 + 3 * q^34 + 4 * q^37 + 2 * q^38 + 9 * q^41 + q^43 - 3 * q^44 + 6 * q^46 - 6 * q^47 - 5 * q^50 + 4 * q^52 + 12 * q^53 + 12 * q^58 + 3 * q^59 - 8 * q^61 + 8 * q^62 + 2 * q^64 - 5 * q^67 + 6 * q^68 + 24 * q^71 - 11 * q^73 + 8 * q^74 + q^76 + 4 * q^79 - 9 * q^82 + 12 * q^83 + 2 * q^86 - 6 * q^88 + 6 * q^89 - 6 * q^92 + 6 * q^94 - 5 * 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 0 0 0 −1.00000 0 0
667.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 0 −1.00000 0 0
 $$n$$: 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.h 2
3.b odd 2 1 882.2.h.c 2
7.b odd 2 1 2646.2.h.i 2
7.c even 3 1 54.2.c.a 2
7.c even 3 1 2646.2.e.b 2
7.d odd 6 1 2646.2.e.c 2
7.d odd 6 1 2646.2.f.g 2
9.c even 3 1 2646.2.e.b 2
9.d odd 6 1 882.2.e.i 2
21.c even 2 1 882.2.h.b 2
21.g even 6 1 882.2.e.g 2
21.g even 6 1 882.2.f.d 2
21.h odd 6 1 18.2.c.a 2
21.h odd 6 1 882.2.e.i 2
28.g odd 6 1 432.2.i.b 2
35.j even 6 1 1350.2.e.c 2
35.l odd 12 2 1350.2.j.a 4
56.k odd 6 1 1728.2.i.f 2
56.p even 6 1 1728.2.i.e 2
63.g even 3 1 162.2.a.b 1
63.g even 3 1 inner 2646.2.h.h 2
63.h even 3 1 54.2.c.a 2
63.i even 6 1 882.2.f.d 2
63.j odd 6 1 18.2.c.a 2
63.k odd 6 1 2646.2.h.i 2
63.k odd 6 1 7938.2.a.i 1
63.l odd 6 1 2646.2.e.c 2
63.n odd 6 1 162.2.a.c 1
63.n odd 6 1 882.2.h.c 2
63.o even 6 1 882.2.e.g 2
63.s even 6 1 882.2.h.b 2
63.s even 6 1 7938.2.a.x 1
63.t odd 6 1 2646.2.f.g 2
84.n even 6 1 144.2.i.c 2
105.o odd 6 1 450.2.e.i 2
105.x even 12 2 450.2.j.e 4
168.s odd 6 1 576.2.i.g 2
168.v even 6 1 576.2.i.a 2
252.o even 6 1 1296.2.a.g 1
252.u odd 6 1 432.2.i.b 2
252.bb even 6 1 144.2.i.c 2
252.bl odd 6 1 1296.2.a.f 1
315.r even 6 1 1350.2.e.c 2
315.v odd 6 1 4050.2.a.c 1
315.bo even 6 1 4050.2.a.v 1
315.br odd 6 1 450.2.e.i 2
315.bt odd 12 2 1350.2.j.a 4
315.bv even 12 2 450.2.j.e 4
315.bx even 12 2 4050.2.c.c 2
315.ch odd 12 2 4050.2.c.r 2
504.w even 6 1 5184.2.a.q 1
504.ba odd 6 1 5184.2.a.p 1
504.bi odd 6 1 576.2.i.g 2
504.bt even 6 1 576.2.i.a 2
504.ce odd 6 1 1728.2.i.f 2
504.cq even 6 1 1728.2.i.e 2
504.cy even 6 1 5184.2.a.o 1
504.db odd 6 1 5184.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.2.c.a 2 21.h odd 6 1
18.2.c.a 2 63.j odd 6 1
54.2.c.a 2 7.c even 3 1
54.2.c.a 2 63.h even 3 1
144.2.i.c 2 84.n even 6 1
144.2.i.c 2 252.bb even 6 1
162.2.a.b 1 63.g even 3 1
162.2.a.c 1 63.n odd 6 1
432.2.i.b 2 28.g odd 6 1
432.2.i.b 2 252.u odd 6 1
450.2.e.i 2 105.o odd 6 1
450.2.e.i 2 315.br odd 6 1
450.2.j.e 4 105.x even 12 2
450.2.j.e 4 315.bv even 12 2
576.2.i.a 2 168.v even 6 1
576.2.i.a 2 504.bt even 6 1
576.2.i.g 2 168.s odd 6 1
576.2.i.g 2 504.bi odd 6 1
882.2.e.g 2 21.g even 6 1
882.2.e.g 2 63.o even 6 1
882.2.e.i 2 9.d odd 6 1
882.2.e.i 2 21.h odd 6 1
882.2.f.d 2 21.g even 6 1
882.2.f.d 2 63.i even 6 1
882.2.h.b 2 21.c even 2 1
882.2.h.b 2 63.s even 6 1
882.2.h.c 2 3.b odd 2 1
882.2.h.c 2 63.n odd 6 1
1296.2.a.f 1 252.bl odd 6 1
1296.2.a.g 1 252.o even 6 1
1350.2.e.c 2 35.j even 6 1
1350.2.e.c 2 315.r even 6 1
1350.2.j.a 4 35.l odd 12 2
1350.2.j.a 4 315.bt odd 12 2
1728.2.i.e 2 56.p even 6 1
1728.2.i.e 2 504.cq even 6 1
1728.2.i.f 2 56.k odd 6 1
1728.2.i.f 2 504.ce odd 6 1
2646.2.e.b 2 7.c even 3 1
2646.2.e.b 2 9.c even 3 1
2646.2.e.c 2 7.d odd 6 1
2646.2.e.c 2 63.l odd 6 1
2646.2.f.g 2 7.d odd 6 1
2646.2.f.g 2 63.t odd 6 1
2646.2.h.h 2 1.a even 1 1 trivial
2646.2.h.h 2 63.g even 3 1 inner
2646.2.h.i 2 7.b odd 2 1
2646.2.h.i 2 63.k odd 6 1
4050.2.a.c 1 315.v odd 6 1
4050.2.a.v 1 315.bo even 6 1
4050.2.c.c 2 315.bx even 12 2
4050.2.c.r 2 315.ch odd 12 2
5184.2.a.o 1 504.cy even 6 1
5184.2.a.p 1 504.ba odd 6 1
5184.2.a.q 1 504.w even 6 1
5184.2.a.r 1 504.db odd 6 1
7938.2.a.i 1 63.k odd 6 1
7938.2.a.x 1 63.s even 6 1

## Hecke kernels

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

 $$T_{5}$$ T5 $$T_{11} - 3$$ T11 - 3 $$T_{13}^{2} + 2T_{13} + 4$$ T13^2 + 2*T13 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2} + 2T + 4$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} - T + 1$$
$23$ $$(T - 6)^{2}$$
$29$ $$T^{2} - 6T + 36$$
$31$ $$T^{2} - 4T + 16$$
$37$ $$T^{2} - 4T + 16$$
$41$ $$T^{2} - 9T + 81$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2} + 6T + 36$$
$53$ $$T^{2} - 12T + 144$$
$59$ $$T^{2} - 3T + 9$$
$61$ $$T^{2} + 8T + 64$$
$67$ $$T^{2} + 5T + 25$$
$71$ $$(T - 12)^{2}$$
$73$ $$T^{2} + 11T + 121$$
$79$ $$T^{2} - 4T + 16$$
$83$ $$T^{2} - 12T + 144$$
$89$ $$T^{2} - 6T + 36$$
$97$ $$T^{2} + 5T + 25$$