# Properties

 Label 2646.2.e.g Level $2646$ Weight $2$ Character orbit 2646.e Analytic conductor $21.128$ Analytic rank $1$ 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(1549,2646)]

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, 2]))

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

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

chi := DirichletCharacter("2646.1549");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2646 = 2 \cdot 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2646.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: $$21.1284163748$$ Analytic rank: $$1$$ 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: $$\Z[a_1, \ldots, a_{13}]$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + (3 \zeta_{6} - 3) q^{5} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + (3*z - 3) * q^5 + q^8 $$q + q^{2} + q^{4} + (3 \zeta_{6} - 3) q^{5} + q^{8} + (3 \zeta_{6} - 3) q^{10} - 3 \zeta_{6} q^{11} - \zeta_{6} q^{13} + q^{16} + (3 \zeta_{6} - 3) q^{17} - 7 \zeta_{6} q^{19} + (3 \zeta_{6} - 3) q^{20} - 3 \zeta_{6} q^{22} + (9 \zeta_{6} - 9) q^{23} - 4 \zeta_{6} q^{25} - \zeta_{6} q^{26} + ( - 3 \zeta_{6} + 3) q^{29} - 8 q^{31} + q^{32} + (3 \zeta_{6} - 3) q^{34} + \zeta_{6} q^{37} - 7 \zeta_{6} q^{38} + (3 \zeta_{6} - 3) q^{40} - 3 \zeta_{6} q^{41} + ( - \zeta_{6} + 1) q^{43} - 3 \zeta_{6} q^{44} + (9 \zeta_{6} - 9) q^{46} - 4 \zeta_{6} q^{50} - \zeta_{6} q^{52} + ( - 3 \zeta_{6} + 3) q^{53} + 9 q^{55} + ( - 3 \zeta_{6} + 3) q^{58} - 2 q^{61} - 8 q^{62} + q^{64} + 3 q^{65} - 4 q^{67} + (3 \zeta_{6} - 3) q^{68} - 12 q^{71} + ( - 11 \zeta_{6} + 11) q^{73} + \zeta_{6} q^{74} - 7 \zeta_{6} q^{76} - 16 q^{79} + (3 \zeta_{6} - 3) q^{80} - 3 \zeta_{6} q^{82} + ( - 9 \zeta_{6} + 9) q^{83} - 9 \zeta_{6} q^{85} + ( - \zeta_{6} + 1) q^{86} - 3 \zeta_{6} q^{88} - 3 \zeta_{6} q^{89} + (9 \zeta_{6} - 9) q^{92} + 21 q^{95} + (\zeta_{6} - 1) q^{97} +O(q^{100})$$ q + q^2 + q^4 + (3*z - 3) * q^5 + q^8 + (3*z - 3) * q^10 - 3*z * q^11 - z * q^13 + q^16 + (3*z - 3) * q^17 - 7*z * q^19 + (3*z - 3) * q^20 - 3*z * q^22 + (9*z - 9) * q^23 - 4*z * q^25 - z * q^26 + (-3*z + 3) * q^29 - 8 * q^31 + q^32 + (3*z - 3) * q^34 + z * q^37 - 7*z * q^38 + (3*z - 3) * q^40 - 3*z * q^41 + (-z + 1) * q^43 - 3*z * q^44 + (9*z - 9) * q^46 - 4*z * q^50 - z * q^52 + (-3*z + 3) * q^53 + 9 * q^55 + (-3*z + 3) * q^58 - 2 * q^61 - 8 * q^62 + q^64 + 3 * q^65 - 4 * q^67 + (3*z - 3) * q^68 - 12 * q^71 + (-11*z + 11) * q^73 + z * q^74 - 7*z * q^76 - 16 * q^79 + (3*z - 3) * q^80 - 3*z * q^82 + (-9*z + 9) * q^83 - 9*z * q^85 + (-z + 1) * q^86 - 3*z * q^88 - 3*z * q^89 + (9*z - 9) * q^92 + 21 * q^95 + (z - 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} - 3 q^{5} + 2 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 - 3 * q^5 + 2 * q^8 $$2 q + 2 q^{2} + 2 q^{4} - 3 q^{5} + 2 q^{8} - 3 q^{10} - 3 q^{11} - q^{13} + 2 q^{16} - 3 q^{17} - 7 q^{19} - 3 q^{20} - 3 q^{22} - 9 q^{23} - 4 q^{25} - q^{26} + 3 q^{29} - 16 q^{31} + 2 q^{32} - 3 q^{34} + q^{37} - 7 q^{38} - 3 q^{40} - 3 q^{41} + q^{43} - 3 q^{44} - 9 q^{46} - 4 q^{50} - q^{52} + 3 q^{53} + 18 q^{55} + 3 q^{58} - 4 q^{61} - 16 q^{62} + 2 q^{64} + 6 q^{65} - 8 q^{67} - 3 q^{68} - 24 q^{71} + 11 q^{73} + q^{74} - 7 q^{76} - 32 q^{79} - 3 q^{80} - 3 q^{82} + 9 q^{83} - 9 q^{85} + q^{86} - 3 q^{88} - 3 q^{89} - 9 q^{92} + 42 q^{95} - q^{97}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 - 3 * q^5 + 2 * q^8 - 3 * q^10 - 3 * q^11 - q^13 + 2 * q^16 - 3 * q^17 - 7 * q^19 - 3 * q^20 - 3 * q^22 - 9 * q^23 - 4 * q^25 - q^26 + 3 * q^29 - 16 * q^31 + 2 * q^32 - 3 * q^34 + q^37 - 7 * q^38 - 3 * q^40 - 3 * q^41 + q^43 - 3 * q^44 - 9 * q^46 - 4 * q^50 - q^52 + 3 * q^53 + 18 * q^55 + 3 * q^58 - 4 * q^61 - 16 * q^62 + 2 * q^64 + 6 * q^65 - 8 * q^67 - 3 * q^68 - 24 * q^71 + 11 * q^73 + q^74 - 7 * q^76 - 32 * q^79 - 3 * q^80 - 3 * q^82 + 9 * q^83 - 9 * q^85 + q^86 - 3 * q^88 - 3 * q^89 - 9 * q^92 + 42 * q^95 - q^97

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1549.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.00000 0 1.00000 −1.50000 2.59808i 0 0 1.00000 0 −1.50000 2.59808i
2125.1 1.00000 0 1.00000 −1.50000 + 2.59808i 0 0 1.00000 0 −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 2646.2.e.g 2
3.b odd 2 1 882.2.e.c 2
7.b odd 2 1 378.2.e.b 2
7.c even 3 1 2646.2.f.a 2
7.c even 3 1 2646.2.h.d 2
7.d odd 6 1 378.2.h.a 2
7.d odd 6 1 2646.2.f.d 2
9.c even 3 1 2646.2.h.d 2
9.d odd 6 1 882.2.h.i 2
21.c even 2 1 126.2.e.a 2
21.g even 6 1 126.2.h.b yes 2
21.g even 6 1 882.2.f.i 2
21.h odd 6 1 882.2.f.g 2
21.h odd 6 1 882.2.h.i 2
28.d even 2 1 3024.2.q.f 2
28.f even 6 1 3024.2.t.a 2
63.g even 3 1 2646.2.f.a 2
63.h even 3 1 inner 2646.2.e.g 2
63.h even 3 1 7938.2.a.be 1
63.i even 6 1 126.2.e.a 2
63.i even 6 1 7938.2.a.m 1
63.j odd 6 1 882.2.e.c 2
63.j odd 6 1 7938.2.a.b 1
63.k odd 6 1 1134.2.g.c 2
63.k odd 6 1 2646.2.f.d 2
63.l odd 6 1 378.2.h.a 2
63.l odd 6 1 1134.2.g.c 2
63.n odd 6 1 882.2.f.g 2
63.o even 6 1 126.2.h.b yes 2
63.o even 6 1 1134.2.g.e 2
63.s even 6 1 882.2.f.i 2
63.s even 6 1 1134.2.g.e 2
63.t odd 6 1 378.2.e.b 2
63.t odd 6 1 7938.2.a.t 1
84.h odd 2 1 1008.2.q.a 2
84.j odd 6 1 1008.2.t.f 2
252.r odd 6 1 1008.2.q.a 2
252.s odd 6 1 1008.2.t.f 2
252.bi even 6 1 3024.2.t.a 2
252.bj even 6 1 3024.2.q.f 2

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

## Hecke kernels

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

 $$T_{5}^{2} + 3T_{5} + 9$$ 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

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 3T + 9$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 3T + 9$$
$13$ $$T^{2} + T + 1$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} + 7T + 49$$
$23$ $$T^{2} + 9T + 81$$
$29$ $$T^{2} - 3T + 9$$
$31$ $$(T + 8)^{2}$$
$37$ $$T^{2} - T + 1$$
$41$ $$T^{2} + 3T + 9$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 3T + 9$$
$59$ $$T^{2}$$
$61$ $$(T + 2)^{2}$$
$67$ $$(T + 4)^{2}$$
$71$ $$(T + 12)^{2}$$
$73$ $$T^{2} - 11T + 121$$
$79$ $$(T + 16)^{2}$$
$83$ $$T^{2} - 9T + 81$$
$89$ $$T^{2} + 3T + 9$$
$97$ $$T^{2} + T + 1$$