# Properties

 Label 2646.2.f.d Level $2646$ Weight $2$ Character orbit 2646.f 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(883,2646)]

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([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("2646.883");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2646 = 2 \cdot 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2646.f (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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + q^{8} +O(q^{10})$$ q + (z - 1) * q^2 - z * q^4 + 3*z * q^5 + q^8 $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + q^{8} - 3 q^{10} + (3 \zeta_{6} - 3) q^{11} + \zeta_{6} q^{13} + (\zeta_{6} - 1) q^{16} - 3 q^{17} - 7 q^{19} + ( - 3 \zeta_{6} + 3) q^{20} - 3 \zeta_{6} q^{22} - 9 \zeta_{6} q^{23} + (4 \zeta_{6} - 4) q^{25} - q^{26} + ( - 3 \zeta_{6} + 3) q^{29} - 8 \zeta_{6} q^{31} - \zeta_{6} q^{32} + ( - 3 \zeta_{6} + 3) q^{34} - q^{37} + ( - 7 \zeta_{6} + 7) q^{38} + 3 \zeta_{6} q^{40} + 3 \zeta_{6} q^{41} + ( - \zeta_{6} + 1) q^{43} + 3 q^{44} + 9 q^{46} - 4 \zeta_{6} q^{50} + ( - \zeta_{6} + 1) q^{52} - 3 q^{53} - 9 q^{55} + 3 \zeta_{6} q^{58} + (2 \zeta_{6} - 2) q^{61} + 8 q^{62} + q^{64} + (3 \zeta_{6} - 3) q^{65} + 4 \zeta_{6} q^{67} + 3 \zeta_{6} q^{68} - 12 q^{71} + 11 q^{73} + ( - \zeta_{6} + 1) q^{74} + 7 \zeta_{6} q^{76} + ( - 16 \zeta_{6} + 16) q^{79} - 3 q^{80} - 3 q^{82} + (9 \zeta_{6} - 9) q^{83} - 9 \zeta_{6} q^{85} + \zeta_{6} q^{86} + (3 \zeta_{6} - 3) q^{88} - 3 q^{89} + (9 \zeta_{6} - 9) q^{92} - 21 \zeta_{6} q^{95} + ( - \zeta_{6} + 1) q^{97} +O(q^{100})$$ q + (z - 1) * q^2 - z * q^4 + 3*z * q^5 + q^8 - 3 * q^10 + (3*z - 3) * q^11 + z * q^13 + (z - 1) * q^16 - 3 * q^17 - 7 * q^19 + (-3*z + 3) * q^20 - 3*z * q^22 - 9*z * q^23 + (4*z - 4) * q^25 - q^26 + (-3*z + 3) * q^29 - 8*z * q^31 - z * q^32 + (-3*z + 3) * q^34 - q^37 + (-7*z + 7) * q^38 + 3*z * q^40 + 3*z * q^41 + (-z + 1) * q^43 + 3 * q^44 + 9 * q^46 - 4*z * q^50 + (-z + 1) * q^52 - 3 * q^53 - 9 * q^55 + 3*z * q^58 + (2*z - 2) * q^61 + 8 * q^62 + q^64 + (3*z - 3) * q^65 + 4*z * q^67 + 3*z * q^68 - 12 * q^71 + 11 * q^73 + (-z + 1) * q^74 + 7*z * q^76 + (-16*z + 16) * q^79 - 3 * q^80 - 3 * q^82 + (9*z - 9) * q^83 - 9*z * q^85 + z * q^86 + (3*z - 3) * q^88 - 3 * q^89 + (9*z - 9) * q^92 - 21*z * q^95 + (-z + 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} + 3 q^{5} + 2 q^{8}+O(q^{10})$$ 2 * q - q^2 - q^4 + 3 * q^5 + 2 * q^8 $$2 q - q^{2} - q^{4} + 3 q^{5} + 2 q^{8} - 6 q^{10} - 3 q^{11} + q^{13} - q^{16} - 6 q^{17} - 14 q^{19} + 3 q^{20} - 3 q^{22} - 9 q^{23} - 4 q^{25} - 2 q^{26} + 3 q^{29} - 8 q^{31} - q^{32} + 3 q^{34} - 2 q^{37} + 7 q^{38} + 3 q^{40} + 3 q^{41} + q^{43} + 6 q^{44} + 18 q^{46} - 4 q^{50} + q^{52} - 6 q^{53} - 18 q^{55} + 3 q^{58} - 2 q^{61} + 16 q^{62} + 2 q^{64} - 3 q^{65} + 4 q^{67} + 3 q^{68} - 24 q^{71} + 22 q^{73} + q^{74} + 7 q^{76} + 16 q^{79} - 6 q^{80} - 6 q^{82} - 9 q^{83} - 9 q^{85} + q^{86} - 3 q^{88} - 6 q^{89} - 9 q^{92} - 21 q^{95} + q^{97}+O(q^{100})$$ 2 * q - q^2 - q^4 + 3 * q^5 + 2 * q^8 - 6 * q^10 - 3 * q^11 + q^13 - q^16 - 6 * q^17 - 14 * q^19 + 3 * q^20 - 3 * q^22 - 9 * q^23 - 4 * q^25 - 2 * q^26 + 3 * q^29 - 8 * q^31 - q^32 + 3 * q^34 - 2 * q^37 + 7 * q^38 + 3 * q^40 + 3 * q^41 + q^43 + 6 * q^44 + 18 * q^46 - 4 * q^50 + q^52 - 6 * q^53 - 18 * q^55 + 3 * q^58 - 2 * q^61 + 16 * q^62 + 2 * q^64 - 3 * q^65 + 4 * q^67 + 3 * q^68 - 24 * q^71 + 22 * q^73 + q^74 + 7 * q^76 + 16 * q^79 - 6 * q^80 - 6 * q^82 - 9 * q^83 - 9 * q^85 + q^86 - 3 * q^88 - 6 * q^89 - 9 * q^92 - 21 * q^95 + q^97

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
883.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 1.50000 + 2.59808i 0 0 1.00000 0 −3.00000
1765.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 1.50000 2.59808i 0 0 1.00000 0 −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 2646.2.f.d 2
3.b odd 2 1 882.2.f.i 2
7.b odd 2 1 2646.2.f.a 2
7.c even 3 1 378.2.e.b 2
7.c even 3 1 378.2.h.a 2
7.d odd 6 1 2646.2.e.g 2
7.d odd 6 1 2646.2.h.d 2
9.c even 3 1 inner 2646.2.f.d 2
9.c even 3 1 7938.2.a.t 1
9.d odd 6 1 882.2.f.i 2
9.d odd 6 1 7938.2.a.m 1
21.c even 2 1 882.2.f.g 2
21.g even 6 1 882.2.e.c 2
21.g even 6 1 882.2.h.i 2
21.h odd 6 1 126.2.e.a 2
21.h odd 6 1 126.2.h.b yes 2
28.g odd 6 1 3024.2.q.f 2
28.g odd 6 1 3024.2.t.a 2
63.g even 3 1 378.2.e.b 2
63.g even 3 1 1134.2.g.c 2
63.h even 3 1 378.2.h.a 2
63.h even 3 1 1134.2.g.c 2
63.i even 6 1 882.2.h.i 2
63.j odd 6 1 126.2.h.b yes 2
63.j odd 6 1 1134.2.g.e 2
63.k odd 6 1 2646.2.e.g 2
63.l odd 6 1 2646.2.f.a 2
63.l odd 6 1 7938.2.a.be 1
63.n odd 6 1 126.2.e.a 2
63.n odd 6 1 1134.2.g.e 2
63.o even 6 1 882.2.f.g 2
63.o even 6 1 7938.2.a.b 1
63.s even 6 1 882.2.e.c 2
63.t odd 6 1 2646.2.h.d 2
84.n even 6 1 1008.2.q.a 2
84.n even 6 1 1008.2.t.f 2
252.o even 6 1 1008.2.q.a 2
252.u odd 6 1 3024.2.t.a 2
252.bb even 6 1 1008.2.t.f 2
252.bl 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 21.h odd 6 1
126.2.e.a 2 63.n odd 6 1
126.2.h.b yes 2 21.h odd 6 1
126.2.h.b yes 2 63.j odd 6 1
378.2.e.b 2 7.c even 3 1
378.2.e.b 2 63.g even 3 1
378.2.h.a 2 7.c even 3 1
378.2.h.a 2 63.h even 3 1
882.2.e.c 2 21.g even 6 1
882.2.e.c 2 63.s even 6 1
882.2.f.g 2 21.c even 2 1
882.2.f.g 2 63.o even 6 1
882.2.f.i 2 3.b odd 2 1
882.2.f.i 2 9.d odd 6 1
882.2.h.i 2 21.g even 6 1
882.2.h.i 2 63.i even 6 1
1008.2.q.a 2 84.n even 6 1
1008.2.q.a 2 252.o even 6 1
1008.2.t.f 2 84.n even 6 1
1008.2.t.f 2 252.bb even 6 1
1134.2.g.c 2 63.g even 3 1
1134.2.g.c 2 63.h even 3 1
1134.2.g.e 2 63.j odd 6 1
1134.2.g.e 2 63.n odd 6 1
2646.2.e.g 2 7.d odd 6 1
2646.2.e.g 2 63.k odd 6 1
2646.2.f.a 2 7.b odd 2 1
2646.2.f.a 2 63.l odd 6 1
2646.2.f.d 2 1.a even 1 1 trivial
2646.2.f.d 2 9.c even 3 1 inner
2646.2.h.d 2 7.d odd 6 1
2646.2.h.d 2 63.t odd 6 1
3024.2.q.f 2 28.g odd 6 1
3024.2.q.f 2 252.bl odd 6 1
3024.2.t.a 2 28.g odd 6 1
3024.2.t.a 2 252.u odd 6 1
7938.2.a.b 1 63.o even 6 1
7938.2.a.m 1 9.d odd 6 1
7938.2.a.t 1 9.c even 3 1
7938.2.a.be 1 63.l odd 6 1

## Hecke kernels

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

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

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