# Properties

 Label 2646.2.d.a Level $2646$ Weight $2$ Character orbit 2646.d Analytic conductor $21.128$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2646,2,Mod(2645,2646)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2646, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

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

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

chi := DirichletCharacter("2646.2645");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2646 = 2 \cdot 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2646.d (of order $$2$$, degree $$1$$, 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: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 378) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - q^{4} + \beta_{3} q^{5} - \beta_1 q^{8}+O(q^{10})$$ q + b1 * q^2 - q^4 + b3 * q^5 - b1 * q^8 $$q + \beta_1 q^{2} - q^{4} + \beta_{3} q^{5} - \beta_1 q^{8} + \beta_{2} q^{10} + q^{16} - 2 \beta_{3} q^{17} - 4 \beta_{2} q^{19} - \beta_{3} q^{20} - 6 \beta_1 q^{23} - 2 q^{25} - 9 \beta_1 q^{29} + 2 \beta_{2} q^{31} + \beta_1 q^{32} - 2 \beta_{2} q^{34} + 4 q^{37} + 4 \beta_{3} q^{38} - \beta_{2} q^{40} + 2 \beta_{3} q^{41} - 8 q^{43} + 6 q^{46} - 2 \beta_{3} q^{47} - 2 \beta_1 q^{50} + 3 \beta_1 q^{53} + 9 q^{58} - 7 \beta_{3} q^{59} - 2 \beta_{2} q^{61} - 2 \beta_{3} q^{62} - q^{64} + 14 q^{67} + 2 \beta_{3} q^{68} + 6 \beta_1 q^{71} - 7 \beta_{2} q^{73} + 4 \beta_1 q^{74} + 4 \beta_{2} q^{76} + 11 q^{79} + \beta_{3} q^{80} + 2 \beta_{2} q^{82} - 10 \beta_{3} q^{83} - 6 q^{85} - 8 \beta_1 q^{86} - 6 \beta_{3} q^{89} + 6 \beta_1 q^{92} - 2 \beta_{2} q^{94} - 12 \beta_1 q^{95} - 4 \beta_{2} q^{97}+O(q^{100})$$ q + b1 * q^2 - q^4 + b3 * q^5 - b1 * q^8 + b2 * q^10 + q^16 - 2*b3 * q^17 - 4*b2 * q^19 - b3 * q^20 - 6*b1 * q^23 - 2 * q^25 - 9*b1 * q^29 + 2*b2 * q^31 + b1 * q^32 - 2*b2 * q^34 + 4 * q^37 + 4*b3 * q^38 - b2 * q^40 + 2*b3 * q^41 - 8 * q^43 + 6 * q^46 - 2*b3 * q^47 - 2*b1 * q^50 + 3*b1 * q^53 + 9 * q^58 - 7*b3 * q^59 - 2*b2 * q^61 - 2*b3 * q^62 - q^64 + 14 * q^67 + 2*b3 * q^68 + 6*b1 * q^71 - 7*b2 * q^73 + 4*b1 * q^74 + 4*b2 * q^76 + 11 * q^79 + b3 * q^80 + 2*b2 * q^82 - 10*b3 * q^83 - 6 * q^85 - 8*b1 * q^86 - 6*b3 * q^89 + 6*b1 * q^92 - 2*b2 * q^94 - 12*b1 * q^95 - 4*b2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4}+O(q^{10})$$ 4 * q - 4 * q^4 $$4 q - 4 q^{4} + 4 q^{16} - 8 q^{25} + 16 q^{37} - 32 q^{43} + 24 q^{46} + 36 q^{58} - 4 q^{64} + 56 q^{67} + 44 q^{79} - 24 q^{85}+O(q^{100})$$ 4 * q - 4 * q^4 + 4 * q^16 - 8 * q^25 + 16 * q^37 - 32 * q^43 + 24 * q^46 + 36 * q^58 - 4 * q^64 + 56 * q^67 + 44 * q^79 - 24 * q^85

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ b1

## Character values

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

 $$n$$ $$785$$ $$1081$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2645.1
 −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i
1.00000i 0 −1.00000 −1.73205 0 0 1.00000i 0 1.73205i
2645.2 1.00000i 0 −1.00000 1.73205 0 0 1.00000i 0 1.73205i
2645.3 1.00000i 0 −1.00000 −1.73205 0 0 1.00000i 0 1.73205i
2645.4 1.00000i 0 −1.00000 1.73205 0 0 1.00000i 0 1.73205i
 $$n$$: 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
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2646.2.d.a 4
3.b odd 2 1 inner 2646.2.d.a 4
7.b odd 2 1 inner 2646.2.d.a 4
7.c even 3 1 378.2.k.c 4
7.d odd 6 1 378.2.k.c 4
21.c even 2 1 inner 2646.2.d.a 4
21.g even 6 1 378.2.k.c 4
21.h odd 6 1 378.2.k.c 4
63.g even 3 1 1134.2.t.c 4
63.h even 3 1 1134.2.l.b 4
63.i even 6 1 1134.2.l.b 4
63.j odd 6 1 1134.2.l.b 4
63.k odd 6 1 1134.2.t.c 4
63.n odd 6 1 1134.2.t.c 4
63.s even 6 1 1134.2.t.c 4
63.t odd 6 1 1134.2.l.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.k.c 4 7.c even 3 1
378.2.k.c 4 7.d odd 6 1
378.2.k.c 4 21.g even 6 1
378.2.k.c 4 21.h odd 6 1
1134.2.l.b 4 63.h even 3 1
1134.2.l.b 4 63.i even 6 1
1134.2.l.b 4 63.j odd 6 1
1134.2.l.b 4 63.t odd 6 1
1134.2.t.c 4 63.g even 3 1
1134.2.t.c 4 63.k odd 6 1
1134.2.t.c 4 63.n odd 6 1
1134.2.t.c 4 63.s even 6 1
2646.2.d.a 4 1.a even 1 1 trivial
2646.2.d.a 4 3.b odd 2 1 inner
2646.2.d.a 4 7.b odd 2 1 inner
2646.2.d.a 4 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 3$$ acting on $$S_{2}^{\mathrm{new}}(2646, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 3)^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 12)^{2}$$
$19$ $$(T^{2} + 48)^{2}$$
$23$ $$(T^{2} + 36)^{2}$$
$29$ $$(T^{2} + 81)^{2}$$
$31$ $$(T^{2} + 12)^{2}$$
$37$ $$(T - 4)^{4}$$
$41$ $$(T^{2} - 12)^{2}$$
$43$ $$(T + 8)^{4}$$
$47$ $$(T^{2} - 12)^{2}$$
$53$ $$(T^{2} + 9)^{2}$$
$59$ $$(T^{2} - 147)^{2}$$
$61$ $$(T^{2} + 12)^{2}$$
$67$ $$(T - 14)^{4}$$
$71$ $$(T^{2} + 36)^{2}$$
$73$ $$(T^{2} + 147)^{2}$$
$79$ $$(T - 11)^{4}$$
$83$ $$(T^{2} - 300)^{2}$$
$89$ $$(T^{2} - 108)^{2}$$
$97$ $$(T^{2} + 48)^{2}$$