# Properties

 Label 2736.2.k.a Level $2736$ Weight $2$ Character orbit 2736.k Analytic conductor $21.847$ 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] = [2736,2,Mod(2431,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))

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

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

chi := DirichletCharacter("2736.2431");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.k (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: $$21.8470699930$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 912) 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 $$\beta = \sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 3 q^{5} + \beta q^{7}+O(q^{10})$$ q - 3 * q^5 + b * q^7 $$q - 3 q^{5} + \beta q^{7} + 3 \beta q^{11} - 4 \beta q^{13} - 3 q^{17} + ( - \beta - 4) q^{19} - 2 \beta q^{23} + 4 q^{25} + 4 q^{31} - 3 \beta q^{35} + 4 \beta q^{37} - 4 \beta q^{41} + 5 \beta q^{43} - 5 \beta q^{47} + 4 q^{49} + 4 \beta q^{53} - 9 \beta q^{55} + 12 q^{59} + 7 q^{61} + 12 \beta q^{65} - 8 q^{67} + 12 q^{71} - 5 q^{73} - 9 q^{77} + 8 q^{79} + 2 \beta q^{83} + 9 q^{85} - 4 \beta q^{89} + 12 q^{91} + (3 \beta + 12) q^{95} + 4 \beta q^{97} +O(q^{100})$$ q - 3 * q^5 + b * q^7 + 3*b * q^11 - 4*b * q^13 - 3 * q^17 + (-b - 4) * q^19 - 2*b * q^23 + 4 * q^25 + 4 * q^31 - 3*b * q^35 + 4*b * q^37 - 4*b * q^41 + 5*b * q^43 - 5*b * q^47 + 4 * q^49 + 4*b * q^53 - 9*b * q^55 + 12 * q^59 + 7 * q^61 + 12*b * q^65 - 8 * q^67 + 12 * q^71 - 5 * q^73 - 9 * q^77 + 8 * q^79 + 2*b * q^83 + 9 * q^85 - 4*b * q^89 + 12 * q^91 + (3*b + 12) * q^95 + 4*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{5}+O(q^{10})$$ 2 * q - 6 * q^5 $$2 q - 6 q^{5} - 6 q^{17} - 8 q^{19} + 8 q^{25} + 8 q^{31} + 8 q^{49} + 24 q^{59} + 14 q^{61} - 16 q^{67} + 24 q^{71} - 10 q^{73} - 18 q^{77} + 16 q^{79} + 18 q^{85} + 24 q^{91} + 24 q^{95}+O(q^{100})$$ 2 * q - 6 * q^5 - 6 * q^17 - 8 * q^19 + 8 * q^25 + 8 * q^31 + 8 * q^49 + 24 * q^59 + 14 * q^61 - 16 * q^67 + 24 * q^71 - 10 * q^73 - 18 * q^77 + 16 * q^79 + 18 * q^85 + 24 * q^91 + 24 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$$.

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2431.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 −3.00000 0 1.73205i 0 0 0
2431.2 0 0 0 −3.00000 0 1.73205i 0 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
76.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.k.a 2
3.b odd 2 1 912.2.k.f yes 2
4.b odd 2 1 2736.2.k.b 2
12.b even 2 1 912.2.k.c 2
19.b odd 2 1 2736.2.k.b 2
24.f even 2 1 3648.2.k.d 2
24.h odd 2 1 3648.2.k.a 2
57.d even 2 1 912.2.k.c 2
76.d even 2 1 inner 2736.2.k.a 2
228.b odd 2 1 912.2.k.f yes 2
456.l odd 2 1 3648.2.k.a 2
456.p even 2 1 3648.2.k.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.k.c 2 12.b even 2 1
912.2.k.c 2 57.d even 2 1
912.2.k.f yes 2 3.b odd 2 1
912.2.k.f yes 2 228.b odd 2 1
2736.2.k.a 2 1.a even 1 1 trivial
2736.2.k.a 2 76.d even 2 1 inner
2736.2.k.b 2 4.b odd 2 1
2736.2.k.b 2 19.b odd 2 1
3648.2.k.a 2 24.h odd 2 1
3648.2.k.a 2 456.l odd 2 1
3648.2.k.d 2 24.f even 2 1
3648.2.k.d 2 456.p even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5} + 3$$ T5 + 3 $$T_{7}^{2} + 3$$ T7^2 + 3 $$T_{11}^{2} + 27$$ T11^2 + 27 $$T_{31} - 4$$ T31 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 3)^{2}$$
$7$ $$T^{2} + 3$$
$11$ $$T^{2} + 27$$
$13$ $$T^{2} + 48$$
$17$ $$(T + 3)^{2}$$
$19$ $$T^{2} + 8T + 19$$
$23$ $$T^{2} + 12$$
$29$ $$T^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 48$$
$41$ $$T^{2} + 48$$
$43$ $$T^{2} + 75$$
$47$ $$T^{2} + 75$$
$53$ $$T^{2} + 48$$
$59$ $$(T - 12)^{2}$$
$61$ $$(T - 7)^{2}$$
$67$ $$(T + 8)^{2}$$
$71$ $$(T - 12)^{2}$$
$73$ $$(T + 5)^{2}$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 12$$
$89$ $$T^{2} + 48$$
$97$ $$T^{2} + 48$$