# Properties

 Label 2736.2.k.c Level $2736$ Weight $2$ Character orbit 2736.k Analytic conductor $21.847$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# 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: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{7}+O(q^{10})$$ q + 2*b * q^7 $$q + 2 \beta q^{7} - 4 \beta q^{13} + ( - \beta - 4) q^{19} - 5 q^{25} + 4 q^{31} - 4 \beta q^{37} - 6 \beta q^{43} - 5 q^{49} - 14 q^{61} + 16 q^{67} + 10 q^{73} - 4 q^{79} + 24 q^{91} - 8 \beta q^{97} +O(q^{100})$$ q + 2*b * q^7 - 4*b * q^13 + (-b - 4) * q^19 - 5 * q^25 + 4 * q^31 - 4*b * q^37 - 6*b * q^43 - 5 * q^49 - 14 * q^61 + 16 * q^67 + 10 * q^73 - 4 * q^79 + 24 * q^91 - 8*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 8 q^{19} - 10 q^{25} + 8 q^{31} - 10 q^{49} - 28 q^{61} + 32 q^{67} + 20 q^{73} - 8 q^{79} + 48 q^{91}+O(q^{100})$$ 2 * q - 8 * q^19 - 10 * q^25 + 8 * q^31 - 10 * q^49 - 28 * q^61 + 32 * q^67 + 20 * q^73 - 8 * q^79 + 48 * q^91

## 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 0 0 3.46410i 0 0 0
2431.2 0 0 0 0 0 3.46410i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
76.d even 2 1 inner
228.b odd 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.c 2
3.b odd 2 1 CM 2736.2.k.c 2
4.b odd 2 1 2736.2.k.d yes 2
12.b even 2 1 2736.2.k.d yes 2
19.b odd 2 1 2736.2.k.d yes 2
57.d even 2 1 2736.2.k.d yes 2
76.d even 2 1 inner 2736.2.k.c 2
228.b odd 2 1 inner 2736.2.k.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2736.2.k.c 2 1.a even 1 1 trivial
2736.2.k.c 2 3.b odd 2 1 CM
2736.2.k.c 2 76.d even 2 1 inner
2736.2.k.c 2 228.b odd 2 1 inner
2736.2.k.d yes 2 4.b odd 2 1
2736.2.k.d yes 2 12.b even 2 1
2736.2.k.d yes 2 19.b odd 2 1
2736.2.k.d yes 2 57.d even 2 1

## Hecke kernels

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

 $$T_{5}$$ T5 $$T_{7}^{2} + 12$$ T7^2 + 12 $$T_{11}$$ T11 $$T_{31} - 4$$ T31 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 12$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 48$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 8T + 19$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 48$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 108$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T + 14)^{2}$$
$67$ $$(T - 16)^{2}$$
$71$ $$T^{2}$$
$73$ $$(T - 10)^{2}$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 192$$