# Properties

 Label 2736.2.k.e 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: $$\Z[a_1, \ldots, a_{19}]$$ 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.

 $$f(q)$$ $$=$$ $$q - 2 \beta q^{7} +O(q^{10})$$ q - 2*b * q^7 $$q - 2 \beta q^{7} + 2 \beta q^{11} - 2 \beta q^{13} + 6 q^{17} + (\beta - 4) q^{19} - 5 q^{25} - 4 \beta q^{29} + 10 q^{31} - 2 \beta q^{37} + 4 \beta q^{41} - 6 \beta q^{43} + 4 \beta q^{47} - 5 q^{49} - 8 \beta q^{53} - 12 q^{59} + 10 q^{61} + 4 q^{67} - 12 q^{71} - 2 q^{73} + 12 q^{77} - 10 q^{79} - 2 \beta q^{83} - 4 \beta q^{89} - 12 q^{91} - 4 \beta q^{97} +O(q^{100})$$ q - 2*b * q^7 + 2*b * q^11 - 2*b * q^13 + 6 * q^17 + (b - 4) * q^19 - 5 * q^25 - 4*b * q^29 + 10 * q^31 - 2*b * q^37 + 4*b * q^41 - 6*b * q^43 + 4*b * q^47 - 5 * q^49 - 8*b * q^53 - 12 * q^59 + 10 * q^61 + 4 * q^67 - 12 * q^71 - 2 * q^73 + 12 * q^77 - 10 * q^79 - 2*b * q^83 - 4*b * q^89 - 12 * q^91 - 4*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 12 q^{17} - 8 q^{19} - 10 q^{25} + 20 q^{31} - 10 q^{49} - 24 q^{59} + 20 q^{61} + 8 q^{67} - 24 q^{71} - 4 q^{73} + 24 q^{77} - 20 q^{79} - 24 q^{91}+O(q^{100})$$ 2 * q + 12 * q^17 - 8 * q^19 - 10 * q^25 + 20 * q^31 - 10 * q^49 - 24 * q^59 + 20 * q^61 + 8 * q^67 - 24 * q^71 - 4 * q^73 + 24 * q^77 - 20 * q^79 - 24 * 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
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.e 2
3.b odd 2 1 912.2.k.e yes 2
4.b odd 2 1 2736.2.k.f 2
12.b even 2 1 912.2.k.b 2
19.b odd 2 1 2736.2.k.f 2
24.f even 2 1 3648.2.k.e 2
24.h odd 2 1 3648.2.k.b 2
57.d even 2 1 912.2.k.b 2
76.d even 2 1 inner 2736.2.k.e 2
228.b odd 2 1 912.2.k.e yes 2
456.l odd 2 1 3648.2.k.b 2
456.p even 2 1 3648.2.k.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.k.b 2 12.b even 2 1
912.2.k.b 2 57.d even 2 1
912.2.k.e yes 2 3.b odd 2 1
912.2.k.e yes 2 228.b odd 2 1
2736.2.k.e 2 1.a even 1 1 trivial
2736.2.k.e 2 76.d even 2 1 inner
2736.2.k.f 2 4.b odd 2 1
2736.2.k.f 2 19.b odd 2 1
3648.2.k.b 2 24.h odd 2 1
3648.2.k.b 2 456.l odd 2 1
3648.2.k.e 2 24.f even 2 1
3648.2.k.e 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 $$S_{2}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5}$$ T5 $$T_{7}^{2} + 12$$ T7^2 + 12 $$T_{11}^{2} + 12$$ T11^2 + 12 $$T_{31} - 10$$ T31 - 10

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 12$$
$11$ $$T^{2} + 12$$
$13$ $$T^{2} + 12$$
$17$ $$(T - 6)^{2}$$
$19$ $$T^{2} + 8T + 19$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 48$$
$31$ $$(T - 10)^{2}$$
$37$ $$T^{2} + 12$$
$41$ $$T^{2} + 48$$
$43$ $$T^{2} + 108$$
$47$ $$T^{2} + 48$$
$53$ $$T^{2} + 192$$
$59$ $$(T + 12)^{2}$$
$61$ $$(T - 10)^{2}$$
$67$ $$(T - 4)^{2}$$
$71$ $$(T + 12)^{2}$$
$73$ $$(T + 2)^{2}$$
$79$ $$(T + 10)^{2}$$
$83$ $$T^{2} + 12$$
$89$ $$T^{2} + 48$$
$97$ $$T^{2} + 48$$