# Properties

 Label 2736.3.o.g Level $2736$ Weight $3$ Character orbit 2736.o Analytic conductor $74.551$ 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] = [2736,3,Mod(721,2736)]

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([0, 0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2736.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2736.o (of order $$2$$, degree $$1$$, 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: $$74.5506003290$$ 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 684) 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 $$\beta = 8\sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{7}+O(q^{10})$$ q + 2 * q^7 $$q + 2 q^{7} - \beta q^{13} + ( - \beta + 13) q^{19} - 25 q^{25} + 3 \beta q^{31} - 5 \beta q^{37} + 22 q^{43} - 45 q^{49} - 74 q^{61} + 4 \beta q^{67} - 46 q^{73} + 5 \beta q^{79} - 2 \beta q^{91} - 14 \beta q^{97} +O(q^{100})$$ q + 2 * q^7 - b * q^13 + (-b + 13) * q^19 - 25 * q^25 + 3*b * q^31 - 5*b * q^37 + 22 * q^43 - 45 * q^49 - 74 * q^61 + 4*b * q^67 - 46 * q^73 + 5*b * q^79 - 2*b * q^91 - 14*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{7}+O(q^{10})$$ 2 * q + 4 * q^7 $$2 q + 4 q^{7} + 26 q^{19} - 50 q^{25} + 44 q^{43} - 90 q^{49} - 148 q^{61} - 92 q^{73}+O(q^{100})$$ 2 * q + 4 * q^7 + 26 * q^19 - 50 * q^25 + 44 * q^43 - 90 * q^49 - 148 * q^61 - 92 * q^73

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
721.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 0 0 2.00000 0 0 0
721.2 0 0 0 0 0 2.00000 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 $$\Q(\sqrt{-3})$$
19.b odd 2 1 inner
57.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.3.o.g 2
3.b odd 2 1 CM 2736.3.o.g 2
4.b odd 2 1 684.3.h.b 2
12.b even 2 1 684.3.h.b 2
19.b odd 2 1 inner 2736.3.o.g 2
57.d even 2 1 inner 2736.3.o.g 2
76.d even 2 1 684.3.h.b 2
228.b odd 2 1 684.3.h.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.3.h.b 2 4.b odd 2 1
684.3.h.b 2 12.b even 2 1
684.3.h.b 2 76.d even 2 1
684.3.h.b 2 228.b odd 2 1
2736.3.o.g 2 1.a even 1 1 trivial
2736.3.o.g 2 3.b odd 2 1 CM
2736.3.o.g 2 19.b odd 2 1 inner
2736.3.o.g 2 57.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5}$$ T5 $$T_{7} - 2$$ T7 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T - 2)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 192$$
$17$ $$T^{2}$$
$19$ $$T^{2} - 26T + 361$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 1728$$
$37$ $$T^{2} + 4800$$
$41$ $$T^{2}$$
$43$ $$(T - 22)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T + 74)^{2}$$
$67$ $$T^{2} + 3072$$
$71$ $$T^{2}$$
$73$ $$(T + 46)^{2}$$
$79$ $$T^{2} + 4800$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 37632$$