# Properties

 Label 2736.2.f.e Level $2736$ Weight $2$ Character orbit 2736.f Analytic conductor $21.847$ Analytic rank $0$ Dimension $4$ CM discriminant -19 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(1025,2736)]

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, 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("2736.1025");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.f (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: $$21.8470699930$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 20x^{2} + 81$$ x^4 + 20*x^2 + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 171) 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{5} - \beta_{3} q^{7}+O(q^{10})$$ q + b1 * q^5 - b3 * q^7 $$q + \beta_1 q^{5} - \beta_{3} q^{7} + ( - \beta_{2} - \beta_1) q^{11} + ( - \beta_{2} + \beta_1) q^{17} + \beta_{3} q^{19} + (\beta_{2} + 2 \beta_1) q^{23} + (\beta_{3} - 5) q^{25} + ( - 3 \beta_{2} + \beta_1) q^{35} + q^{43} + (2 \beta_{2} - \beta_1) q^{47} + 12 q^{49} + (2 \beta_{3} + 7) q^{55} + \beta_{3} q^{61} + 11 q^{73} + (4 \beta_{2} + 5 \beta_1) q^{77} + (3 \beta_{2} + 2 \beta_1) q^{83} + (4 \beta_{3} - 13) q^{85} + (3 \beta_{2} - \beta_1) q^{95}+O(q^{100})$$ q + b1 * q^5 - b3 * q^7 + (-b2 - b1) * q^11 + (-b2 + b1) * q^17 + b3 * q^19 + (b2 + 2*b1) * q^23 + (b3 - 5) * q^25 + (-3*b2 + b1) * q^35 + q^43 + (2*b2 - b1) * q^47 + 12 * q^49 + (2*b3 + 7) * q^55 + b3 * q^61 + 11 * q^73 + (4*b2 + 5*b1) * q^77 + (3*b2 + 2*b1) * q^83 + (4*b3 - 13) * q^85 + (3*b2 - b1) * q^95 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 20 q^{25} + 4 q^{43} + 48 q^{49} + 28 q^{55} + 44 q^{73} - 52 q^{85}+O(q^{100})$$ 4 * q - 20 * q^25 + 4 * q^43 + 48 * q^49 + 28 * q^55 + 44 * q^73 - 52 * q^85

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 20x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 11\nu ) / 3$$ (v^3 + 11*v) / 3 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 10$$ v^2 + 10
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 10$$ b3 - 10 $$\nu^{3}$$ $$=$$ $$3\beta_{2} - 11\beta_1$$ 3*b2 - 11*b1

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1025.1
 − 3.78931i − 2.37510i 2.37510i 3.78931i
0 0 0 3.78931i 0 4.35890 0 0 0
1025.2 0 0 0 2.37510i 0 −4.35890 0 0 0
1025.3 0 0 0 2.37510i 0 −4.35890 0 0 0
1025.4 0 0 0 3.78931i 0 4.35890 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
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
3.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 2736.2.f.e 4
3.b odd 2 1 inner 2736.2.f.e 4
4.b odd 2 1 171.2.d.a 4
12.b even 2 1 171.2.d.a 4
19.b odd 2 1 CM 2736.2.f.e 4
57.d even 2 1 inner 2736.2.f.e 4
76.d even 2 1 171.2.d.a 4
228.b odd 2 1 171.2.d.a 4

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

 $$T_{5}^{4} + 20T_{5}^{2} + 81$$ T5^4 + 20*T5^2 + 81 $$T_{7}^{2} - 19$$ T7^2 - 19 $$T_{11}^{4} + 44T_{11}^{2} + 9$$ T11^4 + 44*T11^2 + 9 $$T_{29}$$ T29

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 20T^{2} + 81$$
$7$ $$(T^{2} - 19)^{2}$$
$11$ $$T^{4} + 44T^{2} + 9$$
$13$ $$T^{4}$$
$17$ $$T^{4} + 68T^{2} + 225$$
$19$ $$(T^{2} - 19)^{2}$$
$23$ $$T^{4} + 92T^{2} + 900$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$(T - 1)^{4}$$
$47$ $$T^{4} + 188T^{2} + 5625$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$(T^{2} - 19)^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$(T - 11)^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4} + 332T^{2} + 8100$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$