# Properties

 Label 2736.3.o.f Level $2736$ Weight $3$ Character orbit 2736.o Self dual yes Analytic conductor $74.551$ Analytic rank $0$ Dimension $2$ CM discriminant -19 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,3,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: yes Analytic conductor: $$74.5506003290$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 19$$ x^2 - 19 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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 $$\beta = \sqrt{19}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} - 5 q^{7}+O(q^{10})$$ q + b * q^5 - 5 * q^7 $$q + \beta q^{5} - 5 q^{7} + 5 \beta q^{11} + 7 \beta q^{17} - 19 q^{19} + 8 \beta q^{23} - 6 q^{25} - 5 \beta q^{35} + 85 q^{43} - 13 \beta q^{47} - 24 q^{49} + 95 q^{55} - 103 q^{61} - 25 q^{73} - 25 \beta q^{77} + 32 \beta q^{83} + 133 q^{85} - 19 \beta q^{95} +O(q^{100})$$ q + b * q^5 - 5 * q^7 + 5*b * q^11 + 7*b * q^17 - 19 * q^19 + 8*b * q^23 - 6 * q^25 - 5*b * q^35 + 85 * q^43 - 13*b * q^47 - 24 * q^49 + 95 * q^55 - 103 * q^61 - 25 * q^73 - 25*b * q^77 + 32*b * q^83 + 133 * q^85 - 19*b * q^95 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 10 q^{7}+O(q^{10})$$ 2 * q - 10 * q^7 $$2 q - 10 q^{7} - 38 q^{19} - 12 q^{25} + 170 q^{43} - 48 q^{49} + 190 q^{55} - 206 q^{61} - 50 q^{73} + 266 q^{85}+O(q^{100})$$ 2 * q - 10 * q^7 - 38 * q^19 - 12 * q^25 + 170 * q^43 - 48 * q^49 + 190 * q^55 - 206 * q^61 - 50 * q^73 + 266 * q^85

## 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
 −4.35890 4.35890
0 0 0 −4.35890 0 −5.00000 0 0 0
721.2 0 0 0 4.35890 0 −5.00000 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
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 2736.3.o.f 2
3.b odd 2 1 inner 2736.3.o.f 2
4.b odd 2 1 171.3.c.d 2
12.b even 2 1 171.3.c.d 2
19.b odd 2 1 CM 2736.3.o.f 2
57.d even 2 1 inner 2736.3.o.f 2
76.d even 2 1 171.3.c.d 2
228.b odd 2 1 171.3.c.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.3.c.d 2 4.b odd 2 1
171.3.c.d 2 12.b even 2 1
171.3.c.d 2 76.d even 2 1
171.3.c.d 2 228.b odd 2 1
2736.3.o.f 2 1.a even 1 1 trivial
2736.3.o.f 2 3.b odd 2 1 inner
2736.3.o.f 2 19.b odd 2 1 CM
2736.3.o.f 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 $$S_{3}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5}^{2} - 19$$ T5^2 - 19 $$T_{7} + 5$$ T7 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 19$$
$7$ $$(T + 5)^{2}$$
$11$ $$T^{2} - 475$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 931$$
$19$ $$(T + 19)^{2}$$
$23$ $$T^{2} - 1216$$
$29$ $$T^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$(T - 85)^{2}$$
$47$ $$T^{2} - 3211$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T + 103)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$(T + 25)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} - 19456$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$