# Properties

 Label 2736.1.o.b Level $2736$ Weight $1$ Character orbit 2736.o Self dual yes Analytic conductor $1.365$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -19 Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,1,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, 1, names="a")

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

chi := DirichletCharacter("2736.721");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ 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: $$1.36544187456$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 76) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.76.1 Artin image: $D_6$ Artin field: Galois closure of 6.2.2495232.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{5} + q^{7}+O(q^{10})$$ q + q^5 + q^7 $$q + q^{5} + q^{7} - q^{11} + q^{17} - q^{19} + 2 q^{23} + q^{35} + q^{43} - q^{47} - q^{55} - q^{61} - q^{73} - q^{77} + 2 q^{83} + q^{85} - q^{95}+O(q^{100})$$ q + q^5 + q^7 - q^11 + q^17 - q^19 + 2 * q^23 + q^35 + q^43 - q^47 - q^55 - q^61 - q^73 - q^77 + 2 * q^83 + q^85 - q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$$.

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$1$$ $$0$$ $$0$$ $$0$$

## 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
0 0 0 1.00000 0 1.00000 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 $$\Q(\sqrt{-19})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.1.o.b 1
3.b odd 2 1 304.1.e.a 1
4.b odd 2 1 684.1.h.a 1
12.b even 2 1 76.1.c.a 1
19.b odd 2 1 CM 2736.1.o.b 1
24.f even 2 1 1216.1.e.a 1
24.h odd 2 1 1216.1.e.b 1
57.d even 2 1 304.1.e.a 1
60.h even 2 1 1900.1.e.a 1
60.l odd 4 2 1900.1.g.a 2
76.d even 2 1 684.1.h.a 1
84.h odd 2 1 3724.1.e.c 1
84.j odd 6 2 3724.1.bc.b 2
84.n even 6 2 3724.1.bc.c 2
228.b odd 2 1 76.1.c.a 1
228.m even 6 2 1444.1.h.a 2
228.n odd 6 2 1444.1.h.a 2
228.u odd 18 6 1444.1.j.a 6
228.v even 18 6 1444.1.j.a 6
456.l odd 2 1 1216.1.e.a 1
456.p even 2 1 1216.1.e.b 1
1140.p odd 2 1 1900.1.e.a 1
1140.w even 4 2 1900.1.g.a 2
1596.p even 2 1 3724.1.e.c 1
1596.bl even 6 2 3724.1.bc.b 2
1596.cb odd 6 2 3724.1.bc.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.1.c.a 1 12.b even 2 1
76.1.c.a 1 228.b odd 2 1
304.1.e.a 1 3.b odd 2 1
304.1.e.a 1 57.d even 2 1
684.1.h.a 1 4.b odd 2 1
684.1.h.a 1 76.d even 2 1
1216.1.e.a 1 24.f even 2 1
1216.1.e.a 1 456.l odd 2 1
1216.1.e.b 1 24.h odd 2 1
1216.1.e.b 1 456.p even 2 1
1444.1.h.a 2 228.m even 6 2
1444.1.h.a 2 228.n odd 6 2
1444.1.j.a 6 228.u odd 18 6
1444.1.j.a 6 228.v even 18 6
1900.1.e.a 1 60.h even 2 1
1900.1.e.a 1 1140.p odd 2 1
1900.1.g.a 2 60.l odd 4 2
1900.1.g.a 2 1140.w even 4 2
2736.1.o.b 1 1.a even 1 1 trivial
2736.1.o.b 1 19.b odd 2 1 CM
3724.1.e.c 1 84.h odd 2 1
3724.1.e.c 1 1596.p even 2 1
3724.1.bc.b 2 84.j odd 6 2
3724.1.bc.b 2 1596.bl even 6 2
3724.1.bc.c 2 84.n even 6 2
3724.1.bc.c 2 1596.cb odd 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 1$$ acting on $$S_{1}^{\mathrm{new}}(2736, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T - 1$$
$11$ $$T + 1$$
$13$ $$T$$
$17$ $$T - 1$$
$19$ $$T + 1$$
$23$ $$T - 2$$
$29$ $$T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T - 1$$
$47$ $$T + 1$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T + 1$$
$67$ $$T$$
$71$ $$T$$
$73$ $$T + 1$$
$79$ $$T$$
$83$ $$T - 2$$
$89$ $$T$$
$97$ $$T$$
show more
show less