# Properties

 Label 2736.1.cd.a Level $2736$ Weight $1$ Character orbit 2736.cd Analytic conductor $1.365$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,1,Mod(145,2736)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2736.145");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2736.cd (of order $$6$$, degree $$2$$, 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: $$1.36544187456$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ 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: $$1$$ Twist minimal: no (minimal twist has level 171) Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.22284891.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - q^{7}+O(q^{10})$$ q - q^7 $$q - q^{7} + (\zeta_{6}^{2} - 1) q^{13} + q^{19} + \zeta_{6} q^{25} + (\zeta_{6}^{2} + \zeta_{6}) q^{31} + (\zeta_{6}^{2} + \zeta_{6}) q^{37} + \zeta_{6}^{2} q^{43} - \zeta_{6} q^{61} + (\zeta_{6}^{2} - 1) q^{67} - \zeta_{6}^{2} q^{73} + (\zeta_{6} + 1) q^{79} + ( - \zeta_{6}^{2} + 1) q^{91} +O(q^{100})$$ q - q^7 + (z^2 - 1) * q^13 + q^19 + z * q^25 + (z^2 + z) * q^31 + (z^2 + z) * q^37 + z^2 * q^43 - z * q^61 + (z^2 - 1) * q^67 - z^2 * q^73 + (z + 1) * q^79 + (-z^2 + 1) * q^91 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{7}+O(q^{10})$$ 2 * q - 2 * q^7 $$2 q - 2 q^{7} - 3 q^{13} + 2 q^{19} + q^{25} - q^{43} - q^{61} - 3 q^{67} + q^{73} + 3 q^{79} + 3 q^{91}+O(q^{100})$$ 2 * q - 2 * q^7 - 3 * q^13 + 2 * q^19 + q^25 - q^43 - q^61 - 3 * q^67 + q^73 + 3 * q^79 + 3 * q^91

## Character values

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

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$\zeta_{6}$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
145.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 0 0 −1.00000 0 0 0
1585.1 0 0 0 0 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
19.d odd 6 1 inner
57.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.1.cd.a 2
3.b odd 2 1 CM 2736.1.cd.a 2
4.b odd 2 1 171.1.p.a 2
12.b even 2 1 171.1.p.a 2
19.d odd 6 1 inner 2736.1.cd.a 2
36.f odd 6 1 1539.1.i.a 2
36.f odd 6 1 1539.1.s.a 2
36.h even 6 1 1539.1.i.a 2
36.h even 6 1 1539.1.s.a 2
57.f even 6 1 inner 2736.1.cd.a 2
76.d even 2 1 3249.1.p.b 2
76.f even 6 1 171.1.p.a 2
76.f even 6 1 3249.1.c.a 2
76.g odd 6 1 3249.1.c.a 2
76.g odd 6 1 3249.1.p.b 2
76.k even 18 3 3249.1.ba.a 6
76.k even 18 3 3249.1.ba.b 6
76.l odd 18 3 3249.1.ba.a 6
76.l odd 18 3 3249.1.ba.b 6
228.b odd 2 1 3249.1.p.b 2
228.m even 6 1 3249.1.c.a 2
228.m even 6 1 3249.1.p.b 2
228.n odd 6 1 171.1.p.a 2
228.n odd 6 1 3249.1.c.a 2
228.u odd 18 3 3249.1.ba.a 6
228.u odd 18 3 3249.1.ba.b 6
228.v even 18 3 3249.1.ba.a 6
228.v even 18 3 3249.1.ba.b 6
684.p odd 6 1 1539.1.i.a 2
684.u even 6 1 1539.1.s.a 2
684.bf odd 6 1 1539.1.s.a 2
684.bn even 6 1 1539.1.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.1.p.a 2 4.b odd 2 1
171.1.p.a 2 12.b even 2 1
171.1.p.a 2 76.f even 6 1
171.1.p.a 2 228.n odd 6 1
1539.1.i.a 2 36.f odd 6 1
1539.1.i.a 2 36.h even 6 1
1539.1.i.a 2 684.p odd 6 1
1539.1.i.a 2 684.bn even 6 1
1539.1.s.a 2 36.f odd 6 1
1539.1.s.a 2 36.h even 6 1
1539.1.s.a 2 684.u even 6 1
1539.1.s.a 2 684.bf odd 6 1
2736.1.cd.a 2 1.a even 1 1 trivial
2736.1.cd.a 2 3.b odd 2 1 CM
2736.1.cd.a 2 19.d odd 6 1 inner
2736.1.cd.a 2 57.f even 6 1 inner
3249.1.c.a 2 76.f even 6 1
3249.1.c.a 2 76.g odd 6 1
3249.1.c.a 2 228.m even 6 1
3249.1.c.a 2 228.n odd 6 1
3249.1.p.b 2 76.d even 2 1
3249.1.p.b 2 76.g odd 6 1
3249.1.p.b 2 228.b odd 2 1
3249.1.p.b 2 228.m even 6 1
3249.1.ba.a 6 76.k even 18 3
3249.1.ba.a 6 76.l odd 18 3
3249.1.ba.a 6 228.u odd 18 3
3249.1.ba.a 6 228.v even 18 3
3249.1.ba.b 6 76.k even 18 3
3249.1.ba.b 6 76.l odd 18 3
3249.1.ba.b 6 228.u odd 18 3
3249.1.ba.b 6 228.v even 18 3

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(2736, [\chi])$$.

## Hecke characteristic polynomials

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