# Properties

 Label 2736.2.bm.f Level $2736$ Weight $2$ Character orbit 2736.bm Analytic conductor $21.847$ 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] = [2736,2,Mod(559,2736)]

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([3, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2736.559");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.bm (of order $$6$$, degree $$2$$, 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: $$2$$ 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: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 1) q^{7}+O(q^{10})$$ q + (-2*z + 1) * q^7 $$q + ( - 2 \zeta_{6} + 1) q^{7} + ( - 3 \zeta_{6} + 6) q^{13} + (2 \zeta_{6} + 3) q^{19} + 5 \zeta_{6} q^{25} - 7 q^{31} + ( - 6 \zeta_{6} + 3) q^{37} + ( - \zeta_{6} - 1) q^{43} + 4 q^{49} - \zeta_{6} q^{61} - 11 \zeta_{6} q^{67} + ( - 17 \zeta_{6} + 17) q^{73} + ( - 17 \zeta_{6} + 17) q^{79} - 9 \zeta_{6} q^{91} + (8 \zeta_{6} + 8) q^{97} +O(q^{100})$$ q + (-2*z + 1) * q^7 + (-3*z + 6) * q^13 + (2*z + 3) * q^19 + 5*z * q^25 - 7 * q^31 + (-6*z + 3) * q^37 + (-z - 1) * q^43 + 4 * q^49 - z * q^61 - 11*z * q^67 + (-17*z + 17) * q^73 + (-17*z + 17) * q^79 - 9*z * q^91 + (8*z + 8) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 9 q^{13} + 8 q^{19} + 5 q^{25} - 14 q^{31} - 3 q^{43} + 8 q^{49} - q^{61} - 11 q^{67} + 17 q^{73} + 17 q^{79} - 9 q^{91} + 24 q^{97}+O(q^{100})$$ 2 * q + 9 * q^13 + 8 * q^19 + 5 * q^25 - 14 * q^31 - 3 * q^43 + 8 * q^49 - q^61 - 11 * q^67 + 17 * q^73 + 17 * q^79 - 9 * q^91 + 24 * q^97

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
559.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 0 0 1.73205i 0 0 0
1855.1 0 0 0 0 0 1.73205i 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 $$\Q(\sqrt{-3})$$
76.f even 6 1 inner
228.n odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.bm.f yes 2
3.b odd 2 1 CM 2736.2.bm.f yes 2
4.b odd 2 1 2736.2.bm.e 2
12.b even 2 1 2736.2.bm.e 2
19.d odd 6 1 2736.2.bm.e 2
57.f even 6 1 2736.2.bm.e 2
76.f even 6 1 inner 2736.2.bm.f yes 2
228.n odd 6 1 inner 2736.2.bm.f yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2736.2.bm.e 2 4.b odd 2 1
2736.2.bm.e 2 12.b even 2 1
2736.2.bm.e 2 19.d odd 6 1
2736.2.bm.e 2 57.f even 6 1
2736.2.bm.f yes 2 1.a even 1 1 trivial
2736.2.bm.f yes 2 3.b odd 2 1 CM
2736.2.bm.f yes 2 76.f even 6 1 inner
2736.2.bm.f yes 2 228.n odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5}$$ T5 $$T_{7}^{2} + 3$$ T7^2 + 3 $$T_{11}$$ T11 $$T_{23}$$ T23 $$T_{31} + 7$$ T31 + 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 3$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 9T + 27$$
$17$ $$T^{2}$$
$19$ $$T^{2} - 8T + 19$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$(T + 7)^{2}$$
$37$ $$T^{2} + 27$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 3T + 3$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + T + 1$$
$67$ $$T^{2} + 11T + 121$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 17T + 289$$
$79$ $$T^{2} - 17T + 289$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} - 24T + 192$$