# Properties

 Label 2736.2.s.f Level $2736$ Weight $2$ Character orbit 2736.s Analytic conductor $21.847$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(577,2736)]

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, 2]))

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

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

chi := DirichletCharacter("2736.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.s (of order $$3$$, 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: $$21.8470699930$$ 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: no (minimal twist has level 456) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{5} + 5 q^{7}+O(q^{10})$$ q + (2*z - 2) * q^5 + 5 * q^7 $$q + (2 \zeta_{6} - 2) q^{5} + 5 q^{7} - 4 q^{11} - 5 \zeta_{6} q^{13} + (2 \zeta_{6} - 5) q^{19} + 6 \zeta_{6} q^{23} + \zeta_{6} q^{25} + 8 \zeta_{6} q^{29} + q^{31} + (10 \zeta_{6} - 10) q^{35} + 7 q^{37} + (11 \zeta_{6} - 11) q^{43} - 10 \zeta_{6} q^{47} + 18 q^{49} + 6 \zeta_{6} q^{53} + ( - 8 \zeta_{6} + 8) q^{55} + (8 \zeta_{6} - 8) q^{59} + \zeta_{6} q^{61} + 10 q^{65} + 5 \zeta_{6} q^{67} + (6 \zeta_{6} - 6) q^{71} + (\zeta_{6} - 1) q^{73} - 20 q^{77} + (13 \zeta_{6} - 13) q^{79} - 4 q^{83} - 12 \zeta_{6} q^{89} - 25 \zeta_{6} q^{91} + ( - 10 \zeta_{6} + 6) q^{95} + (2 \zeta_{6} - 2) q^{97} +O(q^{100})$$ q + (2*z - 2) * q^5 + 5 * q^7 - 4 * q^11 - 5*z * q^13 + (2*z - 5) * q^19 + 6*z * q^23 + z * q^25 + 8*z * q^29 + q^31 + (10*z - 10) * q^35 + 7 * q^37 + (11*z - 11) * q^43 - 10*z * q^47 + 18 * q^49 + 6*z * q^53 + (-8*z + 8) * q^55 + (8*z - 8) * q^59 + z * q^61 + 10 * q^65 + 5*z * q^67 + (6*z - 6) * q^71 + (z - 1) * q^73 - 20 * q^77 + (13*z - 13) * q^79 - 4 * q^83 - 12*z * q^89 - 25*z * q^91 + (-10*z + 6) * q^95 + (2*z - 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} + 10 q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 + 10 * q^7 $$2 q - 2 q^{5} + 10 q^{7} - 8 q^{11} - 5 q^{13} - 8 q^{19} + 6 q^{23} + q^{25} + 8 q^{29} + 2 q^{31} - 10 q^{35} + 14 q^{37} - 11 q^{43} - 10 q^{47} + 36 q^{49} + 6 q^{53} + 8 q^{55} - 8 q^{59} + q^{61} + 20 q^{65} + 5 q^{67} - 6 q^{71} - q^{73} - 40 q^{77} - 13 q^{79} - 8 q^{83} - 12 q^{89} - 25 q^{91} + 2 q^{95} - 2 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 + 10 * q^7 - 8 * q^11 - 5 * q^13 - 8 * q^19 + 6 * q^23 + q^25 + 8 * q^29 + 2 * q^31 - 10 * q^35 + 14 * q^37 - 11 * q^43 - 10 * q^47 + 36 * q^49 + 6 * q^53 + 8 * q^55 - 8 * q^59 + q^61 + 20 * q^65 + 5 * q^67 - 6 * q^71 - q^73 - 40 * q^77 - 13 * q^79 - 8 * q^83 - 12 * q^89 - 25 * q^91 + 2 * q^95 - 2 * 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 −1.00000 1.73205i 0 5.00000 0 0 0
1873.1 0 0 0 −1.00000 + 1.73205i 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.s.f 2
3.b odd 2 1 912.2.q.c 2
4.b odd 2 1 1368.2.s.b 2
12.b even 2 1 456.2.q.c 2
19.c even 3 1 inner 2736.2.s.f 2
57.h odd 6 1 912.2.q.c 2
76.g odd 6 1 1368.2.s.b 2
228.m even 6 1 456.2.q.c 2
228.m even 6 1 8664.2.a.b 1
228.n odd 6 1 8664.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.q.c 2 12.b even 2 1
456.2.q.c 2 228.m even 6 1
912.2.q.c 2 3.b odd 2 1
912.2.q.c 2 57.h odd 6 1
1368.2.s.b 2 4.b odd 2 1
1368.2.s.b 2 76.g odd 6 1
2736.2.s.f 2 1.a even 1 1 trivial
2736.2.s.f 2 19.c even 3 1 inner
8664.2.a.b 1 228.m even 6 1
8664.2.a.h 1 228.n odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} + 2T_{5} + 4$$ T5^2 + 2*T5 + 4 $$T_{7} - 5$$ T7 - 5 $$T_{11} + 4$$ T11 + 4 $$T_{13}^{2} + 5T_{13} + 25$$ T13^2 + 5*T13 + 25

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2T + 4$$
$7$ $$(T - 5)^{2}$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} + 5T + 25$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 8T + 19$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$T^{2} - 8T + 64$$
$31$ $$(T - 1)^{2}$$
$37$ $$(T - 7)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 11T + 121$$
$47$ $$T^{2} + 10T + 100$$
$53$ $$T^{2} - 6T + 36$$
$59$ $$T^{2} + 8T + 64$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} - 5T + 25$$
$71$ $$T^{2} + 6T + 36$$
$73$ $$T^{2} + T + 1$$
$79$ $$T^{2} + 13T + 169$$
$83$ $$(T + 4)^{2}$$
$89$ $$T^{2} + 12T + 144$$
$97$ $$T^{2} + 2T + 4$$