Properties

 Label 2736.2.bm.g Level $2736$ Weight $2$ Character orbit 2736.bm 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(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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 304) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + ( - 3 \zeta_{6} + 3) q^{5} + ( - 4 \zeta_{6} + 2) q^{7}+O(q^{10})$$ q + (-3*z + 3) * q^5 + (-4*z + 2) * q^7 $$q + ( - 3 \zeta_{6} + 3) q^{5} + ( - 4 \zeta_{6} + 2) q^{7} + ( - 4 \zeta_{6} + 2) q^{11} + (3 \zeta_{6} - 6) q^{13} + ( - 3 \zeta_{6} + 3) q^{17} + ( - 2 \zeta_{6} - 3) q^{19} + ( - 3 \zeta_{6} + 6) q^{23} - 4 \zeta_{6} q^{25} + (5 \zeta_{6} - 10) q^{29} + 4 q^{31} + ( - 6 \zeta_{6} - 6) q^{35} + ( - 5 \zeta_{6} - 5) q^{41} + (7 \zeta_{6} + 7) q^{43} + (\zeta_{6} - 2) q^{47} - 5 q^{49} + (\zeta_{6} - 2) q^{53} + ( - 6 \zeta_{6} - 6) q^{55} + (3 \zeta_{6} - 3) q^{59} - 7 \zeta_{6} q^{61} + (18 \zeta_{6} - 9) q^{65} + 5 \zeta_{6} q^{67} + ( - 9 \zeta_{6} + 9) q^{71} + (7 \zeta_{6} - 7) q^{73} - 12 q^{77} + ( - 7 \zeta_{6} + 7) q^{79} + (4 \zeta_{6} - 2) q^{83} - 9 \zeta_{6} q^{85} + (5 \zeta_{6} - 10) q^{89} + 18 \zeta_{6} q^{91} + (9 \zeta_{6} - 15) q^{95} + (5 \zeta_{6} + 5) q^{97} +O(q^{100})$$ q + (-3*z + 3) * q^5 + (-4*z + 2) * q^7 + (-4*z + 2) * q^11 + (3*z - 6) * q^13 + (-3*z + 3) * q^17 + (-2*z - 3) * q^19 + (-3*z + 6) * q^23 - 4*z * q^25 + (5*z - 10) * q^29 + 4 * q^31 + (-6*z - 6) * q^35 + (-5*z - 5) * q^41 + (7*z + 7) * q^43 + (z - 2) * q^47 - 5 * q^49 + (z - 2) * q^53 + (-6*z - 6) * q^55 + (3*z - 3) * q^59 - 7*z * q^61 + (18*z - 9) * q^65 + 5*z * q^67 + (-9*z + 9) * q^71 + (7*z - 7) * q^73 - 12 * q^77 + (-7*z + 7) * q^79 + (4*z - 2) * q^83 - 9*z * q^85 + (5*z - 10) * q^89 + 18*z * q^91 + (9*z - 15) * q^95 + (5*z + 5) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{5}+O(q^{10})$$ 2 * q + 3 * q^5 $$2 q + 3 q^{5} - 9 q^{13} + 3 q^{17} - 8 q^{19} + 9 q^{23} - 4 q^{25} - 15 q^{29} + 8 q^{31} - 18 q^{35} - 15 q^{41} + 21 q^{43} - 3 q^{47} - 10 q^{49} - 3 q^{53} - 18 q^{55} - 3 q^{59} - 7 q^{61} + 5 q^{67} + 9 q^{71} - 7 q^{73} - 24 q^{77} + 7 q^{79} - 9 q^{85} - 15 q^{89} + 18 q^{91} - 21 q^{95} + 15 q^{97}+O(q^{100})$$ 2 * q + 3 * q^5 - 9 * q^13 + 3 * q^17 - 8 * q^19 + 9 * q^23 - 4 * q^25 - 15 * q^29 + 8 * q^31 - 18 * q^35 - 15 * q^41 + 21 * q^43 - 3 * q^47 - 10 * q^49 - 3 * q^53 - 18 * q^55 - 3 * q^59 - 7 * q^61 + 5 * q^67 + 9 * q^71 - 7 * q^73 - 24 * q^77 + 7 * q^79 - 9 * q^85 - 15 * q^89 + 18 * q^91 - 21 * q^95 + 15 * 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 1.50000 2.59808i 0 3.46410i 0 0 0
1855.1 0 0 0 1.50000 + 2.59808i 0 3.46410i 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
76.f even 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.bm.g 2
3.b odd 2 1 304.2.n.a 2
4.b odd 2 1 2736.2.bm.h 2
12.b even 2 1 304.2.n.b yes 2
19.d odd 6 1 2736.2.bm.h 2
24.f even 2 1 1216.2.n.a 2
24.h odd 2 1 1216.2.n.b 2
57.f even 6 1 304.2.n.b yes 2
76.f even 6 1 inner 2736.2.bm.g 2
228.n odd 6 1 304.2.n.a 2
456.s odd 6 1 1216.2.n.b 2
456.v even 6 1 1216.2.n.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.n.a 2 3.b odd 2 1
304.2.n.a 2 228.n odd 6 1
304.2.n.b yes 2 12.b even 2 1
304.2.n.b yes 2 57.f even 6 1
1216.2.n.a 2 24.f even 2 1
1216.2.n.a 2 456.v even 6 1
1216.2.n.b 2 24.h odd 2 1
1216.2.n.b 2 456.s odd 6 1
2736.2.bm.g 2 1.a even 1 1 trivial
2736.2.bm.g 2 76.f even 6 1 inner
2736.2.bm.h 2 4.b odd 2 1
2736.2.bm.h 2 19.d odd 6 1

Hecke kernels

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

 $$T_{5}^{2} - 3T_{5} + 9$$ T5^2 - 3*T5 + 9 $$T_{7}^{2} + 12$$ T7^2 + 12 $$T_{11}^{2} + 12$$ T11^2 + 12 $$T_{23}^{2} - 9T_{23} + 27$$ T23^2 - 9*T23 + 27 $$T_{31} - 4$$ T31 - 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 3T + 9$$
$7$ $$T^{2} + 12$$
$11$ $$T^{2} + 12$$
$13$ $$T^{2} + 9T + 27$$
$17$ $$T^{2} - 3T + 9$$
$19$ $$T^{2} + 8T + 19$$
$23$ $$T^{2} - 9T + 27$$
$29$ $$T^{2} + 15T + 75$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2} + 15T + 75$$
$43$ $$T^{2} - 21T + 147$$
$47$ $$T^{2} + 3T + 3$$
$53$ $$T^{2} + 3T + 3$$
$59$ $$T^{2} + 3T + 9$$
$61$ $$T^{2} + 7T + 49$$
$67$ $$T^{2} - 5T + 25$$
$71$ $$T^{2} - 9T + 81$$
$73$ $$T^{2} + 7T + 49$$
$79$ $$T^{2} - 7T + 49$$
$83$ $$T^{2} + 12$$
$89$ $$T^{2} + 15T + 75$$
$97$ $$T^{2} - 15T + 75$$