# Properties

 Label 2736.2.bm.j Level $2736$ Weight $2$ Character orbit 2736.bm Analytic conductor $21.847$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 912) 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \beta_{2} - 2) q^{5} + ( - \beta_{3} + 2 \beta_{2} - 1) q^{7}+O(q^{10})$$ q + (2*b2 - 2) * q^5 + (-b3 + 2*b2 - 1) * q^7 $$q + (2 \beta_{2} - 2) q^{5} + ( - \beta_{3} + 2 \beta_{2} - 1) q^{7} - 2 \beta_{3} q^{11} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{13} + (2 \beta_{2} - 2) q^{17} + (2 \beta_{2} + 3) q^{19} + (2 \beta_{3} - 2 \beta_1) q^{23} + \beta_{2} q^{25} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{29} + (\beta_{3} - 2 \beta_1 + 3) q^{31} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{35} + ( - \beta_{3} - 2 \beta_{2} + 1) q^{37} + (3 \beta_{2} + 3) q^{43} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 4) q^{47} + ( - 2 \beta_{3} + 4 \beta_1 - 4) q^{49} + ( - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 8) q^{53} + 4 \beta_1 q^{55} + (4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{59} + ( - \beta_{3} + 5 \beta_{2} - \beta_1) q^{61} + ( - 2 \beta_{3} + 4 \beta_{2} - 2) q^{65} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{67} + (4 \beta_{2} - 4) q^{71} + (4 \beta_{3} + 3 \beta_{2} - 2 \beta_1 - 3) q^{73} + ( - 2 \beta_{3} + 4 \beta_1 - 16) q^{77} + (2 \beta_{3} + 5 \beta_{2} - \beta_1 - 5) q^{79} + ( - 4 \beta_{2} + 2) q^{83} - 4 \beta_{2} q^{85} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{89} + ( - 2 \beta_{3} + 11 \beta_{2} - 2 \beta_1) q^{91} + (6 \beta_{2} - 10) q^{95} + ( - 4 \beta_{2} - 4 \beta_1 - 4) q^{97}+O(q^{100})$$ q + (2*b2 - 2) * q^5 + (-b3 + 2*b2 - 1) * q^7 - 2*b3 * q^11 + (b3 - b2 - b1 + 2) * q^13 + (2*b2 - 2) * q^17 + (2*b2 + 3) * q^19 + (2*b3 - 2*b1) * q^23 + b2 * q^25 + (2*b3 + 2*b2 - 2*b1 - 4) * q^29 + (b3 - 2*b1 + 3) * q^31 + (-2*b2 + 2*b1 - 2) * q^35 + (-b3 - 2*b2 + 1) * q^37 + (3*b2 + 3) * q^43 + (-2*b3 + 2*b2 + 2*b1 - 4) * q^47 + (-2*b3 + 4*b1 - 4) * q^49 + (-2*b3 + 4*b2 + 2*b1 - 8) * q^53 + 4*b1 * q^55 + (4*b3 - 2*b2 - 2*b1 + 2) * q^59 + (-b3 + 5*b2 - b1) * q^61 + (-2*b3 + 4*b2 - 2) * q^65 + (2*b3 + b2 + 2*b1) * q^67 + (4*b2 - 4) * q^71 + (4*b3 + 3*b2 - 2*b1 - 3) * q^73 + (-2*b3 + 4*b1 - 16) * q^77 + (2*b3 + 5*b2 - b1 - 5) * q^79 + (-4*b2 + 2) * q^83 - 4*b2 * q^85 + (2*b3 + 2*b2 - 2*b1 - 4) * q^89 + (-2*b3 + 11*b2 - 2*b1) * q^91 + (6*b2 - 10) * q^95 + (-4*b2 - 4*b1 - 4) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{5}+O(q^{10})$$ 4 * q - 4 * q^5 $$4 q - 4 q^{5} + 6 q^{13} - 4 q^{17} + 16 q^{19} + 2 q^{25} - 12 q^{29} + 12 q^{31} - 12 q^{35} + 18 q^{43} - 12 q^{47} - 16 q^{49} - 24 q^{53} + 4 q^{59} + 10 q^{61} + 2 q^{67} - 8 q^{71} - 6 q^{73} - 64 q^{77} - 10 q^{79} - 8 q^{85} - 12 q^{89} + 22 q^{91} - 28 q^{95} - 24 q^{97}+O(q^{100})$$ 4 * q - 4 * q^5 + 6 * q^13 - 4 * q^17 + 16 * q^19 + 2 * q^25 - 12 * q^29 + 12 * q^31 - 12 * q^35 + 18 * q^43 - 12 * q^47 - 16 * q^49 - 24 * q^53 + 4 * q^59 + 10 * q^61 + 2 * q^67 - 8 * q^71 - 6 * q^73 - 64 * q^77 - 10 * q^79 - 8 * q^85 - 12 * q^89 + 22 * q^91 - 28 * q^95 - 24 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{3}$$ v^3
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$\beta_{3}$$ b3

## Character values

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

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$ $$-1$$ $$1$$

## 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
 1.22474 + 0.707107i −1.22474 − 0.707107i −1.22474 + 0.707107i 1.22474 − 0.707107i
0 0 0 −1.00000 + 1.73205i 0 1.09638i 0 0 0
559.2 0 0 0 −1.00000 + 1.73205i 0 4.56048i 0 0 0
1855.1 0 0 0 −1.00000 1.73205i 0 4.56048i 0 0 0
1855.2 0 0 0 −1.00000 1.73205i 0 1.09638i 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.j 4
3.b odd 2 1 912.2.bb.c 4
4.b odd 2 1 2736.2.bm.i 4
12.b even 2 1 912.2.bb.d yes 4
19.d odd 6 1 2736.2.bm.i 4
57.f even 6 1 912.2.bb.d yes 4
76.f even 6 1 inner 2736.2.bm.j 4
228.n odd 6 1 912.2.bb.c 4

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

## Hecke kernels

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

 $$T_{5}^{2} + 2T_{5} + 4$$ T5^2 + 2*T5 + 4 $$T_{7}^{4} + 22T_{7}^{2} + 25$$ T7^4 + 22*T7^2 + 25 $$T_{11}^{2} + 32$$ T11^2 + 32 $$T_{23}^{4} - 32T_{23}^{2} + 1024$$ T23^4 - 32*T23^2 + 1024 $$T_{31}^{2} - 6T_{31} - 15$$ T31^2 - 6*T31 - 15

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 2 T + 4)^{2}$$
$7$ $$T^{4} + 22T^{2} + 25$$
$11$ $$(T^{2} + 32)^{2}$$
$13$ $$T^{4} - 6 T^{3} + 7 T^{2} + 30 T + 25$$
$17$ $$(T^{2} + 2 T + 4)^{2}$$
$19$ $$(T^{2} - 8 T + 19)^{2}$$
$23$ $$T^{4} - 32T^{2} + 1024$$
$29$ $$T^{4} + 12 T^{3} + 28 T^{2} + \cdots + 400$$
$31$ $$(T^{2} - 6 T - 15)^{2}$$
$37$ $$T^{4} + 22T^{2} + 25$$
$41$ $$T^{4}$$
$43$ $$(T^{2} - 9 T + 27)^{2}$$
$47$ $$T^{4} + 12 T^{3} + 28 T^{2} + \cdots + 400$$
$53$ $$T^{4} + 24 T^{3} + 208 T^{2} + \cdots + 256$$
$59$ $$T^{4} - 4 T^{3} + 108 T^{2} + \cdots + 8464$$
$61$ $$T^{4} - 10 T^{3} + 99 T^{2} - 10 T + 1$$
$67$ $$T^{4} - 2 T^{3} + 99 T^{2} + \cdots + 9025$$
$71$ $$(T^{2} + 4 T + 16)^{2}$$
$73$ $$T^{4} + 6 T^{3} + 123 T^{2} + \cdots + 7569$$
$79$ $$T^{4} + 10 T^{3} + 99 T^{2} + 10 T + 1$$
$83$ $$(T^{2} + 12)^{2}$$
$89$ $$T^{4} + 12 T^{3} + 28 T^{2} + \cdots + 400$$
$97$ $$T^{4} + 24 T^{3} + 112 T^{2} + \cdots + 6400$$