# Properties

 Label 2736.2.bm.k Level $2736$ Weight $2$ Character orbit 2736.bm Analytic conductor $21.847$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Learn more

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: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{12}^{2} - 1) q^{5} - 2 \zeta_{12}^{3} q^{7} +O(q^{10})$$ q + (z^2 - 1) * q^5 - 2*z^3 * q^7 $$q + (\zeta_{12}^{2} - 1) q^{5} - 2 \zeta_{12}^{3} q^{7} + 2 \zeta_{12}^{3} q^{11} + (\zeta_{12}^{2} - 2) q^{13} + ( - \zeta_{12}^{2} + 1) q^{17} + ( - 5 \zeta_{12}^{3} + 2 \zeta_{12}) q^{19} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{23} + 4 \zeta_{12}^{2} q^{25} + ( - \zeta_{12}^{2} + 2) q^{29} + (2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{31} + 2 \zeta_{12} q^{35} + ( - 4 \zeta_{12}^{2} + 2) q^{37} + ( - 3 \zeta_{12}^{2} - 3) q^{41} + 9 \zeta_{12} q^{43} + ( - 11 \zeta_{12}^{3} + 11 \zeta_{12}) q^{47} + 3 q^{49} + (5 \zeta_{12}^{2} - 10) q^{53} - 2 \zeta_{12} q^{55} + ( - 10 \zeta_{12}^{3} + 5 \zeta_{12}) q^{59} + 9 \zeta_{12}^{2} q^{61} + ( - 2 \zeta_{12}^{2} + 1) q^{65} + (3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{67} + ( - 18 \zeta_{12}^{3} + 9 \zeta_{12}) q^{71} + ( - \zeta_{12}^{2} + 1) q^{73} + 4 q^{77} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{79} + 8 \zeta_{12}^{3} q^{83} + \zeta_{12}^{2} q^{85} + ( - 9 \zeta_{12}^{2} + 18) q^{89} + (2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{91} + (2 \zeta_{12}^{3} + 3 \zeta_{12}) q^{95} + ( - 9 \zeta_{12}^{2} - 9) q^{97} +O(q^{100})$$ q + (z^2 - 1) * q^5 - 2*z^3 * q^7 + 2*z^3 * q^11 + (z^2 - 2) * q^13 + (-z^2 + 1) * q^17 + (-5*z^3 + 2*z) * q^19 + (-z^3 + z) * q^23 + 4*z^2 * q^25 + (-z^2 + 2) * q^29 + (2*z^3 - 4*z) * q^31 + 2*z * q^35 + (-4*z^2 + 2) * q^37 + (-3*z^2 - 3) * q^41 + 9*z * q^43 + (-11*z^3 + 11*z) * q^47 + 3 * q^49 + (5*z^2 - 10) * q^53 - 2*z * q^55 + (-10*z^3 + 5*z) * q^59 + 9*z^2 * q^61 + (-2*z^2 + 1) * q^65 + (3*z^3 + 3*z) * q^67 + (-18*z^3 + 9*z) * q^71 + (-z^2 + 1) * q^73 + 4 * q^77 + (2*z^3 - z) * q^79 + 8*z^3 * q^83 + z^2 * q^85 + (-9*z^2 + 18) * q^89 + (2*z^3 + 2*z) * q^91 + (2*z^3 + 3*z) * q^95 + (-9*z^2 - 9) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{5}+O(q^{10})$$ 4 * q - 2 * q^5 $$4 q - 2 q^{5} - 6 q^{13} + 2 q^{17} + 8 q^{25} + 6 q^{29} - 18 q^{41} + 12 q^{49} - 30 q^{53} + 18 q^{61} + 2 q^{73} + 16 q^{77} + 2 q^{85} + 54 q^{89} - 54 q^{97}+O(q^{100})$$ 4 * q - 2 * q^5 - 6 * q^13 + 2 * q^17 + 8 * q^25 + 6 * q^29 - 18 * q^41 + 12 * q^49 - 30 * q^53 + 18 * q^61 + 2 * q^73 + 16 * q^77 + 2 * q^85 + 54 * q^89 - 54 * q^97

## Character values

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

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$\zeta_{12}^{2}$$ $$1$$ $$-1$$ $$1$$

## 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
 0.866025 + 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 0 0 −0.500000 + 0.866025i 0 2.00000i 0 0 0
559.2 0 0 0 −0.500000 + 0.866025i 0 2.00000i 0 0 0
1855.1 0 0 0 −0.500000 0.866025i 0 2.00000i 0 0 0
1855.2 0 0 0 −0.500000 0.866025i 0 2.00000i 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
4.b odd 2 1 inner
19.d odd 6 1 inner
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.k 4
3.b odd 2 1 304.2.n.c 4
4.b odd 2 1 inner 2736.2.bm.k 4
12.b even 2 1 304.2.n.c 4
19.d odd 6 1 inner 2736.2.bm.k 4
24.f even 2 1 1216.2.n.c 4
24.h odd 2 1 1216.2.n.c 4
57.f even 6 1 304.2.n.c 4
76.f even 6 1 inner 2736.2.bm.k 4
228.n odd 6 1 304.2.n.c 4
456.s odd 6 1 1216.2.n.c 4
456.v even 6 1 1216.2.n.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.n.c 4 3.b odd 2 1
304.2.n.c 4 12.b even 2 1
304.2.n.c 4 57.f even 6 1
304.2.n.c 4 228.n odd 6 1
1216.2.n.c 4 24.f even 2 1
1216.2.n.c 4 24.h odd 2 1
1216.2.n.c 4 456.s odd 6 1
1216.2.n.c 4 456.v even 6 1
2736.2.bm.k 4 1.a even 1 1 trivial
2736.2.bm.k 4 4.b odd 2 1 inner
2736.2.bm.k 4 19.d odd 6 1 inner
2736.2.bm.k 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 $$S_{2}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5}^{2} + T_{5} + 1$$ T5^2 + T5 + 1 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11}^{2} + 4$$ T11^2 + 4 $$T_{23}^{4} - T_{23}^{2} + 1$$ T23^4 - T23^2 + 1 $$T_{31}^{2} - 12$$ T31^2 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$(T^{2} + 4)^{2}$$
$11$ $$(T^{2} + 4)^{2}$$
$13$ $$(T^{2} + 3 T + 3)^{2}$$
$17$ $$(T^{2} - T + 1)^{2}$$
$19$ $$T^{4} + 26T^{2} + 361$$
$23$ $$T^{4} - T^{2} + 1$$
$29$ $$(T^{2} - 3 T + 3)^{2}$$
$31$ $$(T^{2} - 12)^{2}$$
$37$ $$(T^{2} + 12)^{2}$$
$41$ $$(T^{2} + 9 T + 27)^{2}$$
$43$ $$T^{4} - 81T^{2} + 6561$$
$47$ $$T^{4} - 121 T^{2} + 14641$$
$53$ $$(T^{2} + 15 T + 75)^{2}$$
$59$ $$T^{4} + 75T^{2} + 5625$$
$61$ $$(T^{2} - 9 T + 81)^{2}$$
$67$ $$T^{4} + 27T^{2} + 729$$
$71$ $$T^{4} + 243 T^{2} + 59049$$
$73$ $$(T^{2} - T + 1)^{2}$$
$79$ $$T^{4} + 3T^{2} + 9$$
$83$ $$(T^{2} + 64)^{2}$$
$89$ $$(T^{2} - 27 T + 243)^{2}$$
$97$ $$(T^{2} + 27 T + 243)^{2}$$
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