# Properties

 Label 2736.2.s.bb Level $2736$ Weight $2$ Character orbit 2736.s Analytic conductor $21.847$ Analytic rank $0$ Dimension $8$ Inner twists $4$

# 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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.764411904.5 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 6x^{6} + 21x^{4} - 54x^{2} + 81$$ x^8 - 6*x^6 + 21*x^4 - 54*x^2 + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 171) 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 basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{5} + (\beta_{5} + 1) q^{7}+O(q^{10})$$ q - b1 * q^5 + (b5 + 1) * q^7 $$q - \beta_1 q^{5} + (\beta_{5} + 1) q^{7} + ( - \beta_{7} + \beta_{6} + \cdots + \beta_1) q^{11}+ \cdots + (2 \beta_{4} + 8 \beta_{3}) q^{97}+O(q^{100})$$ q - b1 * q^5 + (b5 + 1) * q^7 + (-b7 + b6 - b2 + b1) * q^11 + (b3 + 1) * q^13 + 2*b1 * q^17 + (-b5 + 2*b4 - 1) * q^19 + (b7 - b2) * q^23 + (2*b5 - 2*b4 - b3 - 1) * q^25 + (2*b7 - 2*b2) * q^29 + (b5 + 7) * q^31 + (-b6 + b1) * q^35 + (2*b5 - 1) * q^37 - 2*b6 * q^41 + (3*b4 + b3) * q^43 + 2*b7 * q^47 + 2*b5 * q^49 - b2 * q^53 + 6*b3 * q^55 + (-b6 - 3*b1) * q^59 + (-5*b3 - 5) * q^61 + (b2 - b1) * q^65 + (3*b5 - 3*b4 - 7*b3 - 7) * q^67 + (-2*b6 + 2*b1) * q^71 + 5*b3 * q^73 + (-4*b7 + 4*b6 - b2 + b1) * q^77 + (3*b4 + 7*b3) * q^79 + (2*b7 - 2*b6 + 2*b2 - 2*b1) * q^83 + (-4*b5 + 4*b4 + 12*b3 + 12) * q^85 + 5*b2 * q^89 + (b5 - b4 + b3 + 1) * q^91 + (-2*b7 + b6 + 4*b2 - b1) * q^95 + (2*b4 + 8*b3) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{7}+O(q^{10})$$ 8 * q + 8 * q^7 $$8 q + 8 q^{7} + 4 q^{13} - 8 q^{19} - 4 q^{25} + 56 q^{31} - 8 q^{37} - 4 q^{43} - 24 q^{55} - 20 q^{61} - 28 q^{67} - 20 q^{73} - 28 q^{79} + 48 q^{85} + 4 q^{91} - 32 q^{97}+O(q^{100})$$ 8 * q + 8 * q^7 + 4 * q^13 - 8 * q^19 - 4 * q^25 + 56 * q^31 - 8 * q^37 - 4 * q^43 - 24 * q^55 - 20 * q^61 - 28 * q^67 - 20 * q^73 - 28 * q^79 + 48 * q^85 + 4 * q^91 - 32 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 6x^{6} + 21x^{4} - 54x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{7} - 21\nu^{5} + 87\nu^{3} - 162\nu ) / 135$$ (-v^7 - 21*v^5 + 87*v^3 - 162*v) / 135 $$\beta_{2}$$ $$=$$ $$( \nu^{7} - 24\nu^{5} + 48\nu^{3} + 27\nu ) / 135$$ (v^7 - 24*v^5 + 48*v^3 + 27*v) / 135 $$\beta_{3}$$ $$=$$ $$( -2\nu^{6} + 3\nu^{4} - 6\nu^{2} - 9 ) / 45$$ (-2*v^6 + 3*v^4 - 6*v^2 - 9) / 45 $$\beta_{4}$$ $$=$$ $$( 2\nu^{6} - 3\nu^{4} + 51\nu^{2} - 81 ) / 45$$ (2*v^6 - 3*v^4 + 51*v^2 - 81) / 45 $$\beta_{5}$$ $$=$$ $$( -\nu^{6} + 6\nu^{4} - 12\nu^{2} + 27 ) / 9$$ (-v^6 + 6*v^4 - 12*v^2 + 27) / 9 $$\beta_{6}$$ $$=$$ $$( -8\nu^{7} + 12\nu^{5} - 114\nu^{3} + 54\nu ) / 135$$ (-8*v^7 + 12*v^5 - 114*v^3 + 54*v) / 135 $$\beta_{7}$$ $$=$$ $$( -22\nu^{7} + 78\nu^{5} - 246\nu^{3} + 486\nu ) / 135$$ (-22*v^7 + 78*v^5 - 246*v^3 + 486*v) / 135
 $$\nu$$ $$=$$ $$( \beta_{7} - 2\beta_{6} + 4\beta_{2} - 2\beta_1 ) / 6$$ (b7 - 2*b6 + 4*b2 - 2*b1) / 6 $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{3} + 2$$ b4 + b3 + 2 $$\nu^{3}$$ $$=$$ $$( \beta_{7} - 3\beta_{6} + 2\beta_1 ) / 2$$ (b7 - 3*b6 + 2*b1) / 2 $$\nu^{4}$$ $$=$$ $$2\beta_{5} + 2\beta_{4} - 3\beta_{3} - 3$$ 2*b5 + 2*b4 - 3*b3 - 3 $$\nu^{5}$$ $$=$$ $$( 2\beta_{7} - 7\beta_{6} - 10\beta_{2} + 2\beta_1 ) / 2$$ (2*b7 - 7*b6 - 10*b2 + 2*b1) / 2 $$\nu^{6}$$ $$=$$ $$3\beta_{5} - 30\beta_{3} - 15$$ 3*b5 - 30*b3 - 15 $$\nu^{7}$$ $$=$$ $$( -9\beta_{7} - 6\beta_{6} - 6\beta_{2} - 30\beta_1 ) / 2$$ (-9*b7 - 6*b6 - 6*b2 - 30*b1) / 2

## Character values

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

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$-1 - \beta_{3}$$ $$1$$ $$1$$ $$1$$

## 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
 −1.27970 + 1.16721i −1.69185 − 0.370982i 1.69185 + 0.370982i 1.27970 − 1.16721i −1.27970 − 1.16721i −1.69185 + 0.370982i 1.69185 − 0.370982i 1.27970 + 1.16721i
0 0 0 −1.65068 2.85906i 0 −1.44949 0 0 0
577.2 0 0 0 −0.524648 0.908716i 0 3.44949 0 0 0
577.3 0 0 0 0.524648 + 0.908716i 0 3.44949 0 0 0
577.4 0 0 0 1.65068 + 2.85906i 0 −1.44949 0 0 0
1873.1 0 0 0 −1.65068 + 2.85906i 0 −1.44949 0 0 0
1873.2 0 0 0 −0.524648 + 0.908716i 0 3.44949 0 0 0
1873.3 0 0 0 0.524648 0.908716i 0 3.44949 0 0 0
1873.4 0 0 0 1.65068 2.85906i 0 −1.44949 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 577.4 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 inner
19.c even 3 1 inner
57.h 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.s.bb 8
3.b odd 2 1 inner 2736.2.s.bb 8
4.b odd 2 1 171.2.f.c 8
12.b even 2 1 171.2.f.c 8
19.c even 3 1 inner 2736.2.s.bb 8
57.h odd 6 1 inner 2736.2.s.bb 8
76.f even 6 1 3249.2.a.be 4
76.g odd 6 1 171.2.f.c 8
76.g odd 6 1 3249.2.a.bd 4
228.m even 6 1 171.2.f.c 8
228.m even 6 1 3249.2.a.bd 4
228.n odd 6 1 3249.2.a.be 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.f.c 8 4.b odd 2 1
171.2.f.c 8 12.b even 2 1
171.2.f.c 8 76.g odd 6 1
171.2.f.c 8 228.m even 6 1
2736.2.s.bb 8 1.a even 1 1 trivial
2736.2.s.bb 8 3.b odd 2 1 inner
2736.2.s.bb 8 19.c even 3 1 inner
2736.2.s.bb 8 57.h odd 6 1 inner
3249.2.a.bd 4 76.g odd 6 1
3249.2.a.bd 4 228.m even 6 1
3249.2.a.be 4 76.f even 6 1
3249.2.a.be 4 228.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 $$S_{2}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5}^{8} + 12T_{5}^{6} + 132T_{5}^{4} + 144T_{5}^{2} + 144$$ T5^8 + 12*T5^6 + 132*T5^4 + 144*T5^2 + 144 $$T_{7}^{2} - 2T_{7} - 5$$ T7^2 - 2*T7 - 5 $$T_{11}^{4} - 36T_{11}^{2} + 108$$ T11^4 - 36*T11^2 + 108 $$T_{13}^{2} - T_{13} + 1$$ T13^2 - T13 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 12 T^{6} + \cdots + 144$$
$7$ $$(T^{2} - 2 T - 5)^{4}$$
$11$ $$(T^{4} - 36 T^{2} + 108)^{2}$$
$13$ $$(T^{2} - T + 1)^{4}$$
$17$ $$T^{8} + 48 T^{6} + \cdots + 36864$$
$19$ $$(T^{2} + 2 T + 19)^{4}$$
$23$ $$T^{8} + 36 T^{6} + \cdots + 90000$$
$29$ $$T^{8} + 144 T^{6} + \cdots + 23040000$$
$31$ $$(T^{2} - 14 T + 43)^{4}$$
$37$ $$(T^{2} + 2 T - 23)^{4}$$
$41$ $$T^{8} + 96 T^{6} + \cdots + 589824$$
$43$ $$(T^{4} + 2 T^{3} + \cdots + 2809)^{2}$$
$47$ $$T^{8} + 96 T^{6} + \cdots + 589824$$
$53$ $$T^{8} + 12 T^{6} + \cdots + 144$$
$59$ $$T^{8} + 132 T^{6} + \cdots + 18766224$$
$61$ $$(T^{2} + 5 T + 25)^{4}$$
$67$ $$(T^{4} + 14 T^{3} + \cdots + 25)^{2}$$
$71$ $$T^{8} + 144 T^{6} + \cdots + 23040000$$
$73$ $$(T^{2} + 5 T + 25)^{4}$$
$79$ $$(T^{4} + 14 T^{3} + \cdots + 25)^{2}$$
$83$ $$(T^{4} - 144 T^{2} + 1728)^{2}$$
$89$ $$T^{8} + 300 T^{6} + \cdots + 56250000$$
$97$ $$(T^{4} + 16 T^{3} + \cdots + 1600)^{2}$$