Properties

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

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(1025,2736)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

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

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

chi := DirichletCharacter("2736.1025");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.f (of order $$2$$, degree $$1$$, 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: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 6x^{2} + 4$$ x^4 + 6*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 171) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{5} + 2 q^{7}+O(q^{10})$$ q - b1 * q^5 + 2 * q^7 $$q - \beta_1 q^{5} + 2 q^{7} - \beta_1 q^{11} + 2 \beta_{2} q^{13} - 3 \beta_1 q^{17} + ( - \beta_{2} + 3) q^{19} + 5 \beta_1 q^{23} + 3 q^{25} + \beta_{3} q^{29} + 2 \beta_{2} q^{31} - 2 \beta_1 q^{35} - \beta_{3} q^{41} - 4 q^{43} - 5 \beta_1 q^{47} - 3 q^{49} + 3 \beta_{3} q^{53} - 2 q^{55} + 2 \beta_{3} q^{59} + 8 q^{61} + 2 \beta_{3} q^{65} + 2 \beta_{2} q^{67} + 6 q^{73} - 2 \beta_1 q^{77} - 4 \beta_{2} q^{79} - \beta_1 q^{83} - 6 q^{85} - 3 \beta_{3} q^{89} + 4 \beta_{2} q^{91} + ( - \beta_{3} - 3 \beta_1) q^{95} + 2 \beta_{2} q^{97}+O(q^{100})$$ q - b1 * q^5 + 2 * q^7 - b1 * q^11 + 2*b2 * q^13 - 3*b1 * q^17 + (-b2 + 3) * q^19 + 5*b1 * q^23 + 3 * q^25 + b3 * q^29 + 2*b2 * q^31 - 2*b1 * q^35 - b3 * q^41 - 4 * q^43 - 5*b1 * q^47 - 3 * q^49 + 3*b3 * q^53 - 2 * q^55 + 2*b3 * q^59 + 8 * q^61 + 2*b3 * q^65 + 2*b2 * q^67 + 6 * q^73 - 2*b1 * q^77 - 4*b2 * q^79 - b1 * q^83 - 6 * q^85 - 3*b3 * q^89 + 4*b2 * q^91 + (-b3 - 3*b1) * q^95 + 2*b2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{7}+O(q^{10})$$ 4 * q + 8 * q^7 $$4 q + 8 q^{7} + 12 q^{19} + 12 q^{25} - 16 q^{43} - 12 q^{49} - 8 q^{55} + 32 q^{61} + 24 q^{73} - 24 q^{85}+O(q^{100})$$ 4 * q + 8 * q^7 + 12 * q^19 + 12 * q^25 - 16 * q^43 - 12 * q^49 - 8 * q^55 + 32 * q^61 + 24 * q^73 - 24 * q^85

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 4\nu ) / 2$$ (v^3 + 4*v) / 2 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 8\nu ) / 2$$ (v^3 + 8*v) / 2 $$\beta_{3}$$ $$=$$ $$2\nu^{2} + 6$$ 2*v^2 + 6
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 6 ) / 2$$ (b3 - 6) / 2 $$\nu^{3}$$ $$=$$ $$-2\beta_{2} + 4\beta_1$$ -2*b2 + 4*b1

Character values

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

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1025.1
 − 2.28825i 0.874032i − 0.874032i 2.28825i
0 0 0 1.41421i 0 2.00000 0 0 0
1025.2 0 0 0 1.41421i 0 2.00000 0 0 0
1025.3 0 0 0 1.41421i 0 2.00000 0 0 0
1025.4 0 0 0 1.41421i 0 2.00000 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
3.b odd 2 1 inner
19.b odd 2 1 inner
57.d even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.f.f 4
3.b odd 2 1 inner 2736.2.f.f 4
4.b odd 2 1 171.2.d.b 4
12.b even 2 1 171.2.d.b 4
19.b odd 2 1 inner 2736.2.f.f 4
57.d even 2 1 inner 2736.2.f.f 4
76.d even 2 1 171.2.d.b 4
228.b odd 2 1 171.2.d.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.d.b 4 4.b odd 2 1
171.2.d.b 4 12.b even 2 1
171.2.d.b 4 76.d even 2 1
171.2.d.b 4 228.b odd 2 1
2736.2.f.f 4 1.a even 1 1 trivial
2736.2.f.f 4 3.b odd 2 1 inner
2736.2.f.f 4 19.b odd 2 1 inner
2736.2.f.f 4 57.d even 2 1 inner

Hecke kernels

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

 $$T_{5}^{2} + 2$$ T5^2 + 2 $$T_{7} - 2$$ T7 - 2 $$T_{11}^{2} + 2$$ T11^2 + 2 $$T_{29}^{2} - 20$$ T29^2 - 20

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 2)^{2}$$
$7$ $$(T - 2)^{4}$$
$11$ $$(T^{2} + 2)^{2}$$
$13$ $$(T^{2} + 40)^{2}$$
$17$ $$(T^{2} + 18)^{2}$$
$19$ $$(T^{2} - 6 T + 19)^{2}$$
$23$ $$(T^{2} + 50)^{2}$$
$29$ $$(T^{2} - 20)^{2}$$
$31$ $$(T^{2} + 40)^{2}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} - 20)^{2}$$
$43$ $$(T + 4)^{4}$$
$47$ $$(T^{2} + 50)^{2}$$
$53$ $$(T^{2} - 180)^{2}$$
$59$ $$(T^{2} - 80)^{2}$$
$61$ $$(T - 8)^{4}$$
$67$ $$(T^{2} + 40)^{2}$$
$71$ $$T^{4}$$
$73$ $$(T - 6)^{4}$$
$79$ $$(T^{2} + 160)^{2}$$
$83$ $$(T^{2} + 2)^{2}$$
$89$ $$(T^{2} - 180)^{2}$$
$97$ $$(T^{2} + 40)^{2}$$