# Properties

 Label 2736.2.f.b Level $2736$ Weight $2$ Character orbit 2736.f 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(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: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 342) 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 $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} - 2 q^{7}+O(q^{10})$$ q + b * q^5 - 2 * q^7 $$q + \beta q^{5} - 2 q^{7} - \beta q^{11} + \beta q^{17} + ( - 3 \beta - 1) q^{19} - \beta q^{23} + 3 q^{25} + 6 q^{29} - 2 \beta q^{35} + 6 \beta q^{37} + 6 q^{41} + 4 q^{43} + 5 \beta q^{47} - 3 q^{49} + 6 q^{53} + 2 q^{55} - 4 q^{61} + 6 \beta q^{67} + 12 q^{71} - 10 q^{73} + 2 \beta q^{77} + 6 \beta q^{79} + 11 \beta q^{83} - 2 q^{85} + 6 q^{89} + ( - \beta + 6) q^{95} - 6 \beta q^{97} +O(q^{100})$$ q + b * q^5 - 2 * q^7 - b * q^11 + b * q^17 + (-3*b - 1) * q^19 - b * q^23 + 3 * q^25 + 6 * q^29 - 2*b * q^35 + 6*b * q^37 + 6 * q^41 + 4 * q^43 + 5*b * q^47 - 3 * q^49 + 6 * q^53 + 2 * q^55 - 4 * q^61 + 6*b * q^67 + 12 * q^71 - 10 * q^73 + 2*b * q^77 + 6*b * q^79 + 11*b * q^83 - 2 * q^85 + 6 * q^89 + (-b + 6) * q^95 - 6*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{7}+O(q^{10})$$ 2 * q - 4 * q^7 $$2 q - 4 q^{7} - 2 q^{19} + 6 q^{25} + 12 q^{29} + 12 q^{41} + 8 q^{43} - 6 q^{49} + 12 q^{53} + 4 q^{55} - 8 q^{61} + 24 q^{71} - 20 q^{73} - 4 q^{85} + 12 q^{89} + 12 q^{95}+O(q^{100})$$ 2 * q - 4 * q^7 - 2 * q^19 + 6 * q^25 + 12 * q^29 + 12 * q^41 + 8 * q^43 - 6 * q^49 + 12 * q^53 + 4 * q^55 - 8 * q^61 + 24 * q^71 - 20 * q^73 - 4 * q^85 + 12 * q^89 + 12 * q^95

## 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
 − 1.41421i 1.41421i
0 0 0 1.41421i 0 −2.00000 0 0 0
1025.2 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
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.b 2
3.b odd 2 1 2736.2.f.a 2
4.b odd 2 1 342.2.b.a 2
12.b even 2 1 342.2.b.b yes 2
19.b odd 2 1 2736.2.f.a 2
57.d even 2 1 inner 2736.2.f.b 2
76.d even 2 1 342.2.b.b yes 2
228.b odd 2 1 342.2.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.2.b.a 2 4.b odd 2 1
342.2.b.a 2 228.b odd 2 1
342.2.b.b yes 2 12.b even 2 1
342.2.b.b yes 2 76.d even 2 1
2736.2.f.a 2 3.b odd 2 1
2736.2.f.a 2 19.b odd 2 1
2736.2.f.b 2 1.a even 1 1 trivial
2736.2.f.b 2 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 $$T_{29} - 6$$ T29 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2$$
$7$ $$(T + 2)^{2}$$
$11$ $$T^{2} + 2$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 2$$
$19$ $$T^{2} + 2T + 19$$
$23$ $$T^{2} + 2$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 72$$
$41$ $$(T - 6)^{2}$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} + 50$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2}$$
$61$ $$(T + 4)^{2}$$
$67$ $$T^{2} + 72$$
$71$ $$(T - 12)^{2}$$
$73$ $$(T + 10)^{2}$$
$79$ $$T^{2} + 72$$
$83$ $$T^{2} + 242$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 72$$