# Properties

 Label 2700.2.a.w Level $2700$ Weight $2$ Character orbit 2700.a Self dual yes Analytic conductor $21.560$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2700,2,Mod(1,2700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2700, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2700.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2700 = 2^{2} \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2700.a (trivial)

## 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: yes Analytic conductor: $$21.5596085457$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 10$$ x^2 - 10 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 540) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = \sqrt{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{7}+O(q^{10})$$ q + q^7 $$q + q^{7} - \beta q^{11} + 3 q^{13} - 2 \beta q^{17} + 3 q^{19} + \beta q^{23} + 3 \beta q^{29} - 2 q^{31} + q^{37} - \beta q^{41} + 10 q^{43} + 2 \beta q^{47} - 6 q^{49} + 3 \beta q^{53} - 2 \beta q^{59} - q^{61} + 11 q^{67} + 3 \beta q^{71} + 13 q^{73} - \beta q^{77} - 3 q^{79} - 5 \beta q^{83} - 4 \beta q^{89} + 3 q^{91} - q^{97} +O(q^{100})$$ q + q^7 - b * q^11 + 3 * q^13 - 2*b * q^17 + 3 * q^19 + b * q^23 + 3*b * q^29 - 2 * q^31 + q^37 - b * q^41 + 10 * q^43 + 2*b * q^47 - 6 * q^49 + 3*b * q^53 - 2*b * q^59 - q^61 + 11 * q^67 + 3*b * q^71 + 13 * q^73 - b * q^77 - 3 * q^79 - 5*b * q^83 - 4*b * q^89 + 3 * q^91 - q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{7}+O(q^{10})$$ 2 * q + 2 * q^7 $$2 q + 2 q^{7} + 6 q^{13} + 6 q^{19} - 4 q^{31} + 2 q^{37} + 20 q^{43} - 12 q^{49} - 2 q^{61} + 22 q^{67} + 26 q^{73} - 6 q^{79} + 6 q^{91} - 2 q^{97}+O(q^{100})$$ 2 * q + 2 * q^7 + 6 * q^13 + 6 * q^19 - 4 * q^31 + 2 * q^37 + 20 * q^43 - 12 * q^49 - 2 * q^61 + 22 * q^67 + 26 * q^73 - 6 * q^79 + 6 * q^91 - 2 * q^97

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 3.16228 −3.16228
0 0 0 0 0 1.00000 0 0 0
1.2 0 0 0 0 0 1.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2700.2.a.w 2
3.b odd 2 1 inner 2700.2.a.w 2
5.b even 2 1 2700.2.a.v 2
5.c odd 4 2 540.2.d.c 4
15.d odd 2 1 2700.2.a.v 2
15.e even 4 2 540.2.d.c 4
20.e even 4 2 2160.2.f.l 4
45.k odd 12 4 1620.2.r.g 8
45.l even 12 4 1620.2.r.g 8
60.l odd 4 2 2160.2.f.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.2.d.c 4 5.c odd 4 2
540.2.d.c 4 15.e even 4 2
1620.2.r.g 8 45.k odd 12 4
1620.2.r.g 8 45.l even 12 4
2160.2.f.l 4 20.e even 4 2
2160.2.f.l 4 60.l odd 4 2
2700.2.a.v 2 5.b even 2 1
2700.2.a.v 2 15.d odd 2 1
2700.2.a.w 2 1.a even 1 1 trivial
2700.2.a.w 2 3.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2700))$$:

 $$T_{7} - 1$$ T7 - 1 $$T_{11}^{2} - 10$$ T11^2 - 10 $$T_{13} - 3$$ T13 - 3 $$T_{17}^{2} - 40$$ T17^2 - 40

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2} - 10$$
$13$ $$(T - 3)^{2}$$
$17$ $$T^{2} - 40$$
$19$ $$(T - 3)^{2}$$
$23$ $$T^{2} - 10$$
$29$ $$T^{2} - 90$$
$31$ $$(T + 2)^{2}$$
$37$ $$(T - 1)^{2}$$
$41$ $$T^{2} - 10$$
$43$ $$(T - 10)^{2}$$
$47$ $$T^{2} - 40$$
$53$ $$T^{2} - 90$$
$59$ $$T^{2} - 40$$
$61$ $$(T + 1)^{2}$$
$67$ $$(T - 11)^{2}$$
$71$ $$T^{2} - 90$$
$73$ $$(T - 13)^{2}$$
$79$ $$(T + 3)^{2}$$
$83$ $$T^{2} - 250$$
$89$ $$T^{2} - 160$$
$97$ $$(T + 1)^{2}$$