# Properties

 Label 540.2.d.a Level $540$ Weight $2$ Character orbit 540.d Analytic conductor $4.312$ 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] = [540,2,Mod(109,540)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(540, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("540.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$540 = 2^{2} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 540.d (of order $$2$$, degree $$1$$, 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: $$4.31192170915$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (i - 2) q^{5} - 2 i q^{7} +O(q^{10})$$ q + (i - 2) * q^5 - 2*i * q^7 $$q + (i - 2) q^{5} - 2 i q^{7} + 2 q^{11} - 6 i q^{13} + i q^{17} + 3 q^{19} - 7 i q^{23} + ( - 4 i + 3) q^{25} + 6 q^{29} + q^{31} + (4 i + 2) q^{35} - 8 i q^{37} - 10 q^{41} + 2 i q^{43} + 8 i q^{47} + 3 q^{49} - 9 i q^{53} + (2 i - 4) q^{55} - 10 q^{59} + 5 q^{61} + (12 i + 6) q^{65} + 8 i q^{67} + 6 q^{71} + 2 i q^{73} - 4 i q^{77} - 15 q^{79} - 13 i q^{83} + ( - 2 i - 1) q^{85} - 2 q^{89} - 12 q^{91} + (3 i - 6) q^{95} + 8 i q^{97} +O(q^{100})$$ q + (i - 2) * q^5 - 2*i * q^7 + 2 * q^11 - 6*i * q^13 + i * q^17 + 3 * q^19 - 7*i * q^23 + (-4*i + 3) * q^25 + 6 * q^29 + q^31 + (4*i + 2) * q^35 - 8*i * q^37 - 10 * q^41 + 2*i * q^43 + 8*i * q^47 + 3 * q^49 - 9*i * q^53 + (2*i - 4) * q^55 - 10 * q^59 + 5 * q^61 + (12*i + 6) * q^65 + 8*i * q^67 + 6 * q^71 + 2*i * q^73 - 4*i * q^77 - 15 * q^79 - 13*i * q^83 + (-2*i - 1) * q^85 - 2 * q^89 - 12 * q^91 + (3*i - 6) * q^95 + 8*i * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{5}+O(q^{10})$$ 2 * q - 4 * q^5 $$2 q - 4 q^{5} + 4 q^{11} + 6 q^{19} + 6 q^{25} + 12 q^{29} + 2 q^{31} + 4 q^{35} - 20 q^{41} + 6 q^{49} - 8 q^{55} - 20 q^{59} + 10 q^{61} + 12 q^{65} + 12 q^{71} - 30 q^{79} - 2 q^{85} - 4 q^{89} - 24 q^{91} - 12 q^{95}+O(q^{100})$$ 2 * q - 4 * q^5 + 4 * q^11 + 6 * q^19 + 6 * q^25 + 12 * q^29 + 2 * q^31 + 4 * q^35 - 20 * q^41 + 6 * q^49 - 8 * q^55 - 20 * q^59 + 10 * q^61 + 12 * q^65 + 12 * q^71 - 30 * q^79 - 2 * q^85 - 4 * q^89 - 24 * q^91 - 12 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/540\mathbb{Z}\right)^\times$$.

 $$n$$ $$217$$ $$271$$ $$461$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 − 1.00000i 1.00000i
0 0 0 −2.00000 1.00000i 0 2.00000i 0 0 0
109.2 0 0 0 −2.00000 + 1.00000i 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 540.2.d.a 2
3.b odd 2 1 540.2.d.b yes 2
4.b odd 2 1 2160.2.f.a 2
5.b even 2 1 inner 540.2.d.a 2
5.c odd 4 1 2700.2.a.h 1
5.c odd 4 1 2700.2.a.p 1
9.c even 3 2 1620.2.r.f 4
9.d odd 6 2 1620.2.r.a 4
12.b even 2 1 2160.2.f.h 2
15.d odd 2 1 540.2.d.b yes 2
15.e even 4 1 2700.2.a.e 1
15.e even 4 1 2700.2.a.o 1
20.d odd 2 1 2160.2.f.a 2
45.h odd 6 2 1620.2.r.a 4
45.j even 6 2 1620.2.r.f 4
60.h even 2 1 2160.2.f.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.2.d.a 2 1.a even 1 1 trivial
540.2.d.a 2 5.b even 2 1 inner
540.2.d.b yes 2 3.b odd 2 1
540.2.d.b yes 2 15.d odd 2 1
1620.2.r.a 4 9.d odd 6 2
1620.2.r.a 4 45.h odd 6 2
1620.2.r.f 4 9.c even 3 2
1620.2.r.f 4 45.j even 6 2
2160.2.f.a 2 4.b odd 2 1
2160.2.f.a 2 20.d odd 2 1
2160.2.f.h 2 12.b even 2 1
2160.2.f.h 2 60.h even 2 1
2700.2.a.e 1 15.e even 4 1
2700.2.a.h 1 5.c odd 4 1
2700.2.a.o 1 15.e even 4 1
2700.2.a.p 1 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(540, [\chi])$$:

 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11} - 2$$ T11 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 4T + 5$$
$7$ $$T^{2} + 4$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 1$$
$19$ $$(T - 3)^{2}$$
$23$ $$T^{2} + 49$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T - 1)^{2}$$
$37$ $$T^{2} + 64$$
$41$ $$(T + 10)^{2}$$
$43$ $$T^{2} + 4$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 81$$
$59$ $$(T + 10)^{2}$$
$61$ $$(T - 5)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$(T - 6)^{2}$$
$73$ $$T^{2} + 4$$
$79$ $$(T + 15)^{2}$$
$83$ $$T^{2} + 169$$
$89$ $$(T + 2)^{2}$$
$97$ $$T^{2} + 64$$