# Properties

 Label 540.2.d.c Level $540$ Weight $2$ Character orbit 540.d Analytic conductor $4.312$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Learn more

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: $$4$$ Coefficient field: $$\Q(i, \sqrt{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 25$$ x^4 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{5} + \beta_{2} q^{7}+O(q^{10})$$ q + b1 * q^5 + b2 * q^7 $$q + \beta_1 q^{5} + \beta_{2} q^{7} + ( - \beta_{3} + \beta_1) q^{11} - 3 \beta_{2} q^{13} + (2 \beta_{3} + 2 \beta_1) q^{17} - 3 q^{19} + (\beta_{3} + \beta_1) q^{23} + 5 \beta_{2} q^{25} + ( - 3 \beta_{3} + 3 \beta_1) q^{29} - 2 q^{31} + \beta_{3} q^{35} + \beta_{2} q^{37} + ( - \beta_{3} + \beta_1) q^{41} - 10 \beta_{2} q^{43} + ( - 2 \beta_{3} - 2 \beta_1) q^{47} + 6 q^{49} + (3 \beta_{3} + 3 \beta_1) q^{53} + (5 \beta_{2} + 5) q^{55} + (2 \beta_{3} - 2 \beta_1) q^{59} - q^{61} - 3 \beta_{3} q^{65} + 11 \beta_{2} q^{67} + (3 \beta_{3} - 3 \beta_1) q^{71} - 13 \beta_{2} q^{73} + (\beta_{3} + \beta_1) q^{77} + 3 q^{79} + ( - 5 \beta_{3} - 5 \beta_1) q^{83} + (10 \beta_{2} - 10) q^{85} + (4 \beta_{3} - 4 \beta_1) q^{89} + 3 q^{91} - 3 \beta_1 q^{95} - \beta_{2} q^{97}+O(q^{100})$$ q + b1 * q^5 + b2 * q^7 + (-b3 + b1) * q^11 - 3*b2 * q^13 + (2*b3 + 2*b1) * q^17 - 3 * q^19 + (b3 + b1) * q^23 + 5*b2 * q^25 + (-3*b3 + 3*b1) * q^29 - 2 * q^31 + b3 * q^35 + b2 * q^37 + (-b3 + b1) * q^41 - 10*b2 * q^43 + (-2*b3 - 2*b1) * q^47 + 6 * q^49 + (3*b3 + 3*b1) * q^53 + (5*b2 + 5) * q^55 + (2*b3 - 2*b1) * q^59 - q^61 - 3*b3 * q^65 + 11*b2 * q^67 + (3*b3 - 3*b1) * q^71 - 13*b2 * q^73 + (b3 + b1) * q^77 + 3 * q^79 + (-5*b3 - 5*b1) * q^83 + (10*b2 - 10) * q^85 + (4*b3 - 4*b1) * q^89 + 3 * q^91 - 3*b1 * q^95 - b2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 12 q^{19} - 8 q^{31} + 24 q^{49} + 20 q^{55} - 4 q^{61} + 12 q^{79} - 40 q^{85} + 12 q^{91}+O(q^{100})$$ 4 * q - 12 * q^19 - 8 * q^31 + 24 * q^49 + 20 * q^55 - 4 * q^61 + 12 * q^79 - 40 * q^85 + 12 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 5$$ (v^2) / 5 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 5$$ (v^3) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$5\beta_{2}$$ 5*b2 $$\nu^{3}$$ $$=$$ $$5\beta_{3}$$ 5*b3

## 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
 −1.58114 − 1.58114i −1.58114 + 1.58114i 1.58114 − 1.58114i 1.58114 + 1.58114i
0 0 0 −1.58114 1.58114i 0 1.00000i 0 0 0
109.2 0 0 0 −1.58114 + 1.58114i 0 1.00000i 0 0 0
109.3 0 0 0 1.58114 1.58114i 0 1.00000i 0 0 0
109.4 0 0 0 1.58114 + 1.58114i 0 1.00000i 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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 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.c 4
3.b odd 2 1 inner 540.2.d.c 4
4.b odd 2 1 2160.2.f.l 4
5.b even 2 1 inner 540.2.d.c 4
5.c odd 4 1 2700.2.a.v 2
5.c odd 4 1 2700.2.a.w 2
9.c even 3 2 1620.2.r.g 8
9.d odd 6 2 1620.2.r.g 8
12.b even 2 1 2160.2.f.l 4
15.d odd 2 1 inner 540.2.d.c 4
15.e even 4 1 2700.2.a.v 2
15.e even 4 1 2700.2.a.w 2
20.d odd 2 1 2160.2.f.l 4
45.h odd 6 2 1620.2.r.g 8
45.j even 6 2 1620.2.r.g 8
60.h even 2 1 2160.2.f.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.2.d.c 4 1.a even 1 1 trivial
540.2.d.c 4 3.b odd 2 1 inner
540.2.d.c 4 5.b even 2 1 inner
540.2.d.c 4 15.d odd 2 1 inner
1620.2.r.g 8 9.c even 3 2
1620.2.r.g 8 9.d odd 6 2
1620.2.r.g 8 45.h odd 6 2
1620.2.r.g 8 45.j even 6 2
2160.2.f.l 4 4.b odd 2 1
2160.2.f.l 4 12.b even 2 1
2160.2.f.l 4 20.d odd 2 1
2160.2.f.l 4 60.h even 2 1
2700.2.a.v 2 5.c odd 4 1
2700.2.a.v 2 15.e even 4 1
2700.2.a.w 2 5.c odd 4 1
2700.2.a.w 2 15.e even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 1$$ T7^2 + 1 $$T_{11}^{2} - 10$$ T11^2 - 10

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 25$$
$7$ $$(T^{2} + 1)^{2}$$
$11$ $$(T^{2} - 10)^{2}$$
$13$ $$(T^{2} + 9)^{2}$$
$17$ $$(T^{2} + 40)^{2}$$
$19$ $$(T + 3)^{4}$$
$23$ $$(T^{2} + 10)^{2}$$
$29$ $$(T^{2} - 90)^{2}$$
$31$ $$(T + 2)^{4}$$
$37$ $$(T^{2} + 1)^{2}$$
$41$ $$(T^{2} - 10)^{2}$$
$43$ $$(T^{2} + 100)^{2}$$
$47$ $$(T^{2} + 40)^{2}$$
$53$ $$(T^{2} + 90)^{2}$$
$59$ $$(T^{2} - 40)^{2}$$
$61$ $$(T + 1)^{4}$$
$67$ $$(T^{2} + 121)^{2}$$
$71$ $$(T^{2} - 90)^{2}$$
$73$ $$(T^{2} + 169)^{2}$$
$79$ $$(T - 3)^{4}$$
$83$ $$(T^{2} + 250)^{2}$$
$89$ $$(T^{2} - 160)^{2}$$
$97$ $$(T^{2} + 1)^{2}$$
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