# Properties

 Label 1620.1.p.a Level $1620$ Weight $1$ Character orbit 1620.p Analytic conductor $0.808$ Analytic rank $0$ Dimension $4$ Projective image $D_{6}$ CM discriminant -15 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,1,Mod(379,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 4, 3]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1620.379");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1620.p (of order $$6$$, degree $$2$$, 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: $$0.808485320465$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 540) Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.1166400.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - \zeta_{12}^{3} q^{2} - q^{4} + \zeta_{12}^{5} q^{5} + \zeta_{12}^{3} q^{8} +O(q^{10})$$ q - z^3 * q^2 - q^4 + z^5 * q^5 + z^3 * q^8 $$q - \zeta_{12}^{3} q^{2} - q^{4} + \zeta_{12}^{5} q^{5} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{10} + q^{16} - \zeta_{12}^{3} q^{17} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{19} - \zeta_{12}^{5} q^{20} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{23} - \zeta_{12}^{4} q^{25} + ( - \zeta_{12}^{4} + 1) q^{31} - \zeta_{12}^{3} q^{32} - q^{34} + (\zeta_{12}^{5} - \zeta_{12}) q^{38} - \zeta_{12}^{2} q^{40} + (\zeta_{12}^{4} - 1) q^{46} + \zeta_{12}^{2} q^{49} - \zeta_{12} q^{50} + \zeta_{12}^{3} q^{53} + \zeta_{12}^{4} q^{61} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{62} - q^{64} + \zeta_{12}^{3} q^{68} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{76} + ( - \zeta_{12}^{2} - 1) q^{79} + \zeta_{12}^{5} q^{80} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{83} + \zeta_{12}^{2} q^{85} + (\zeta_{12}^{3} + \zeta_{12}) q^{92} + (\zeta_{12}^{3} + \zeta_{12}) q^{95} - \zeta_{12}^{5} q^{98} +O(q^{100})$$ q - z^3 * q^2 - q^4 + z^5 * q^5 + z^3 * q^8 + z^2 * q^10 + q^16 - z^3 * q^17 + (-z^4 - z^2) * q^19 - z^5 * q^20 + (-z^3 - z) * q^23 - z^4 * q^25 + (-z^4 + 1) * q^31 - z^3 * q^32 - q^34 + (z^5 - z) * q^38 - z^2 * q^40 + (z^4 - 1) * q^46 + z^2 * q^49 - z * q^50 + z^3 * q^53 + z^4 * q^61 + (-z^3 - z) * q^62 - q^64 + z^3 * q^68 + (z^4 + z^2) * q^76 + (-z^2 - 1) * q^79 + z^5 * q^80 + (-z^5 - z^3) * q^83 + z^2 * q^85 + (z^3 + z) * q^92 + (z^3 + z) * q^95 - z^5 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4}+O(q^{10})$$ 4 * q - 4 * q^4 $$4 q - 4 q^{4} + 2 q^{10} + 4 q^{16} + 2 q^{25} + 6 q^{31} - 4 q^{34} - 2 q^{40} - 6 q^{46} + 2 q^{49} - 2 q^{61} - 4 q^{64} - 6 q^{79} + 2 q^{85}+O(q^{100})$$ 4 * q - 4 * q^4 + 2 * q^10 + 4 * q^16 + 2 * q^25 + 6 * q^31 - 4 * q^34 - 2 * q^40 - 6 * q^46 + 2 * q^49 - 2 * q^61 - 4 * q^64 - 6 * q^79 + 2 * q^85

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$$.

 $$n$$ $$811$$ $$1297$$ $$1541$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
379.1
 0.866025 + 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
1.00000i 0 −1.00000 −0.866025 + 0.500000i 0 0 1.00000i 0 0.500000 + 0.866025i
379.2 1.00000i 0 −1.00000 0.866025 0.500000i 0 0 1.00000i 0 0.500000 + 0.866025i
919.1 1.00000i 0 −1.00000 0.866025 + 0.500000i 0 0 1.00000i 0 0.500000 0.866025i
919.2 1.00000i 0 −1.00000 −0.866025 0.500000i 0 0 1.00000i 0 0.500000 0.866025i
 $$n$$: 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
5.b even 2 1 inner
36.f odd 6 1 inner
36.h even 6 1 inner
180.n even 6 1 inner
180.p odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.1.p.a 4
3.b odd 2 1 inner 1620.1.p.a 4
4.b odd 2 1 1620.1.p.d 4
5.b even 2 1 inner 1620.1.p.a 4
9.c even 3 1 540.1.f.a 4
9.c even 3 1 1620.1.p.d 4
9.d odd 6 1 540.1.f.a 4
9.d odd 6 1 1620.1.p.d 4
12.b even 2 1 1620.1.p.d 4
15.d odd 2 1 CM 1620.1.p.a 4
20.d odd 2 1 1620.1.p.d 4
36.f odd 6 1 540.1.f.a 4
36.f odd 6 1 inner 1620.1.p.a 4
36.h even 6 1 540.1.f.a 4
36.h even 6 1 inner 1620.1.p.a 4
45.h odd 6 1 540.1.f.a 4
45.h odd 6 1 1620.1.p.d 4
45.j even 6 1 540.1.f.a 4
45.j even 6 1 1620.1.p.d 4
45.k odd 12 1 2700.1.c.a 2
45.k odd 12 1 2700.1.c.b 2
45.l even 12 1 2700.1.c.a 2
45.l even 12 1 2700.1.c.b 2
60.h even 2 1 1620.1.p.d 4
180.n even 6 1 540.1.f.a 4
180.n even 6 1 inner 1620.1.p.a 4
180.p odd 6 1 540.1.f.a 4
180.p odd 6 1 inner 1620.1.p.a 4
180.v odd 12 1 2700.1.c.a 2
180.v odd 12 1 2700.1.c.b 2
180.x even 12 1 2700.1.c.a 2
180.x even 12 1 2700.1.c.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.1.f.a 4 9.c even 3 1
540.1.f.a 4 9.d odd 6 1
540.1.f.a 4 36.f odd 6 1
540.1.f.a 4 36.h even 6 1
540.1.f.a 4 45.h odd 6 1
540.1.f.a 4 45.j even 6 1
540.1.f.a 4 180.n even 6 1
540.1.f.a 4 180.p odd 6 1
1620.1.p.a 4 1.a even 1 1 trivial
1620.1.p.a 4 3.b odd 2 1 inner
1620.1.p.a 4 5.b even 2 1 inner
1620.1.p.a 4 15.d odd 2 1 CM
1620.1.p.a 4 36.f odd 6 1 inner
1620.1.p.a 4 36.h even 6 1 inner
1620.1.p.a 4 180.n even 6 1 inner
1620.1.p.a 4 180.p odd 6 1 inner
1620.1.p.d 4 4.b odd 2 1
1620.1.p.d 4 9.c even 3 1
1620.1.p.d 4 9.d odd 6 1
1620.1.p.d 4 12.b even 2 1
1620.1.p.d 4 20.d odd 2 1
1620.1.p.d 4 45.h odd 6 1
1620.1.p.d 4 45.j even 6 1
1620.1.p.d 4 60.h even 2 1
2700.1.c.a 2 45.k odd 12 1
2700.1.c.a 2 45.l even 12 1
2700.1.c.a 2 180.v odd 12 1
2700.1.c.a 2 180.x even 12 1
2700.1.c.b 2 45.k odd 12 1
2700.1.c.b 2 45.l even 12 1
2700.1.c.b 2 180.v odd 12 1
2700.1.c.b 2 180.x even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1620, [\chi])$$:

 $$T_{13}$$ T13 $$T_{17}^{2} + 1$$ T17^2 + 1 $$T_{31}^{2} - 3T_{31} + 3$$ T31^2 - 3*T31 + 3

## Hecke characteristic polynomials

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