# Properties

 Label 1620.2.a.a Level $1620$ Weight $2$ Character orbit 1620.a Self dual yes Analytic conductor $12.936$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,2,Mod(1,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, 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("1620.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1620.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: $$12.9357651274$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q - q^{5} - 4 q^{7}+O(q^{10})$$ q - q^5 - 4 * q^7 $$q - q^{5} - 4 q^{7} - 3 q^{11} - 4 q^{13} + 5 q^{19} + 6 q^{23} + q^{25} + 9 q^{29} + 5 q^{31} + 4 q^{35} + 2 q^{37} + 9 q^{41} - 10 q^{43} + 6 q^{47} + 9 q^{49} + 12 q^{53} + 3 q^{55} - 9 q^{59} - 10 q^{61} + 4 q^{65} + 2 q^{67} - 3 q^{71} - 4 q^{73} + 12 q^{77} - 4 q^{79} - 6 q^{83} + 9 q^{89} + 16 q^{91} - 5 q^{95} + 2 q^{97}+O(q^{100})$$ q - q^5 - 4 * q^7 - 3 * q^11 - 4 * q^13 + 5 * q^19 + 6 * q^23 + q^25 + 9 * q^29 + 5 * q^31 + 4 * q^35 + 2 * q^37 + 9 * q^41 - 10 * q^43 + 6 * q^47 + 9 * q^49 + 12 * q^53 + 3 * q^55 - 9 * q^59 - 10 * q^61 + 4 * q^65 + 2 * q^67 - 3 * q^71 - 4 * q^73 + 12 * q^77 - 4 * q^79 - 6 * q^83 + 9 * q^89 + 16 * q^91 - 5 * q^95 + 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
 0
0 0 0 −1.00000 0 −4.00000 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.a.a 1
3.b odd 2 1 1620.2.a.d yes 1
4.b odd 2 1 6480.2.a.m 1
5.b even 2 1 8100.2.a.m 1
5.c odd 4 2 8100.2.d.b 2
9.c even 3 2 1620.2.i.l 2
9.d odd 6 2 1620.2.i.e 2
12.b even 2 1 6480.2.a.y 1
15.d odd 2 1 8100.2.a.n 1
15.e even 4 2 8100.2.d.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.2.a.a 1 1.a even 1 1 trivial
1620.2.a.d yes 1 3.b odd 2 1
1620.2.i.e 2 9.d odd 6 2
1620.2.i.l 2 9.c even 3 2
6480.2.a.m 1 4.b odd 2 1
6480.2.a.y 1 12.b even 2 1
8100.2.a.m 1 5.b even 2 1
8100.2.a.n 1 15.d odd 2 1
8100.2.d.b 2 5.c odd 4 2
8100.2.d.g 2 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1620))$$:

 $$T_{7} + 4$$ T7 + 4 $$T_{11} + 3$$ T11 + 3 $$T_{17}$$ T17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T + 4$$
$11$ $$T + 3$$
$13$ $$T + 4$$
$17$ $$T$$
$19$ $$T - 5$$
$23$ $$T - 6$$
$29$ $$T - 9$$
$31$ $$T - 5$$
$37$ $$T - 2$$
$41$ $$T - 9$$
$43$ $$T + 10$$
$47$ $$T - 6$$
$53$ $$T - 12$$
$59$ $$T + 9$$
$61$ $$T + 10$$
$67$ $$T - 2$$
$71$ $$T + 3$$
$73$ $$T + 4$$
$79$ $$T + 4$$
$83$ $$T + 6$$
$89$ $$T - 9$$
$97$ $$T - 2$$