# Properties

 Label 1620.2.d.b Level $1620$ Weight $2$ Character orbit 1620.d Analytic conductor $12.936$ 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] = [1620,2,Mod(649,1620)]

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, 1]))

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

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

chi := DirichletCharacter("1620.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1620.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: $$12.9357651274$$ 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: $$2$$ 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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{5} + \beta q^{7}+O(q^{10})$$ q + (b + 1) * q^5 + b * q^7 $$q + (\beta + 1) q^{5} + \beta q^{7} - q^{11} + \beta q^{17} + 3 q^{19} + 2 \beta q^{23} + (2 \beta - 3) q^{25} - 3 q^{29} + q^{31} + (\beta - 4) q^{35} - 5 \beta q^{37} - q^{41} + 5 \beta q^{43} + 5 \beta q^{47} + 3 q^{49} + 3 \beta q^{53} + ( - \beta - 1) q^{55} - 13 q^{59} - 10 q^{61} - 4 \beta q^{67} - 9 q^{71} - 7 \beta q^{73} - \beta q^{77} + 5 \beta q^{83} + (\beta - 4) q^{85} + 13 q^{89} + (3 \beta + 3) q^{95} + 5 \beta q^{97} +O(q^{100})$$ q + (b + 1) * q^5 + b * q^7 - q^11 + b * q^17 + 3 * q^19 + 2*b * q^23 + (2*b - 3) * q^25 - 3 * q^29 + q^31 + (b - 4) * q^35 - 5*b * q^37 - q^41 + 5*b * q^43 + 5*b * q^47 + 3 * q^49 + 3*b * q^53 + (-b - 1) * q^55 - 13 * q^59 - 10 * q^61 - 4*b * q^67 - 9 * q^71 - 7*b * q^73 - b * q^77 + 5*b * q^83 + (b - 4) * q^85 + 13 * q^89 + (3*b + 3) * q^95 + 5*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5}+O(q^{10})$$ 2 * q + 2 * q^5 $$2 q + 2 q^{5} - 2 q^{11} + 6 q^{19} - 6 q^{25} - 6 q^{29} + 2 q^{31} - 8 q^{35} - 2 q^{41} + 6 q^{49} - 2 q^{55} - 26 q^{59} - 20 q^{61} - 18 q^{71} - 8 q^{85} + 26 q^{89} + 6 q^{95}+O(q^{100})$$ 2 * q + 2 * q^5 - 2 * q^11 + 6 * q^19 - 6 * q^25 - 6 * q^29 + 2 * q^31 - 8 * q^35 - 2 * q^41 + 6 * q^49 - 2 * q^55 - 26 * q^59 - 20 * q^61 - 18 * q^71 - 8 * q^85 + 26 * q^89 + 6 * q^95

## Character values

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

 $$n$$ $$811$$ $$1297$$ $$1541$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 − 1.00000i 1.00000i
0 0 0 1.00000 2.00000i 0 2.00000i 0 0 0
649.2 0 0 0 1.00000 + 2.00000i 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 1620.2.d.b yes 2
3.b odd 2 1 1620.2.d.a 2
5.b even 2 1 inner 1620.2.d.b yes 2
5.c odd 4 1 8100.2.a.c 1
5.c odd 4 1 8100.2.a.k 1
9.c even 3 2 1620.2.r.b 4
9.d odd 6 2 1620.2.r.e 4
15.d odd 2 1 1620.2.d.a 2
15.e even 4 1 8100.2.a.d 1
15.e even 4 1 8100.2.a.l 1
45.h odd 6 2 1620.2.r.e 4
45.j even 6 2 1620.2.r.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.2.d.a 2 3.b odd 2 1
1620.2.d.a 2 15.d odd 2 1
1620.2.d.b yes 2 1.a even 1 1 trivial
1620.2.d.b yes 2 5.b even 2 1 inner
1620.2.r.b 4 9.c even 3 2
1620.2.r.b 4 45.j even 6 2
1620.2.r.e 4 9.d odd 6 2
1620.2.r.e 4 45.h odd 6 2
8100.2.a.c 1 5.c odd 4 1
8100.2.a.d 1 15.e even 4 1
8100.2.a.k 1 5.c odd 4 1
8100.2.a.l 1 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 $$S_{2}^{\mathrm{new}}(1620, [\chi])$$:

 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11} + 1$$ T11 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2T + 5$$
$7$ $$T^{2} + 4$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 4$$
$19$ $$(T - 3)^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$(T + 3)^{2}$$
$31$ $$(T - 1)^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$(T + 1)^{2}$$
$43$ $$T^{2} + 100$$
$47$ $$T^{2} + 100$$
$53$ $$T^{2} + 36$$
$59$ $$(T + 13)^{2}$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$(T + 9)^{2}$$
$73$ $$T^{2} + 196$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 100$$
$89$ $$(T - 13)^{2}$$
$97$ $$T^{2} + 100$$