# Properties

 Label 1620.2.i.j Level $1620$ Weight $2$ Character orbit 1620.i 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(541,1620)]

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([0, 4, 0]))

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

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

chi := DirichletCharacter("1620.541");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1620.i (of order $$3$$, 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: $$12.9357651274$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 540) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7}+O(q^{10})$$ q + z * q^5 + (-z + 1) * q^7 $$q + \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} + ( - 6 \zeta_{6} + 6) q^{11} + \zeta_{6} q^{13} - q^{19} + 6 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + ( - 6 \zeta_{6} + 6) q^{29} - 8 \zeta_{6} q^{31} + q^{35} - 7 q^{37} - 6 \zeta_{6} q^{41} + ( - 4 \zeta_{6} + 4) q^{43} + ( - 12 \zeta_{6} + 12) q^{47} + 6 \zeta_{6} q^{49} + 6 q^{53} + 6 q^{55} + (11 \zeta_{6} - 11) q^{61} + (\zeta_{6} - 1) q^{65} + 7 \zeta_{6} q^{67} + 6 q^{71} + 11 q^{73} - 6 \zeta_{6} q^{77} + ( - \zeta_{6} + 1) q^{79} + ( - 6 \zeta_{6} + 6) q^{83} + 12 q^{89} + q^{91} - \zeta_{6} q^{95} + ( - 13 \zeta_{6} + 13) q^{97} +O(q^{100})$$ q + z * q^5 + (-z + 1) * q^7 + (-6*z + 6) * q^11 + z * q^13 - q^19 + 6*z * q^23 + (z - 1) * q^25 + (-6*z + 6) * q^29 - 8*z * q^31 + q^35 - 7 * q^37 - 6*z * q^41 + (-4*z + 4) * q^43 + (-12*z + 12) * q^47 + 6*z * q^49 + 6 * q^53 + 6 * q^55 + (11*z - 11) * q^61 + (z - 1) * q^65 + 7*z * q^67 + 6 * q^71 + 11 * q^73 - 6*z * q^77 + (-z + 1) * q^79 + (-6*z + 6) * q^83 + 12 * q^89 + q^91 - z * q^95 + (-13*z + 13) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{5} + q^{7}+O(q^{10})$$ 2 * q + q^5 + q^7 $$2 q + q^{5} + q^{7} + 6 q^{11} + q^{13} - 2 q^{19} + 6 q^{23} - q^{25} + 6 q^{29} - 8 q^{31} + 2 q^{35} - 14 q^{37} - 6 q^{41} + 4 q^{43} + 12 q^{47} + 6 q^{49} + 12 q^{53} + 12 q^{55} - 11 q^{61} - q^{65} + 7 q^{67} + 12 q^{71} + 22 q^{73} - 6 q^{77} + q^{79} + 6 q^{83} + 24 q^{89} + 2 q^{91} - q^{95} + 13 q^{97}+O(q^{100})$$ 2 * q + q^5 + q^7 + 6 * q^11 + q^13 - 2 * q^19 + 6 * q^23 - q^25 + 6 * q^29 - 8 * q^31 + 2 * q^35 - 14 * q^37 - 6 * q^41 + 4 * q^43 + 12 * q^47 + 6 * q^49 + 12 * q^53 + 12 * q^55 - 11 * q^61 - q^65 + 7 * q^67 + 12 * q^71 + 22 * q^73 - 6 * q^77 + q^79 + 6 * q^83 + 24 * q^89 + 2 * q^91 - q^95 + 13 * q^97

## Character values

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

 $$n$$ $$811$$ $$1297$$ $$1541$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
541.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 0.500000 + 0.866025i 0 0.500000 0.866025i 0 0 0
1081.1 0 0 0 0.500000 0.866025i 0 0.500000 + 0.866025i 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.i.j 2
3.b odd 2 1 1620.2.i.d 2
9.c even 3 1 540.2.a.b 1
9.c even 3 1 inner 1620.2.i.j 2
9.d odd 6 1 540.2.a.e yes 1
9.d odd 6 1 1620.2.i.d 2
36.f odd 6 1 2160.2.a.h 1
36.h even 6 1 2160.2.a.s 1
45.h odd 6 1 2700.2.a.m 1
45.j even 6 1 2700.2.a.k 1
45.k odd 12 2 2700.2.d.a 2
45.l even 12 2 2700.2.d.k 2
72.j odd 6 1 8640.2.a.l 1
72.l even 6 1 8640.2.a.s 1
72.n even 6 1 8640.2.a.br 1
72.p odd 6 1 8640.2.a.bu 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.2.a.b 1 9.c even 3 1
540.2.a.e yes 1 9.d odd 6 1
1620.2.i.d 2 3.b odd 2 1
1620.2.i.d 2 9.d odd 6 1
1620.2.i.j 2 1.a even 1 1 trivial
1620.2.i.j 2 9.c even 3 1 inner
2160.2.a.h 1 36.f odd 6 1
2160.2.a.s 1 36.h even 6 1
2700.2.a.k 1 45.j even 6 1
2700.2.a.m 1 45.h odd 6 1
2700.2.d.a 2 45.k odd 12 2
2700.2.d.k 2 45.l even 12 2
8640.2.a.l 1 72.j odd 6 1
8640.2.a.s 1 72.l even 6 1
8640.2.a.br 1 72.n even 6 1
8640.2.a.bu 1 72.p odd 6 1

## Hecke kernels

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

 $$T_{7}^{2} - T_{7} + 1$$ T7^2 - T7 + 1 $$T_{11}^{2} - 6T_{11} + 36$$ T11^2 - 6*T11 + 36 $$T_{17}$$ T17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2} - T + 1$$
$11$ $$T^{2} - 6T + 36$$
$13$ $$T^{2} - T + 1$$
$17$ $$T^{2}$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$T^{2} - 6T + 36$$
$31$ $$T^{2} + 8T + 64$$
$37$ $$(T + 7)^{2}$$
$41$ $$T^{2} + 6T + 36$$
$43$ $$T^{2} - 4T + 16$$
$47$ $$T^{2} - 12T + 144$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 11T + 121$$
$67$ $$T^{2} - 7T + 49$$
$71$ $$(T - 6)^{2}$$
$73$ $$(T - 11)^{2}$$
$79$ $$T^{2} - T + 1$$
$83$ $$T^{2} - 6T + 36$$
$89$ $$(T - 12)^{2}$$
$97$ $$T^{2} - 13T + 169$$