# Properties

 Label 1620.2.i.i 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7}+O(q^{10})$$ q + z * q^5 + (2*z - 2) * q^7 $$q + \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7} - 2 \zeta_{6} q^{13} + 3 q^{17} + 5 q^{19} + 3 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + (6 \zeta_{6} - 6) q^{29} - 5 \zeta_{6} q^{31} - 2 q^{35} + 2 q^{37} + 12 \zeta_{6} q^{41} + (8 \zeta_{6} - 8) q^{43} + (12 \zeta_{6} - 12) q^{47} + 3 \zeta_{6} q^{49} + 3 q^{53} + 6 \zeta_{6} q^{59} + ( - 7 \zeta_{6} + 7) q^{61} + ( - 2 \zeta_{6} + 2) q^{65} - 2 \zeta_{6} q^{67} - 12 q^{71} - 16 q^{73} + ( - \zeta_{6} + 1) q^{79} + (15 \zeta_{6} - 15) q^{83} + 3 \zeta_{6} q^{85} + 12 q^{89} + 4 q^{91} + 5 \zeta_{6} q^{95} + ( - 16 \zeta_{6} + 16) q^{97} +O(q^{100})$$ q + z * q^5 + (2*z - 2) * q^7 - 2*z * q^13 + 3 * q^17 + 5 * q^19 + 3*z * q^23 + (z - 1) * q^25 + (6*z - 6) * q^29 - 5*z * q^31 - 2 * q^35 + 2 * q^37 + 12*z * q^41 + (8*z - 8) * q^43 + (12*z - 12) * q^47 + 3*z * q^49 + 3 * q^53 + 6*z * q^59 + (-7*z + 7) * q^61 + (-2*z + 2) * q^65 - 2*z * q^67 - 12 * q^71 - 16 * q^73 + (-z + 1) * q^79 + (15*z - 15) * q^83 + 3*z * q^85 + 12 * q^89 + 4 * q^91 + 5*z * q^95 + (-16*z + 16) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{5} - 2 q^{7}+O(q^{10})$$ 2 * q + q^5 - 2 * q^7 $$2 q + q^{5} - 2 q^{7} - 2 q^{13} + 6 q^{17} + 10 q^{19} + 3 q^{23} - q^{25} - 6 q^{29} - 5 q^{31} - 4 q^{35} + 4 q^{37} + 12 q^{41} - 8 q^{43} - 12 q^{47} + 3 q^{49} + 6 q^{53} + 6 q^{59} + 7 q^{61} + 2 q^{65} - 2 q^{67} - 24 q^{71} - 32 q^{73} + q^{79} - 15 q^{83} + 3 q^{85} + 24 q^{89} + 8 q^{91} + 5 q^{95} + 16 q^{97}+O(q^{100})$$ 2 * q + q^5 - 2 * q^7 - 2 * q^13 + 6 * q^17 + 10 * q^19 + 3 * q^23 - q^25 - 6 * q^29 - 5 * q^31 - 4 * q^35 + 4 * q^37 + 12 * q^41 - 8 * q^43 - 12 * q^47 + 3 * q^49 + 6 * q^53 + 6 * q^59 + 7 * q^61 + 2 * q^65 - 2 * q^67 - 24 * q^71 - 32 * q^73 + q^79 - 15 * q^83 + 3 * q^85 + 24 * q^89 + 8 * q^91 + 5 * q^95 + 16 * 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 −1.00000 + 1.73205i 0 0 0
1081.1 0 0 0 0.500000 0.866025i 0 −1.00000 1.73205i 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.i 2
3.b odd 2 1 1620.2.i.a 2
9.c even 3 1 540.2.a.c 1
9.c even 3 1 inner 1620.2.i.i 2
9.d odd 6 1 540.2.a.f yes 1
9.d odd 6 1 1620.2.i.a 2
36.f odd 6 1 2160.2.a.c 1
36.h even 6 1 2160.2.a.o 1
45.h odd 6 1 2700.2.a.g 1
45.j even 6 1 2700.2.a.f 1
45.k odd 12 2 2700.2.d.e 2
45.l even 12 2 2700.2.d.f 2
72.j odd 6 1 8640.2.a.w 1
72.l even 6 1 8640.2.a.g 1
72.n even 6 1 8640.2.a.bz 1
72.p odd 6 1 8640.2.a.bl 1

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

## Hecke kernels

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

 $$T_{7}^{2} + 2T_{7} + 4$$ T7^2 + 2*T7 + 4 $$T_{11}$$ T11 $$T_{17} - 3$$ T17 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2} + 2T + 4$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 2T + 4$$
$17$ $$(T - 3)^{2}$$
$19$ $$(T - 5)^{2}$$
$23$ $$T^{2} - 3T + 9$$
$29$ $$T^{2} + 6T + 36$$
$31$ $$T^{2} + 5T + 25$$
$37$ $$(T - 2)^{2}$$
$41$ $$T^{2} - 12T + 144$$
$43$ $$T^{2} + 8T + 64$$
$47$ $$T^{2} + 12T + 144$$
$53$ $$(T - 3)^{2}$$
$59$ $$T^{2} - 6T + 36$$
$61$ $$T^{2} - 7T + 49$$
$67$ $$T^{2} + 2T + 4$$
$71$ $$(T + 12)^{2}$$
$73$ $$(T + 16)^{2}$$
$79$ $$T^{2} - T + 1$$
$83$ $$T^{2} + 15T + 225$$
$89$ $$(T - 12)^{2}$$
$97$ $$T^{2} - 16T + 256$$