# Properties

 Label 720.2.q.e Level $720$ Weight $2$ Character orbit 720.q Analytic conductor $5.749$ 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] = [720,2,Mod(241,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 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("720.241");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 720.q (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: $$5.74922894553$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 360) 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} + 1) q^{3} + \zeta_{6} q^{5} + 3 \zeta_{6} q^{9}+O(q^{10})$$ q + (z + 1) * q^3 + z * q^5 + 3*z * q^9 $$q + (\zeta_{6} + 1) q^{3} + \zeta_{6} q^{5} + 3 \zeta_{6} q^{9} + (5 \zeta_{6} - 5) q^{11} + (2 \zeta_{6} - 1) q^{15} + 3 q^{17} - 5 q^{19} + 6 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + (6 \zeta_{6} - 3) q^{27} + ( - 10 \zeta_{6} + 10) q^{29} - 2 \zeta_{6} q^{31} + (5 \zeta_{6} - 10) q^{33} + 4 q^{37} + 3 \zeta_{6} q^{41} + ( - 3 \zeta_{6} + 3) q^{43} + (3 \zeta_{6} - 3) q^{45} + ( - 4 \zeta_{6} + 4) q^{47} + 7 \zeta_{6} q^{49} + (3 \zeta_{6} + 3) q^{51} - 6 q^{53} - 5 q^{55} + ( - 5 \zeta_{6} - 5) q^{57} - 3 \zeta_{6} q^{59} + (2 \zeta_{6} - 2) q^{61} - 11 \zeta_{6} q^{67} + (12 \zeta_{6} - 6) q^{69} + 14 q^{71} - 15 q^{73} + (\zeta_{6} - 2) q^{75} + ( - 10 \zeta_{6} + 10) q^{79} + (9 \zeta_{6} - 9) q^{81} + (12 \zeta_{6} - 12) q^{83} + 3 \zeta_{6} q^{85} + ( - 10 \zeta_{6} + 20) q^{87} + 14 q^{89} + ( - 4 \zeta_{6} + 2) q^{93} - 5 \zeta_{6} q^{95} + ( - 13 \zeta_{6} + 13) q^{97} - 15 q^{99} +O(q^{100})$$ q + (z + 1) * q^3 + z * q^5 + 3*z * q^9 + (5*z - 5) * q^11 + (2*z - 1) * q^15 + 3 * q^17 - 5 * q^19 + 6*z * q^23 + (z - 1) * q^25 + (6*z - 3) * q^27 + (-10*z + 10) * q^29 - 2*z * q^31 + (5*z - 10) * q^33 + 4 * q^37 + 3*z * q^41 + (-3*z + 3) * q^43 + (3*z - 3) * q^45 + (-4*z + 4) * q^47 + 7*z * q^49 + (3*z + 3) * q^51 - 6 * q^53 - 5 * q^55 + (-5*z - 5) * q^57 - 3*z * q^59 + (2*z - 2) * q^61 - 11*z * q^67 + (12*z - 6) * q^69 + 14 * q^71 - 15 * q^73 + (z - 2) * q^75 + (-10*z + 10) * q^79 + (9*z - 9) * q^81 + (12*z - 12) * q^83 + 3*z * q^85 + (-10*z + 20) * q^87 + 14 * q^89 + (-4*z + 2) * q^93 - 5*z * q^95 + (-13*z + 13) * q^97 - 15 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{3} + q^{5} + 3 q^{9}+O(q^{10})$$ 2 * q + 3 * q^3 + q^5 + 3 * q^9 $$2 q + 3 q^{3} + q^{5} + 3 q^{9} - 5 q^{11} + 6 q^{17} - 10 q^{19} + 6 q^{23} - q^{25} + 10 q^{29} - 2 q^{31} - 15 q^{33} + 8 q^{37} + 3 q^{41} + 3 q^{43} - 3 q^{45} + 4 q^{47} + 7 q^{49} + 9 q^{51} - 12 q^{53} - 10 q^{55} - 15 q^{57} - 3 q^{59} - 2 q^{61} - 11 q^{67} + 28 q^{71} - 30 q^{73} - 3 q^{75} + 10 q^{79} - 9 q^{81} - 12 q^{83} + 3 q^{85} + 30 q^{87} + 28 q^{89} - 5 q^{95} + 13 q^{97} - 30 q^{99}+O(q^{100})$$ 2 * q + 3 * q^3 + q^5 + 3 * q^9 - 5 * q^11 + 6 * q^17 - 10 * q^19 + 6 * q^23 - q^25 + 10 * q^29 - 2 * q^31 - 15 * q^33 + 8 * q^37 + 3 * q^41 + 3 * q^43 - 3 * q^45 + 4 * q^47 + 7 * q^49 + 9 * q^51 - 12 * q^53 - 10 * q^55 - 15 * q^57 - 3 * q^59 - 2 * q^61 - 11 * q^67 + 28 * q^71 - 30 * q^73 - 3 * q^75 + 10 * q^79 - 9 * q^81 - 12 * q^83 + 3 * q^85 + 30 * q^87 + 28 * q^89 - 5 * q^95 + 13 * q^97 - 30 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$$.

 $$n$$ $$181$$ $$271$$ $$577$$ $$641$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
241.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.50000 + 0.866025i 0 0.500000 + 0.866025i 0 0 0 1.50000 + 2.59808i 0
481.1 0 1.50000 0.866025i 0 0.500000 0.866025i 0 0 0 1.50000 2.59808i 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 720.2.q.e 2
3.b odd 2 1 2160.2.q.c 2
4.b odd 2 1 360.2.q.a 2
9.c even 3 1 inner 720.2.q.e 2
9.c even 3 1 6480.2.a.e 1
9.d odd 6 1 2160.2.q.c 2
9.d odd 6 1 6480.2.a.q 1
12.b even 2 1 1080.2.q.a 2
36.f odd 6 1 360.2.q.a 2
36.f odd 6 1 3240.2.a.b 1
36.h even 6 1 1080.2.q.a 2
36.h even 6 1 3240.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.q.a 2 4.b odd 2 1
360.2.q.a 2 36.f odd 6 1
720.2.q.e 2 1.a even 1 1 trivial
720.2.q.e 2 9.c even 3 1 inner
1080.2.q.a 2 12.b even 2 1
1080.2.q.a 2 36.h even 6 1
2160.2.q.c 2 3.b odd 2 1
2160.2.q.c 2 9.d odd 6 1
3240.2.a.b 1 36.f odd 6 1
3240.2.a.f 1 36.h even 6 1
6480.2.a.e 1 9.c even 3 1
6480.2.a.q 1 9.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(720, [\chi])$$:

 $$T_{7}$$ T7 $$T_{11}^{2} + 5T_{11} + 25$$ T11^2 + 5*T11 + 25

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 3T + 3$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 5T + 25$$
$13$ $$T^{2}$$
$17$ $$(T - 3)^{2}$$
$19$ $$(T + 5)^{2}$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$T^{2} - 10T + 100$$
$31$ $$T^{2} + 2T + 4$$
$37$ $$(T - 4)^{2}$$
$41$ $$T^{2} - 3T + 9$$
$43$ $$T^{2} - 3T + 9$$
$47$ $$T^{2} - 4T + 16$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} + 3T + 9$$
$61$ $$T^{2} + 2T + 4$$
$67$ $$T^{2} + 11T + 121$$
$71$ $$(T - 14)^{2}$$
$73$ $$(T + 15)^{2}$$
$79$ $$T^{2} - 10T + 100$$
$83$ $$T^{2} + 12T + 144$$
$89$ $$(T - 14)^{2}$$
$97$ $$T^{2} - 13T + 169$$