# Properties

 Label 720.2.q.g Level $720$ Weight $2$ Character orbit 720.q Analytic conductor $5.749$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - \beta_{2} + \beta_1) q^{3} + \beta_{2} q^{5} + ( - 2 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{2} - 2 \beta_1 + 1) q^{9}+O(q^{10})$$ q + (-b3 - b2 + b1) * q^3 + b2 * q^5 + (-2*b3 - b2 + b1 + 1) * q^7 + (-b2 - 2*b1 + 1) * q^9 $$q + ( - \beta_{3} - \beta_{2} + \beta_1) q^{3} + \beta_{2} q^{5} + ( - 2 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{7} + ( - \beta_{2} - 2 \beta_1 + 1) q^{9} + ( - \beta_{2} + \beta_1 + 1) q^{15} - 2 q^{17} + ( - 2 \beta_{3} + 4 \beta_1 + 2) q^{19} + ( - 4 \beta_{2} - 2 \beta_1 + 1) q^{21} + ( - \beta_{3} - 5 \beta_{2} - \beta_1) q^{23} + (\beta_{2} - 1) q^{25} + (\beta_{3} - 5) q^{27} + ( - 4 \beta_{3} + 3 \beta_{2} + \cdots - 3) q^{29}+ \cdots + (2 \beta_{2} - 2) q^{97}+O(q^{100})$$ q + (-b3 - b2 + b1) * q^3 + b2 * q^5 + (-2*b3 - b2 + b1 + 1) * q^7 + (-b2 - 2*b1 + 1) * q^9 + (-b2 + b1 + 1) * q^15 - 2 * q^17 + (-2*b3 + 4*b1 + 2) * q^19 + (-4*b2 - 2*b1 + 1) * q^21 + (-b3 - 5*b2 - b1) * q^23 + (b2 - 1) * q^25 + (b3 - 5) * q^27 + (-4*b3 + 3*b2 + 2*b1 - 3) * q^29 + (-2*b3 + 6*b2 - 2*b1) * q^31 + (-b3 + 2*b1 + 1) * q^35 - 6 * q^37 + (2*b3 + 5*b2 + 2*b1) * q^41 + (-8*b3 - 2*b2 + 4*b1 + 2) * q^43 + (-2*b3 + 1) * q^45 + (2*b3 + 7*b2 - b1 - 7) * q^47 + (-2*b3 - 2*b1) * q^49 + (2*b3 + 2*b2 - 2*b1) * q^51 + (-4*b3 + 8*b1 - 2) * q^53 + (-4*b3 - 6*b2 + 8) * q^57 + (2*b3 - 6*b2 + 2*b1) * q^59 + (-3*b2 + 3) * q^61 + (b3 + 3*b2 - 3*b1 - 8) * q^63 + (-5*b3 - b2 - 5*b1) * q^67 + (2*b3 + 3*b2 - 6*b1 - 7) * q^69 + (4*b3 - 8*b1) * q^71 + (-4*b3 + 8*b1 + 4) * q^73 + (b3 + 1) * q^75 + (4*b3 + 2*b2 - 2*b1 - 2) * q^79 + (4*b3 + 7*b2 - 4*b1) * q^81 + (2*b3 + 3*b2 - b1 - 3) * q^83 - 2*b2 * q^85 + (5*b3 - 8*b2 - 4*b1 + 7) * q^87 + (-4*b3 + 8*b1 - 7) * q^89 + (4*b3 - 10*b2 + 4*b1 + 2) * q^93 + (2*b3 + 2*b2 + 2*b1) * q^95 + (2*b2 - 2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 $$4 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + 2 q^{15} - 8 q^{17} + 8 q^{19} - 4 q^{21} - 10 q^{23} - 2 q^{25} - 20 q^{27} - 6 q^{29} + 12 q^{31} + 4 q^{35} - 24 q^{37} + 10 q^{41} + 4 q^{43} + 4 q^{45} - 14 q^{47} + 4 q^{51} - 8 q^{53} + 20 q^{57} - 12 q^{59} + 6 q^{61} - 26 q^{63} - 2 q^{67} - 22 q^{69} + 16 q^{73} + 4 q^{75} - 4 q^{79} + 14 q^{81} - 6 q^{83} - 4 q^{85} + 12 q^{87} - 28 q^{89} - 12 q^{93} + 4 q^{95} - 4 q^{97}+O(q^{100})$$ 4 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 + 2 * q^15 - 8 * q^17 + 8 * q^19 - 4 * q^21 - 10 * q^23 - 2 * q^25 - 20 * q^27 - 6 * q^29 + 12 * q^31 + 4 * q^35 - 24 * q^37 + 10 * q^41 + 4 * q^43 + 4 * q^45 - 14 * q^47 + 4 * q^51 - 8 * q^53 + 20 * q^57 - 12 * q^59 + 6 * q^61 - 26 * q^63 - 2 * q^67 - 22 * q^69 + 16 * q^73 + 4 * q^75 - 4 * q^79 + 14 * q^81 - 6 * q^83 - 4 * q^85 + 12 * q^87 - 28 * q^89 - 12 * q^93 + 4 * q^95 - 4 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

 $$n$$ $$181$$ $$271$$ $$577$$ $$641$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\beta_{2}$$

## 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
 −1.22474 − 0.707107i 1.22474 + 0.707107i −1.22474 + 0.707107i 1.22474 − 0.707107i
0 −1.72474 0.158919i 0 0.500000 + 0.866025i 0 −0.724745 + 1.25529i 0 2.94949 + 0.548188i 0
241.2 0 0.724745 1.57313i 0 0.500000 + 0.866025i 0 1.72474 2.98735i 0 −1.94949 2.28024i 0
481.1 0 −1.72474 + 0.158919i 0 0.500000 0.866025i 0 −0.724745 1.25529i 0 2.94949 0.548188i 0
481.2 0 0.724745 + 1.57313i 0 0.500000 0.866025i 0 1.72474 + 2.98735i 0 −1.94949 + 2.28024i 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.g 4
3.b odd 2 1 2160.2.q.g 4
4.b odd 2 1 360.2.q.c 4
9.c even 3 1 inner 720.2.q.g 4
9.c even 3 1 6480.2.a.bc 2
9.d odd 6 1 2160.2.q.g 4
9.d odd 6 1 6480.2.a.bl 2
12.b even 2 1 1080.2.q.c 4
36.f odd 6 1 360.2.q.c 4
36.f odd 6 1 3240.2.a.j 2
36.h even 6 1 1080.2.q.c 4
36.h even 6 1 3240.2.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.q.c 4 4.b odd 2 1
360.2.q.c 4 36.f odd 6 1
720.2.q.g 4 1.a even 1 1 trivial
720.2.q.g 4 9.c even 3 1 inner
1080.2.q.c 4 12.b even 2 1
1080.2.q.c 4 36.h even 6 1
2160.2.q.g 4 3.b odd 2 1
2160.2.q.g 4 9.d odd 6 1
3240.2.a.j 2 36.f odd 6 1
3240.2.a.o 2 36.h even 6 1
6480.2.a.bc 2 9.c even 3 1
6480.2.a.bl 2 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 $$S_{2}^{\mathrm{new}}(720, [\chi])$$:

 $$T_{7}^{4} - 2T_{7}^{3} + 9T_{7}^{2} + 10T_{7} + 25$$ T7^4 - 2*T7^3 + 9*T7^2 + 10*T7 + 25 $$T_{11}$$ T11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 2 T^{3} + \cdots + 9$$
$5$ $$(T^{2} - T + 1)^{2}$$
$7$ $$T^{4} - 2 T^{3} + \cdots + 25$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$(T + 2)^{4}$$
$19$ $$(T^{2} - 4 T - 20)^{2}$$
$23$ $$T^{4} + 10 T^{3} + \cdots + 361$$
$29$ $$T^{4} + 6 T^{3} + \cdots + 225$$
$31$ $$T^{4} - 12 T^{3} + \cdots + 144$$
$37$ $$(T + 6)^{4}$$
$41$ $$T^{4} - 10 T^{3} + \cdots + 1$$
$43$ $$T^{4} - 4 T^{3} + \cdots + 8464$$
$47$ $$T^{4} + 14 T^{3} + \cdots + 1849$$
$53$ $$(T^{2} + 4 T - 92)^{2}$$
$59$ $$T^{4} + 12 T^{3} + \cdots + 144$$
$61$ $$(T^{2} - 3 T + 9)^{2}$$
$67$ $$T^{4} + 2 T^{3} + \cdots + 22201$$
$71$ $$(T^{2} - 96)^{2}$$
$73$ $$(T^{2} - 8 T - 80)^{2}$$
$79$ $$T^{4} + 4 T^{3} + \cdots + 400$$
$83$ $$T^{4} + 6 T^{3} + \cdots + 9$$
$89$ $$(T^{2} + 14 T - 47)^{2}$$
$97$ $$(T^{2} + 2 T + 4)^{2}$$