# Properties

 Label 720.2.a.i Level $720$ Weight $2$ Character orbit 720.a Self dual yes Analytic conductor $5.749$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,2,Mod(1,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

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

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

chi := DirichletCharacter("720.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 720.a (trivial)

## 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: yes Analytic conductor: $$5.74922894553$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 360) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{5} - 2 q^{7}+O(q^{10})$$ q + q^5 - 2 * q^7 $$q + q^{5} - 2 q^{7} + 2 q^{11} + 4 q^{13} + 2 q^{17} - 4 q^{19} + 8 q^{23} + q^{25} + 10 q^{29} - 4 q^{31} - 2 q^{35} + 8 q^{43} + 8 q^{47} - 3 q^{49} - 6 q^{53} + 2 q^{55} - 14 q^{59} - 14 q^{61} + 4 q^{65} + 4 q^{67} + 12 q^{71} + 6 q^{73} - 4 q^{77} + 12 q^{79} + 4 q^{83} + 2 q^{85} + 12 q^{89} - 8 q^{91} - 4 q^{95} - 14 q^{97}+O(q^{100})$$ q + q^5 - 2 * q^7 + 2 * q^11 + 4 * q^13 + 2 * q^17 - 4 * q^19 + 8 * q^23 + q^25 + 10 * q^29 - 4 * q^31 - 2 * q^35 + 8 * q^43 + 8 * q^47 - 3 * q^49 - 6 * q^53 + 2 * q^55 - 14 * q^59 - 14 * q^61 + 4 * q^65 + 4 * q^67 + 12 * q^71 + 6 * q^73 - 4 * q^77 + 12 * q^79 + 4 * q^83 + 2 * q^85 + 12 * q^89 - 8 * q^91 - 4 * q^95 - 14 * q^97

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 1.00000 0 −2.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$3$$ $$+1$$
$$5$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.a.i 1
3.b odd 2 1 720.2.a.a 1
4.b odd 2 1 360.2.a.d yes 1
5.b even 2 1 3600.2.a.bh 1
5.c odd 4 2 3600.2.f.q 2
8.b even 2 1 2880.2.a.e 1
8.d odd 2 1 2880.2.a.n 1
12.b even 2 1 360.2.a.c 1
15.d odd 2 1 3600.2.a.bd 1
15.e even 4 2 3600.2.f.g 2
20.d odd 2 1 1800.2.a.f 1
20.e even 4 2 1800.2.f.d 2
24.f even 2 1 2880.2.a.bd 1
24.h odd 2 1 2880.2.a.w 1
36.f odd 6 2 3240.2.q.d 2
36.h even 6 2 3240.2.q.n 2
60.h even 2 1 1800.2.a.i 1
60.l odd 4 2 1800.2.f.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.a.c 1 12.b even 2 1
360.2.a.d yes 1 4.b odd 2 1
720.2.a.a 1 3.b odd 2 1
720.2.a.i 1 1.a even 1 1 trivial
1800.2.a.f 1 20.d odd 2 1
1800.2.a.i 1 60.h even 2 1
1800.2.f.d 2 20.e even 4 2
1800.2.f.h 2 60.l odd 4 2
2880.2.a.e 1 8.b even 2 1
2880.2.a.n 1 8.d odd 2 1
2880.2.a.w 1 24.h odd 2 1
2880.2.a.bd 1 24.f even 2 1
3240.2.q.d 2 36.f odd 6 2
3240.2.q.n 2 36.h even 6 2
3600.2.a.bd 1 15.d odd 2 1
3600.2.a.bh 1 5.b even 2 1
3600.2.f.g 2 15.e even 4 2
3600.2.f.q 2 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(720))$$:

 $$T_{7} + 2$$ T7 + 2 $$T_{11} - 2$$ T11 - 2 $$T_{13} - 4$$ T13 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T + 2$$
$11$ $$T - 2$$
$13$ $$T - 4$$
$17$ $$T - 2$$
$19$ $$T + 4$$
$23$ $$T - 8$$
$29$ $$T - 10$$
$31$ $$T + 4$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T - 8$$
$47$ $$T - 8$$
$53$ $$T + 6$$
$59$ $$T + 14$$
$61$ $$T + 14$$
$67$ $$T - 4$$
$71$ $$T - 12$$
$73$ $$T - 6$$
$79$ $$T - 12$$
$83$ $$T - 4$$
$89$ $$T - 12$$
$97$ $$T + 14$$