# Properties

 Label 720.4.a.bc Level $720$ Weight $4$ Character orbit 720.a Self dual yes Analytic conductor $42.481$ 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] = [720,4,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, 4, names="a")

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

chi := DirichletCharacter("720.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$42.4813752041$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 45) 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)

 $$f(q)$$ $$=$$ $$q + 5 q^{5} + 30 q^{7}+O(q^{10})$$ q + 5 * q^5 + 30 * q^7 $$q + 5 q^{5} + 30 q^{7} + 50 q^{11} - 20 q^{13} + 10 q^{17} + 44 q^{19} + 120 q^{23} + 25 q^{25} + 50 q^{29} - 108 q^{31} + 150 q^{35} - 40 q^{37} - 400 q^{41} - 280 q^{43} - 280 q^{47} + 557 q^{49} + 610 q^{53} + 250 q^{55} + 50 q^{59} - 518 q^{61} - 100 q^{65} + 180 q^{67} + 700 q^{71} - 410 q^{73} + 1500 q^{77} + 516 q^{79} + 660 q^{83} + 50 q^{85} + 1500 q^{89} - 600 q^{91} + 220 q^{95} - 1630 q^{97}+O(q^{100})$$ q + 5 * q^5 + 30 * q^7 + 50 * q^11 - 20 * q^13 + 10 * q^17 + 44 * q^19 + 120 * q^23 + 25 * q^25 + 50 * q^29 - 108 * q^31 + 150 * q^35 - 40 * q^37 - 400 * q^41 - 280 * q^43 - 280 * q^47 + 557 * q^49 + 610 * q^53 + 250 * q^55 + 50 * q^59 - 518 * q^61 - 100 * q^65 + 180 * q^67 + 700 * q^71 - 410 * q^73 + 1500 * q^77 + 516 * q^79 + 660 * q^83 + 50 * q^85 + 1500 * q^89 - 600 * q^91 + 220 * q^95 - 1630 * 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 5.00000 0 30.0000 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 720.4.a.bc 1
3.b odd 2 1 720.4.a.o 1
4.b odd 2 1 45.4.a.a 1
12.b even 2 1 45.4.a.e yes 1
20.d odd 2 1 225.4.a.h 1
20.e even 4 2 225.4.b.a 2
28.d even 2 1 2205.4.a.a 1
36.f odd 6 2 405.4.e.n 2
36.h even 6 2 405.4.e.b 2
60.h even 2 1 225.4.a.a 1
60.l odd 4 2 225.4.b.b 2
84.h odd 2 1 2205.4.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.a.a 1 4.b odd 2 1
45.4.a.e yes 1 12.b even 2 1
225.4.a.a 1 60.h even 2 1
225.4.a.h 1 20.d odd 2 1
225.4.b.a 2 20.e even 4 2
225.4.b.b 2 60.l odd 4 2
405.4.e.b 2 36.h even 6 2
405.4.e.n 2 36.f odd 6 2
720.4.a.o 1 3.b odd 2 1
720.4.a.bc 1 1.a even 1 1 trivial
2205.4.a.a 1 28.d even 2 1
2205.4.a.t 1 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(720))$$:

 $$T_{7} - 30$$ T7 - 30 $$T_{11} - 50$$ T11 - 50

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 5$$
$7$ $$T - 30$$
$11$ $$T - 50$$
$13$ $$T + 20$$
$17$ $$T - 10$$
$19$ $$T - 44$$
$23$ $$T - 120$$
$29$ $$T - 50$$
$31$ $$T + 108$$
$37$ $$T + 40$$
$41$ $$T + 400$$
$43$ $$T + 280$$
$47$ $$T + 280$$
$53$ $$T - 610$$
$59$ $$T - 50$$
$61$ $$T + 518$$
$67$ $$T - 180$$
$71$ $$T - 700$$
$73$ $$T + 410$$
$79$ $$T - 516$$
$83$ $$T - 660$$
$89$ $$T - 1500$$
$97$ $$T + 1630$$