# Properties

 Label 720.4.a.t 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 90) 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)

 $$f(q)$$ $$=$$ $$q + 5 q^{5} - 14 q^{7}+O(q^{10})$$ q + 5 * q^5 - 14 * q^7 $$q + 5 q^{5} - 14 q^{7} + 6 q^{11} + 68 q^{13} - 78 q^{17} - 44 q^{19} + 120 q^{23} + 25 q^{25} - 126 q^{29} + 244 q^{31} - 70 q^{35} - 304 q^{37} + 480 q^{41} - 104 q^{43} + 600 q^{47} - 147 q^{49} + 258 q^{53} + 30 q^{55} + 534 q^{59} + 362 q^{61} + 340 q^{65} + 268 q^{67} - 972 q^{71} + 470 q^{73} - 84 q^{77} - 1244 q^{79} + 396 q^{83} - 390 q^{85} + 972 q^{89} - 952 q^{91} - 220 q^{95} - 46 q^{97}+O(q^{100})$$ q + 5 * q^5 - 14 * q^7 + 6 * q^11 + 68 * q^13 - 78 * q^17 - 44 * q^19 + 120 * q^23 + 25 * q^25 - 126 * q^29 + 244 * q^31 - 70 * q^35 - 304 * q^37 + 480 * q^41 - 104 * q^43 + 600 * q^47 - 147 * q^49 + 258 * q^53 + 30 * q^55 + 534 * q^59 + 362 * q^61 + 340 * q^65 + 268 * q^67 - 972 * q^71 + 470 * q^73 - 84 * q^77 - 1244 * q^79 + 396 * q^83 - 390 * q^85 + 972 * q^89 - 952 * q^91 - 220 * q^95 - 46 * 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 −14.0000 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.t 1
3.b odd 2 1 720.4.a.e 1
4.b odd 2 1 90.4.a.e yes 1
12.b even 2 1 90.4.a.b 1
20.d odd 2 1 450.4.a.c 1
20.e even 4 2 450.4.c.f 2
36.f odd 6 2 810.4.e.a 2
36.h even 6 2 810.4.e.u 2
60.h even 2 1 450.4.a.m 1
60.l odd 4 2 450.4.c.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.4.a.b 1 12.b even 2 1
90.4.a.e yes 1 4.b odd 2 1
450.4.a.c 1 20.d odd 2 1
450.4.a.m 1 60.h even 2 1
450.4.c.f 2 20.e even 4 2
450.4.c.g 2 60.l odd 4 2
720.4.a.e 1 3.b odd 2 1
720.4.a.t 1 1.a even 1 1 trivial
810.4.e.a 2 36.f odd 6 2
810.4.e.u 2 36.h even 6 2

## Hecke kernels

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

 $$T_{7} + 14$$ T7 + 14 $$T_{11} - 6$$ T11 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 5$$
$7$ $$T + 14$$
$11$ $$T - 6$$
$13$ $$T - 68$$
$17$ $$T + 78$$
$19$ $$T + 44$$
$23$ $$T - 120$$
$29$ $$T + 126$$
$31$ $$T - 244$$
$37$ $$T + 304$$
$41$ $$T - 480$$
$43$ $$T + 104$$
$47$ $$T - 600$$
$53$ $$T - 258$$
$59$ $$T - 534$$
$61$ $$T - 362$$
$67$ $$T - 268$$
$71$ $$T + 972$$
$73$ $$T - 470$$
$79$ $$T + 1244$$
$83$ $$T - 396$$
$89$ $$T - 972$$
$97$ $$T + 46$$