# Properties

 Label 720.4.a.r Level $720$ Weight $4$ Character orbit 720.a Self dual yes Analytic conductor $42.481$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) 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)

 $$f(q)$$ $$=$$ $$q + 5 q^{5} - 20 q^{7}+O(q^{10})$$ q + 5 * q^5 - 20 * q^7 $$q + 5 q^{5} - 20 q^{7} - 24 q^{11} + 74 q^{13} - 54 q^{17} + 124 q^{19} - 120 q^{23} + 25 q^{25} + 78 q^{29} - 200 q^{31} - 100 q^{35} - 70 q^{37} - 330 q^{41} - 92 q^{43} - 24 q^{47} + 57 q^{49} - 450 q^{53} - 120 q^{55} + 24 q^{59} - 322 q^{61} + 370 q^{65} + 196 q^{67} - 288 q^{71} - 430 q^{73} + 480 q^{77} + 520 q^{79} + 156 q^{83} - 270 q^{85} - 1026 q^{89} - 1480 q^{91} + 620 q^{95} - 286 q^{97}+O(q^{100})$$ q + 5 * q^5 - 20 * q^7 - 24 * q^11 + 74 * q^13 - 54 * q^17 + 124 * q^19 - 120 * q^23 + 25 * q^25 + 78 * q^29 - 200 * q^31 - 100 * q^35 - 70 * q^37 - 330 * q^41 - 92 * q^43 - 24 * q^47 + 57 * q^49 - 450 * q^53 - 120 * q^55 + 24 * q^59 - 322 * q^61 + 370 * q^65 + 196 * q^67 - 288 * q^71 - 430 * q^73 + 480 * q^77 + 520 * q^79 + 156 * q^83 - 270 * q^85 - 1026 * q^89 - 1480 * q^91 + 620 * q^95 - 286 * 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 −20.0000 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.r 1
3.b odd 2 1 240.4.a.f 1
4.b odd 2 1 45.4.a.b 1
12.b even 2 1 15.4.a.b 1
15.d odd 2 1 1200.4.a.o 1
15.e even 4 2 1200.4.f.m 2
20.d odd 2 1 225.4.a.g 1
20.e even 4 2 225.4.b.d 2
24.f even 2 1 960.4.a.bi 1
24.h odd 2 1 960.4.a.l 1
28.d even 2 1 2205.4.a.c 1
36.f odd 6 2 405.4.e.k 2
36.h even 6 2 405.4.e.d 2
60.h even 2 1 75.4.a.a 1
60.l odd 4 2 75.4.b.a 2
84.h odd 2 1 735.4.a.i 1
132.d odd 2 1 1815.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.b 1 12.b even 2 1
45.4.a.b 1 4.b odd 2 1
75.4.a.a 1 60.h even 2 1
75.4.b.a 2 60.l odd 4 2
225.4.a.g 1 20.d odd 2 1
225.4.b.d 2 20.e even 4 2
240.4.a.f 1 3.b odd 2 1
405.4.e.d 2 36.h even 6 2
405.4.e.k 2 36.f odd 6 2
720.4.a.r 1 1.a even 1 1 trivial
735.4.a.i 1 84.h odd 2 1
960.4.a.l 1 24.h odd 2 1
960.4.a.bi 1 24.f even 2 1
1200.4.a.o 1 15.d odd 2 1
1200.4.f.m 2 15.e even 4 2
1815.4.a.a 1 132.d odd 2 1
2205.4.a.c 1 28.d even 2 1

## Hecke kernels

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

 $$T_{7} + 20$$ T7 + 20 $$T_{11} + 24$$ T11 + 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 5$$
$7$ $$T + 20$$
$11$ $$T + 24$$
$13$ $$T - 74$$
$17$ $$T + 54$$
$19$ $$T - 124$$
$23$ $$T + 120$$
$29$ $$T - 78$$
$31$ $$T + 200$$
$37$ $$T + 70$$
$41$ $$T + 330$$
$43$ $$T + 92$$
$47$ $$T + 24$$
$53$ $$T + 450$$
$59$ $$T - 24$$
$61$ $$T + 322$$
$67$ $$T - 196$$
$71$ $$T + 288$$
$73$ $$T + 430$$
$79$ $$T - 520$$
$83$ $$T - 156$$
$89$ $$T + 1026$$
$97$ $$T + 286$$