# Properties

 Label 720.4.a.j 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 10) 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)

 $$f(q)$$ $$=$$ $$q - 5 q^{5} + 4 q^{7}+O(q^{10})$$ q - 5 * q^5 + 4 * q^7 $$q - 5 q^{5} + 4 q^{7} + 12 q^{11} - 58 q^{13} - 66 q^{17} + 100 q^{19} + 132 q^{23} + 25 q^{25} + 90 q^{29} - 152 q^{31} - 20 q^{35} - 34 q^{37} + 438 q^{41} - 32 q^{43} - 204 q^{47} - 327 q^{49} - 222 q^{53} - 60 q^{55} + 420 q^{59} + 902 q^{61} + 290 q^{65} + 1024 q^{67} + 432 q^{71} + 362 q^{73} + 48 q^{77} + 160 q^{79} + 72 q^{83} + 330 q^{85} - 810 q^{89} - 232 q^{91} - 500 q^{95} + 1106 q^{97}+O(q^{100})$$ q - 5 * q^5 + 4 * q^7 + 12 * q^11 - 58 * q^13 - 66 * q^17 + 100 * q^19 + 132 * q^23 + 25 * q^25 + 90 * q^29 - 152 * q^31 - 20 * q^35 - 34 * q^37 + 438 * q^41 - 32 * q^43 - 204 * q^47 - 327 * q^49 - 222 * q^53 - 60 * q^55 + 420 * q^59 + 902 * q^61 + 290 * q^65 + 1024 * q^67 + 432 * q^71 + 362 * q^73 + 48 * q^77 + 160 * q^79 + 72 * q^83 + 330 * q^85 - 810 * q^89 - 232 * q^91 - 500 * q^95 + 1106 * 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 4.00000 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 720.4.a.j 1
3.b odd 2 1 80.4.a.f 1
4.b odd 2 1 90.4.a.a 1
12.b even 2 1 10.4.a.a 1
15.d odd 2 1 400.4.a.b 1
15.e even 4 2 400.4.c.c 2
20.d odd 2 1 450.4.a.q 1
20.e even 4 2 450.4.c.d 2
24.f even 2 1 320.4.a.m 1
24.h odd 2 1 320.4.a.b 1
36.f odd 6 2 810.4.e.w 2
36.h even 6 2 810.4.e.c 2
48.i odd 4 2 1280.4.d.g 2
48.k even 4 2 1280.4.d.j 2
60.h even 2 1 50.4.a.c 1
60.l odd 4 2 50.4.b.a 2
84.h odd 2 1 490.4.a.o 1
84.j odd 6 2 490.4.e.a 2
84.n even 6 2 490.4.e.i 2
120.i odd 2 1 1600.4.a.bx 1
120.m even 2 1 1600.4.a.d 1
132.d odd 2 1 1210.4.a.b 1
156.h even 2 1 1690.4.a.a 1
420.o odd 2 1 2450.4.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.a.a 1 12.b even 2 1
50.4.a.c 1 60.h even 2 1
50.4.b.a 2 60.l odd 4 2
80.4.a.f 1 3.b odd 2 1
90.4.a.a 1 4.b odd 2 1
320.4.a.b 1 24.h odd 2 1
320.4.a.m 1 24.f even 2 1
400.4.a.b 1 15.d odd 2 1
400.4.c.c 2 15.e even 4 2
450.4.a.q 1 20.d odd 2 1
450.4.c.d 2 20.e even 4 2
490.4.a.o 1 84.h odd 2 1
490.4.e.a 2 84.j odd 6 2
490.4.e.i 2 84.n even 6 2
720.4.a.j 1 1.a even 1 1 trivial
810.4.e.c 2 36.h even 6 2
810.4.e.w 2 36.f odd 6 2
1210.4.a.b 1 132.d odd 2 1
1280.4.d.g 2 48.i odd 4 2
1280.4.d.j 2 48.k even 4 2
1600.4.a.d 1 120.m even 2 1
1600.4.a.bx 1 120.i odd 2 1
1690.4.a.a 1 156.h even 2 1
2450.4.a.b 1 420.o odd 2 1

## Hecke kernels

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

 $$T_{7} - 4$$ T7 - 4 $$T_{11} - 12$$ T11 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 5$$
$7$ $$T - 4$$
$11$ $$T - 12$$
$13$ $$T + 58$$
$17$ $$T + 66$$
$19$ $$T - 100$$
$23$ $$T - 132$$
$29$ $$T - 90$$
$31$ $$T + 152$$
$37$ $$T + 34$$
$41$ $$T - 438$$
$43$ $$T + 32$$
$47$ $$T + 204$$
$53$ $$T + 222$$
$59$ $$T - 420$$
$61$ $$T - 902$$
$67$ $$T - 1024$$
$71$ $$T - 432$$
$73$ $$T - 362$$
$79$ $$T - 160$$
$83$ $$T - 72$$
$89$ $$T + 810$$
$97$ $$T - 1106$$