# Properties

 Label 360.4.a.l Level $360$ Weight $4$ Character orbit 360.a Self dual yes Analytic conductor $21.241$ 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] = [360,4,Mod(1,360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(360, 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("360.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$360 = 2^{3} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 360.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: $$21.2406876021$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 120) 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 $$q + 5 q^{5} + 4 q^{7} - 72 q^{11} - 6 q^{13} - 38 q^{17} + 52 q^{19} - 152 q^{23} + 25 q^{25} + 78 q^{29} + 120 q^{31} + 20 q^{35} - 150 q^{37} - 362 q^{41} - 484 q^{43} - 280 q^{47} - 327 q^{49} + 670 q^{53} - 360 q^{55} - 696 q^{59} + 222 q^{61} - 30 q^{65} - 4 q^{67} - 96 q^{71} + 178 q^{73} - 288 q^{77} - 632 q^{79} + 612 q^{83} - 190 q^{85} - 994 q^{89} - 24 q^{91} + 260 q^{95} + 1634 q^{97}+O(q^{100})$$ q + 5 * q^5 + 4 * q^7 - 72 * q^11 - 6 * q^13 - 38 * q^17 + 52 * q^19 - 152 * q^23 + 25 * q^25 + 78 * q^29 + 120 * q^31 + 20 * q^35 - 150 * q^37 - 362 * q^41 - 484 * q^43 - 280 * q^47 - 327 * q^49 + 670 * q^53 - 360 * q^55 - 696 * q^59 + 222 * q^61 - 30 * q^65 - 4 * q^67 - 96 * q^71 + 178 * q^73 - 288 * q^77 - 632 * q^79 + 612 * q^83 - 190 * q^85 - 994 * q^89 - 24 * q^91 + 260 * q^95 + 1634 * 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 360.4.a.l 1
3.b odd 2 1 120.4.a.a 1
4.b odd 2 1 720.4.a.v 1
5.b even 2 1 1800.4.a.n 1
5.c odd 4 2 1800.4.f.a 2
12.b even 2 1 240.4.a.h 1
15.d odd 2 1 600.4.a.l 1
15.e even 4 2 600.4.f.i 2
24.f even 2 1 960.4.a.o 1
24.h odd 2 1 960.4.a.bf 1
60.h even 2 1 1200.4.a.k 1
60.l odd 4 2 1200.4.f.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.a.a 1 3.b odd 2 1
240.4.a.h 1 12.b even 2 1
360.4.a.l 1 1.a even 1 1 trivial
600.4.a.l 1 15.d odd 2 1
600.4.f.i 2 15.e even 4 2
720.4.a.v 1 4.b odd 2 1
960.4.a.o 1 24.f even 2 1
960.4.a.bf 1 24.h odd 2 1
1200.4.a.k 1 60.h even 2 1
1200.4.f.a 2 60.l odd 4 2
1800.4.a.n 1 5.b even 2 1
1800.4.f.a 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(360))$$:

 $$T_{7} - 4$$ T7 - 4 $$T_{11} + 72$$ T11 + 72

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 5$$
$7$ $$T - 4$$
$11$ $$T + 72$$
$13$ $$T + 6$$
$17$ $$T + 38$$
$19$ $$T - 52$$
$23$ $$T + 152$$
$29$ $$T - 78$$
$31$ $$T - 120$$
$37$ $$T + 150$$
$41$ $$T + 362$$
$43$ $$T + 484$$
$47$ $$T + 280$$
$53$ $$T - 670$$
$59$ $$T + 696$$
$61$ $$T - 222$$
$67$ $$T + 4$$
$71$ $$T + 96$$
$73$ $$T - 178$$
$79$ $$T + 632$$
$83$ $$T - 612$$
$89$ $$T + 994$$
$97$ $$T - 1634$$