Properties

 Label 360.4.a.n Level $360$ Weight $4$ Character orbit 360.a Self dual yes Analytic conductor $21.241$ 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] = [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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + 5 q^{5} + 20 q^{7}+O(q^{10})$$ q + 5 * q^5 + 20 * q^7 $$q + 5 q^{5} + 20 q^{7} + 56 q^{11} - 86 q^{13} + 106 q^{17} + 4 q^{19} - 136 q^{23} + 25 q^{25} + 206 q^{29} - 152 q^{31} + 100 q^{35} + 282 q^{37} + 246 q^{41} + 412 q^{43} - 40 q^{47} + 57 q^{49} + 126 q^{53} + 280 q^{55} - 56 q^{59} - 2 q^{61} - 430 q^{65} - 388 q^{67} + 672 q^{71} + 1170 q^{73} + 1120 q^{77} + 408 q^{79} - 668 q^{83} + 530 q^{85} - 66 q^{89} - 1720 q^{91} + 20 q^{95} - 926 q^{97}+O(q^{100})$$ q + 5 * q^5 + 20 * q^7 + 56 * q^11 - 86 * q^13 + 106 * q^17 + 4 * q^19 - 136 * q^23 + 25 * q^25 + 206 * q^29 - 152 * q^31 + 100 * q^35 + 282 * q^37 + 246 * q^41 + 412 * q^43 - 40 * q^47 + 57 * q^49 + 126 * q^53 + 280 * q^55 - 56 * q^59 - 2 * q^61 - 430 * q^65 - 388 * q^67 + 672 * q^71 + 1170 * q^73 + 1120 * q^77 + 408 * q^79 - 668 * q^83 + 530 * q^85 - 66 * q^89 - 1720 * q^91 + 20 * q^95 - 926 * 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 360.4.a.n 1
3.b odd 2 1 120.4.a.b 1
4.b odd 2 1 720.4.a.q 1
5.b even 2 1 1800.4.a.f 1
5.c odd 4 2 1800.4.f.v 2
12.b even 2 1 240.4.a.g 1
15.d odd 2 1 600.4.a.i 1
15.e even 4 2 600.4.f.a 2
24.f even 2 1 960.4.a.k 1
24.h odd 2 1 960.4.a.bj 1
60.h even 2 1 1200.4.a.p 1
60.l odd 4 2 1200.4.f.t 2

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

 $$T_{7} - 20$$ T7 - 20 $$T_{11} - 56$$ T11 - 56

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 5$$
$7$ $$T - 20$$
$11$ $$T - 56$$
$13$ $$T + 86$$
$17$ $$T - 106$$
$19$ $$T - 4$$
$23$ $$T + 136$$
$29$ $$T - 206$$
$31$ $$T + 152$$
$37$ $$T - 282$$
$41$ $$T - 246$$
$43$ $$T - 412$$
$47$ $$T + 40$$
$53$ $$T - 126$$
$59$ $$T + 56$$
$61$ $$T + 2$$
$67$ $$T + 388$$
$71$ $$T - 672$$
$73$ $$T - 1170$$
$79$ $$T - 408$$
$83$ $$T + 668$$
$89$ $$T + 66$$
$97$ $$T + 926$$