# Properties

 Label 3600.2.a.bs Level $3600$ Weight $2$ Character orbit 3600.a Self dual yes Analytic conductor $28.746$ Analytic rank $1$ Dimension $2$ CM discriminant -15 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,2,Mod(1,3600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3600.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: $$28.7461447277$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 45) Fricke sign: $$+1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q+O(q^{10})$$ q $$q - \beta q^{17} - 4 q^{19} + 2 \beta q^{23} - 8 q^{31} - 2 \beta q^{47} - 7 q^{49} + \beta q^{53} + 2 q^{61} - 16 q^{79} - 4 \beta q^{83} +O(q^{100})$$ q - b * q^17 - 4 * q^19 + 2*b * q^23 - 8 * q^31 - 2*b * q^47 - 7 * q^49 + b * q^53 + 2 * q^61 - 16 * q^79 - 4*b * q^83 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 8 q^{19} - 16 q^{31} - 14 q^{49} + 4 q^{61} - 32 q^{79}+O(q^{100})$$ 2 * q - 8 * q^19 - 16 * q^31 - 14 * q^49 + 4 * q^61 - 32 * q^79

## 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
 1.61803 −0.618034
0 0 0 0 0 0 0 0 0
1.2 0 0 0 0 0 0 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.2.a.bs 2
3.b odd 2 1 inner 3600.2.a.bs 2
4.b odd 2 1 225.2.a.f 2
5.b even 2 1 inner 3600.2.a.bs 2
5.c odd 4 2 720.2.f.d 2
12.b even 2 1 225.2.a.f 2
15.d odd 2 1 CM 3600.2.a.bs 2
15.e even 4 2 720.2.f.d 2
20.d odd 2 1 225.2.a.f 2
20.e even 4 2 45.2.b.a 2
40.i odd 4 2 2880.2.f.j 2
40.k even 4 2 2880.2.f.k 2
60.h even 2 1 225.2.a.f 2
60.l odd 4 2 45.2.b.a 2
120.q odd 4 2 2880.2.f.k 2
120.w even 4 2 2880.2.f.j 2
140.j odd 4 2 2205.2.d.a 2
180.v odd 12 4 405.2.j.c 4
180.x even 12 4 405.2.j.c 4
420.w even 4 2 2205.2.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.b.a 2 20.e even 4 2
45.2.b.a 2 60.l odd 4 2
225.2.a.f 2 4.b odd 2 1
225.2.a.f 2 12.b even 2 1
225.2.a.f 2 20.d odd 2 1
225.2.a.f 2 60.h even 2 1
405.2.j.c 4 180.v odd 12 4
405.2.j.c 4 180.x even 12 4
720.2.f.d 2 5.c odd 4 2
720.2.f.d 2 15.e even 4 2
2205.2.d.a 2 140.j odd 4 2
2205.2.d.a 2 420.w even 4 2
2880.2.f.j 2 40.i odd 4 2
2880.2.f.j 2 120.w even 4 2
2880.2.f.k 2 40.k even 4 2
2880.2.f.k 2 120.q odd 4 2
3600.2.a.bs 2 1.a even 1 1 trivial
3600.2.a.bs 2 3.b odd 2 1 inner
3600.2.a.bs 2 5.b even 2 1 inner
3600.2.a.bs 2 15.d odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3600))$$:

 $$T_{7}$$ T7 $$T_{11}$$ T11 $$T_{13}$$ T13 $$T_{17}^{2} - 20$$ T17^2 - 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 20$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} - 80$$
$29$ $$T^{2}$$
$31$ $$(T + 8)^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2} - 80$$
$53$ $$T^{2} - 20$$
$59$ $$T^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$(T + 16)^{2}$$
$83$ $$T^{2} - 320$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$