# Properties

 Label 3600.2.a.p Level $3600$ Weight $2$ Character orbit 3600.a Self dual yes Analytic conductor $28.746$ 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] = [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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1800) 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)

 $$f(q)$$ $$=$$ $$q - q^{7}+O(q^{10})$$ q - q^7 $$q - q^{7} - 4 q^{11} - q^{13} + 4 q^{17} - q^{19} + 4 q^{23} + 4 q^{29} + 5 q^{31} - 6 q^{37} + 12 q^{41} - 5 q^{43} - 8 q^{47} - 6 q^{49} - 12 q^{53} - 8 q^{59} + 7 q^{61} - 13 q^{67} - 12 q^{71} - 6 q^{73} + 4 q^{77} - 12 q^{79} + 8 q^{83} + q^{91} - 13 q^{97}+O(q^{100})$$ q - q^7 - 4 * q^11 - q^13 + 4 * q^17 - q^19 + 4 * q^23 + 4 * q^29 + 5 * q^31 - 6 * q^37 + 12 * q^41 - 5 * q^43 - 8 * q^47 - 6 * q^49 - 12 * q^53 - 8 * q^59 + 7 * q^61 - 13 * q^67 - 12 * q^71 - 6 * q^73 + 4 * q^77 - 12 * q^79 + 8 * q^83 + q^91 - 13 * 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 0 0 −1.00000 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 3600.2.a.p 1
3.b odd 2 1 3600.2.a.r 1
4.b odd 2 1 1800.2.a.p yes 1
5.b even 2 1 3600.2.a.w 1
5.c odd 4 2 3600.2.f.d 2
12.b even 2 1 1800.2.a.o yes 1
15.d odd 2 1 3600.2.a.y 1
15.e even 4 2 3600.2.f.s 2
20.d odd 2 1 1800.2.a.l yes 1
20.e even 4 2 1800.2.f.i 2
60.h even 2 1 1800.2.a.k 1
60.l odd 4 2 1800.2.f.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1800.2.a.k 1 60.h even 2 1
1800.2.a.l yes 1 20.d odd 2 1
1800.2.a.o yes 1 12.b even 2 1
1800.2.a.p yes 1 4.b odd 2 1
1800.2.f.b 2 60.l odd 4 2
1800.2.f.i 2 20.e even 4 2
3600.2.a.p 1 1.a even 1 1 trivial
3600.2.a.r 1 3.b odd 2 1
3600.2.a.w 1 5.b even 2 1
3600.2.a.y 1 15.d odd 2 1
3600.2.f.d 2 5.c odd 4 2
3600.2.f.s 2 15.e even 4 2

## Hecke kernels

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

 $$T_{7} + 1$$ T7 + 1 $$T_{11} + 4$$ T11 + 4 $$T_{13} + 1$$ T13 + 1 $$T_{17} - 4$$ T17 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 1$$
$11$ $$T + 4$$
$13$ $$T + 1$$
$17$ $$T - 4$$
$19$ $$T + 1$$
$23$ $$T - 4$$
$29$ $$T - 4$$
$31$ $$T - 5$$
$37$ $$T + 6$$
$41$ $$T - 12$$
$43$ $$T + 5$$
$47$ $$T + 8$$
$53$ $$T + 12$$
$59$ $$T + 8$$
$61$ $$T - 7$$
$67$ $$T + 13$$
$71$ $$T + 12$$
$73$ $$T + 6$$
$79$ $$T + 12$$
$83$ $$T - 8$$
$89$ $$T$$
$97$ $$T + 13$$