# Properties

 Label 3600.2.a.s 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 300) 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 $$q - q^{7} + 6 q^{11} - 5 q^{13} - 6 q^{17} - 5 q^{19} + 6 q^{23} + 6 q^{29} + q^{31} - 2 q^{37} - q^{43} - 6 q^{47} - 6 q^{49} - 12 q^{53} - 6 q^{59} - 13 q^{61} + 11 q^{67} - 2 q^{73} - 6 q^{77} - 8 q^{79} + 6 q^{83} + 5 q^{91} + 7 q^{97}+O(q^{100})$$ q - q^7 + 6 * q^11 - 5 * q^13 - 6 * q^17 - 5 * q^19 + 6 * q^23 + 6 * q^29 + q^31 - 2 * q^37 - q^43 - 6 * q^47 - 6 * q^49 - 12 * q^53 - 6 * q^59 - 13 * q^61 + 11 * q^67 - 2 * q^73 - 6 * q^77 - 8 * q^79 + 6 * q^83 + 5 * q^91 + 7 * 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.s 1
3.b odd 2 1 1200.2.a.n 1
4.b odd 2 1 900.2.a.e 1
5.b even 2 1 3600.2.a.z 1
5.c odd 4 2 3600.2.f.v 2
12.b even 2 1 300.2.a.b 1
15.d odd 2 1 1200.2.a.f 1
15.e even 4 2 1200.2.f.a 2
20.d odd 2 1 900.2.a.c 1
20.e even 4 2 900.2.d.a 2
24.f even 2 1 4800.2.a.ce 1
24.h odd 2 1 4800.2.a.p 1
60.h even 2 1 300.2.a.c yes 1
60.l odd 4 2 300.2.d.a 2
120.i odd 2 1 4800.2.a.cf 1
120.m even 2 1 4800.2.a.o 1
120.q odd 4 2 4800.2.f.b 2
120.w even 4 2 4800.2.f.bi 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.a.b 1 12.b even 2 1
300.2.a.c yes 1 60.h even 2 1
300.2.d.a 2 60.l odd 4 2
900.2.a.c 1 20.d odd 2 1
900.2.a.e 1 4.b odd 2 1
900.2.d.a 2 20.e even 4 2
1200.2.a.f 1 15.d odd 2 1
1200.2.a.n 1 3.b odd 2 1
1200.2.f.a 2 15.e even 4 2
3600.2.a.s 1 1.a even 1 1 trivial
3600.2.a.z 1 5.b even 2 1
3600.2.f.v 2 5.c odd 4 2
4800.2.a.o 1 120.m even 2 1
4800.2.a.p 1 24.h odd 2 1
4800.2.a.ce 1 24.f even 2 1
4800.2.a.cf 1 120.i odd 2 1
4800.2.f.b 2 120.q odd 4 2
4800.2.f.bi 2 120.w 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 $$T_{11} - 6$$ T11 - 6 $$T_{13} + 5$$ T13 + 5 $$T_{17} + 6$$ T17 + 6

## Hecke characteristic polynomials

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