# Properties

 Label 3600.2.a.bq 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 600) 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)

 $$f(q)$$ $$=$$ $$q + 5 q^{7}+O(q^{10})$$ q + 5 * q^7 $$q + 5 q^{7} - 6 q^{11} - 3 q^{13} + 2 q^{17} - q^{19} - 2 q^{23} - 6 q^{29} - 3 q^{31} - 6 q^{37} - 4 q^{41} - 11 q^{43} - 10 q^{47} + 18 q^{49} + 8 q^{53} - 6 q^{59} + 3 q^{61} + q^{67} - 12 q^{71} + 10 q^{73} - 30 q^{77} + 8 q^{79} - 6 q^{83} + 16 q^{89} - 15 q^{91} - 7 q^{97}+O(q^{100})$$ q + 5 * q^7 - 6 * q^11 - 3 * q^13 + 2 * q^17 - q^19 - 2 * q^23 - 6 * q^29 - 3 * q^31 - 6 * q^37 - 4 * q^41 - 11 * q^43 - 10 * q^47 + 18 * q^49 + 8 * q^53 - 6 * q^59 + 3 * q^61 + q^67 - 12 * q^71 + 10 * q^73 - 30 * q^77 + 8 * q^79 - 6 * q^83 + 16 * q^89 - 15 * 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 5.00000 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.bq 1
3.b odd 2 1 1200.2.a.i 1
4.b odd 2 1 1800.2.a.a 1
5.b even 2 1 3600.2.a.a 1
5.c odd 4 2 3600.2.f.b 2
12.b even 2 1 600.2.a.f yes 1
15.d odd 2 1 1200.2.a.j 1
15.e even 4 2 1200.2.f.i 2
20.d odd 2 1 1800.2.a.x 1
20.e even 4 2 1800.2.f.k 2
24.f even 2 1 4800.2.a.b 1
24.h odd 2 1 4800.2.a.cs 1
60.h even 2 1 600.2.a.e 1
60.l odd 4 2 600.2.f.a 2
120.i odd 2 1 4800.2.a.a 1
120.m even 2 1 4800.2.a.ct 1
120.q odd 4 2 4800.2.f.bj 2
120.w even 4 2 4800.2.f.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.a.e 1 60.h even 2 1
600.2.a.f yes 1 12.b even 2 1
600.2.f.a 2 60.l odd 4 2
1200.2.a.i 1 3.b odd 2 1
1200.2.a.j 1 15.d odd 2 1
1200.2.f.i 2 15.e even 4 2
1800.2.a.a 1 4.b odd 2 1
1800.2.a.x 1 20.d odd 2 1
1800.2.f.k 2 20.e even 4 2
3600.2.a.a 1 5.b even 2 1
3600.2.a.bq 1 1.a even 1 1 trivial
3600.2.f.b 2 5.c odd 4 2
4800.2.a.a 1 120.i odd 2 1
4800.2.a.b 1 24.f even 2 1
4800.2.a.cs 1 24.h odd 2 1
4800.2.a.ct 1 120.m even 2 1
4800.2.f.a 2 120.w even 4 2
4800.2.f.bj 2 120.q odd 4 2

## Hecke kernels

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

 $$T_{7} - 5$$ T7 - 5 $$T_{11} + 6$$ T11 + 6 $$T_{13} + 3$$ T13 + 3 $$T_{17} - 2$$ T17 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 5$$
$11$ $$T + 6$$
$13$ $$T + 3$$
$17$ $$T - 2$$
$19$ $$T + 1$$
$23$ $$T + 2$$
$29$ $$T + 6$$
$31$ $$T + 3$$
$37$ $$T + 6$$
$41$ $$T + 4$$
$43$ $$T + 11$$
$47$ $$T + 10$$
$53$ $$T - 8$$
$59$ $$T + 6$$
$61$ $$T - 3$$
$67$ $$T - 1$$
$71$ $$T + 12$$
$73$ $$T - 10$$
$79$ $$T - 8$$
$83$ $$T + 6$$
$89$ $$T - 16$$
$97$ $$T + 7$$