Properties

 Label 3600.2.a.be 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 20) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{7}+O(q^{10})$$ q + 2 * q^7 $$q + 2 q^{7} - 2 q^{13} - 6 q^{17} + 4 q^{19} - 6 q^{23} - 6 q^{29} + 4 q^{31} - 2 q^{37} - 6 q^{41} - 10 q^{43} + 6 q^{47} - 3 q^{49} - 6 q^{53} + 12 q^{59} + 2 q^{61} + 2 q^{67} - 12 q^{71} - 2 q^{73} - 8 q^{79} - 6 q^{83} + 6 q^{89} - 4 q^{91} - 2 q^{97}+O(q^{100})$$ q + 2 * q^7 - 2 * q^13 - 6 * q^17 + 4 * q^19 - 6 * q^23 - 6 * q^29 + 4 * q^31 - 2 * q^37 - 6 * q^41 - 10 * q^43 + 6 * q^47 - 3 * q^49 - 6 * q^53 + 12 * q^59 + 2 * q^61 + 2 * q^67 - 12 * q^71 - 2 * q^73 - 8 * q^79 - 6 * q^83 + 6 * q^89 - 4 * q^91 - 2 * 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 2.00000 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.be 1
3.b odd 2 1 400.2.a.c 1
4.b odd 2 1 900.2.a.b 1
5.b even 2 1 720.2.a.h 1
5.c odd 4 2 3600.2.f.j 2
12.b even 2 1 100.2.a.a 1
15.d odd 2 1 80.2.a.b 1
15.e even 4 2 400.2.c.b 2
20.d odd 2 1 180.2.a.a 1
20.e even 4 2 900.2.d.c 2
24.f even 2 1 1600.2.a.c 1
24.h odd 2 1 1600.2.a.w 1
40.e odd 2 1 2880.2.a.m 1
40.f even 2 1 2880.2.a.f 1
60.h even 2 1 20.2.a.a 1
60.l odd 4 2 100.2.c.a 2
84.h odd 2 1 4900.2.a.e 1
105.g even 2 1 3920.2.a.h 1
120.i odd 2 1 320.2.a.a 1
120.m even 2 1 320.2.a.f 1
120.q odd 4 2 1600.2.c.d 2
120.w even 4 2 1600.2.c.e 2
140.c even 2 1 8820.2.a.g 1
165.d even 2 1 9680.2.a.ba 1
180.n even 6 2 1620.2.i.h 2
180.p odd 6 2 1620.2.i.b 2
240.t even 4 2 1280.2.d.c 2
240.bm odd 4 2 1280.2.d.g 2
420.o odd 2 1 980.2.a.h 1
420.w even 4 2 4900.2.e.f 2
420.ba even 6 2 980.2.i.i 2
420.be odd 6 2 980.2.i.c 2
660.g odd 2 1 2420.2.a.a 1
780.d even 2 1 3380.2.a.c 1
780.bb odd 4 2 3380.2.f.b 2
1020.b even 2 1 5780.2.a.f 1
1020.ba even 4 2 5780.2.c.a 2
1140.p odd 2 1 7220.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 60.h even 2 1
80.2.a.b 1 15.d odd 2 1
100.2.a.a 1 12.b even 2 1
100.2.c.a 2 60.l odd 4 2
180.2.a.a 1 20.d odd 2 1
320.2.a.a 1 120.i odd 2 1
320.2.a.f 1 120.m even 2 1
400.2.a.c 1 3.b odd 2 1
400.2.c.b 2 15.e even 4 2
720.2.a.h 1 5.b even 2 1
900.2.a.b 1 4.b odd 2 1
900.2.d.c 2 20.e even 4 2
980.2.a.h 1 420.o odd 2 1
980.2.i.c 2 420.be odd 6 2
980.2.i.i 2 420.ba even 6 2
1280.2.d.c 2 240.t even 4 2
1280.2.d.g 2 240.bm odd 4 2
1600.2.a.c 1 24.f even 2 1
1600.2.a.w 1 24.h odd 2 1
1600.2.c.d 2 120.q odd 4 2
1600.2.c.e 2 120.w even 4 2
1620.2.i.b 2 180.p odd 6 2
1620.2.i.h 2 180.n even 6 2
2420.2.a.a 1 660.g odd 2 1
2880.2.a.f 1 40.f even 2 1
2880.2.a.m 1 40.e odd 2 1
3380.2.a.c 1 780.d even 2 1
3380.2.f.b 2 780.bb odd 4 2
3600.2.a.be 1 1.a even 1 1 trivial
3600.2.f.j 2 5.c odd 4 2
3920.2.a.h 1 105.g even 2 1
4900.2.a.e 1 84.h odd 2 1
4900.2.e.f 2 420.w even 4 2
5780.2.a.f 1 1020.b even 2 1
5780.2.c.a 2 1020.ba even 4 2
7220.2.a.f 1 1140.p odd 2 1
8820.2.a.g 1 140.c even 2 1
9680.2.a.ba 1 165.d even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

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