# Properties

 Label 3600.2.a.v Level $3600$ Weight $2$ Character orbit 3600.a Self dual yes Analytic conductor $28.746$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 24) 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)

 $$f(q)$$ $$=$$ $$q+O(q^{10})$$ q $$q + 4 q^{11} + 2 q^{13} + 2 q^{17} + 4 q^{19} + 8 q^{23} - 6 q^{29} - 8 q^{31} - 6 q^{37} + 6 q^{41} + 4 q^{43} - 7 q^{49} - 2 q^{53} + 4 q^{59} - 2 q^{61} - 4 q^{67} + 8 q^{71} - 10 q^{73} + 8 q^{79} + 4 q^{83} + 6 q^{89} - 2 q^{97}+O(q^{100})$$ q + 4 * q^11 + 2 * q^13 + 2 * q^17 + 4 * q^19 + 8 * q^23 - 6 * q^29 - 8 * q^31 - 6 * q^37 + 6 * q^41 + 4 * q^43 - 7 * q^49 - 2 * q^53 + 4 * q^59 - 2 * q^61 - 4 * q^67 + 8 * q^71 - 10 * q^73 + 8 * q^79 + 4 * q^83 + 6 * q^89 - 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 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

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.v 1
3.b odd 2 1 1200.2.a.d 1
4.b odd 2 1 1800.2.a.m 1
5.b even 2 1 144.2.a.b 1
5.c odd 4 2 3600.2.f.r 2
12.b even 2 1 600.2.a.h 1
15.d odd 2 1 48.2.a.a 1
15.e even 4 2 1200.2.f.b 2
20.d odd 2 1 72.2.a.a 1
20.e even 4 2 1800.2.f.c 2
24.f even 2 1 4800.2.a.q 1
24.h odd 2 1 4800.2.a.cc 1
35.c odd 2 1 7056.2.a.q 1
40.e odd 2 1 576.2.a.d 1
40.f even 2 1 576.2.a.b 1
45.h odd 6 2 1296.2.i.m 2
45.j even 6 2 1296.2.i.e 2
60.h even 2 1 24.2.a.a 1
60.l odd 4 2 600.2.f.e 2
80.k odd 4 2 2304.2.d.i 2
80.q even 4 2 2304.2.d.k 2
105.g even 2 1 2352.2.a.i 1
105.o odd 6 2 2352.2.q.l 2
105.p even 6 2 2352.2.q.r 2
120.i odd 2 1 192.2.a.b 1
120.m even 2 1 192.2.a.d 1
120.q odd 4 2 4800.2.f.d 2
120.w even 4 2 4800.2.f.bg 2
140.c even 2 1 3528.2.a.d 1
140.p odd 6 2 3528.2.s.j 2
140.s even 6 2 3528.2.s.y 2
165.d even 2 1 5808.2.a.s 1
180.n even 6 2 648.2.i.g 2
180.p odd 6 2 648.2.i.b 2
195.e odd 2 1 8112.2.a.be 1
220.g even 2 1 8712.2.a.u 1
240.t even 4 2 768.2.d.e 2
240.bm odd 4 2 768.2.d.d 2
420.o odd 2 1 1176.2.a.i 1
420.ba even 6 2 1176.2.q.i 2
420.be odd 6 2 1176.2.q.a 2
660.g odd 2 1 2904.2.a.c 1
780.d even 2 1 4056.2.a.i 1
780.bb odd 4 2 4056.2.c.e 2
840.b odd 2 1 9408.2.a.h 1
840.u even 2 1 9408.2.a.cc 1
1020.b even 2 1 6936.2.a.p 1
1140.p odd 2 1 8664.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.a.a 1 60.h even 2 1
48.2.a.a 1 15.d odd 2 1
72.2.a.a 1 20.d odd 2 1
144.2.a.b 1 5.b even 2 1
192.2.a.b 1 120.i odd 2 1
192.2.a.d 1 120.m even 2 1
576.2.a.b 1 40.f even 2 1
576.2.a.d 1 40.e odd 2 1
600.2.a.h 1 12.b even 2 1
600.2.f.e 2 60.l odd 4 2
648.2.i.b 2 180.p odd 6 2
648.2.i.g 2 180.n even 6 2
768.2.d.d 2 240.bm odd 4 2
768.2.d.e 2 240.t even 4 2
1176.2.a.i 1 420.o odd 2 1
1176.2.q.a 2 420.be odd 6 2
1176.2.q.i 2 420.ba even 6 2
1200.2.a.d 1 3.b odd 2 1
1200.2.f.b 2 15.e even 4 2
1296.2.i.e 2 45.j even 6 2
1296.2.i.m 2 45.h odd 6 2
1800.2.a.m 1 4.b odd 2 1
1800.2.f.c 2 20.e even 4 2
2304.2.d.i 2 80.k odd 4 2
2304.2.d.k 2 80.q even 4 2
2352.2.a.i 1 105.g even 2 1
2352.2.q.l 2 105.o odd 6 2
2352.2.q.r 2 105.p even 6 2
2904.2.a.c 1 660.g odd 2 1
3528.2.a.d 1 140.c even 2 1
3528.2.s.j 2 140.p odd 6 2
3528.2.s.y 2 140.s even 6 2
3600.2.a.v 1 1.a even 1 1 trivial
3600.2.f.r 2 5.c odd 4 2
4056.2.a.i 1 780.d even 2 1
4056.2.c.e 2 780.bb odd 4 2
4800.2.a.q 1 24.f even 2 1
4800.2.a.cc 1 24.h odd 2 1
4800.2.f.d 2 120.q odd 4 2
4800.2.f.bg 2 120.w even 4 2
5808.2.a.s 1 165.d even 2 1
6936.2.a.p 1 1020.b even 2 1
7056.2.a.q 1 35.c odd 2 1
8112.2.a.be 1 195.e odd 2 1
8664.2.a.j 1 1140.p odd 2 1
8712.2.a.u 1 220.g even 2 1
9408.2.a.h 1 840.b odd 2 1
9408.2.a.cc 1 840.u even 2 1

## Hecke kernels

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

 $$T_{7}$$ T7 $$T_{11} - 4$$ T11 - 4 $$T_{13} - 2$$ T13 - 2 $$T_{17} - 2$$ T17 - 2

## Hecke characteristic polynomials

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