# Properties

 Label 3600.1.j.b Level $3600$ Weight $1$ Character orbit 3600.j Analytic conductor $1.797$ Analytic rank $0$ Dimension $4$ Projective image $D_{6}$ CM discriminant -3 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,1,Mod(1999,3600)]

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([1, 0, 0, 1]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3600.1999");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3600.j (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$1.79663404548$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.2.4320000.2

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + (\zeta_{12}^{5} - \zeta_{12}) q^{7}+O(q^{10})$$ q + (z^5 - z) * q^7 $$q + (\zeta_{12}^{5} - \zeta_{12}) q^{7} - \zeta_{12}^{3} q^{13} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{19} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{31} + 2 \zeta_{12}^{3} q^{37} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{43} + ( - \zeta_{12}^{4} + \zeta_{12}^{2} + 1) q^{49} - q^{61} + (\zeta_{12}^{5} - \zeta_{12}) q^{67} + 2 \zeta_{12}^{3} q^{73} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{91} + \zeta_{12}^{3} q^{97} +O(q^{100})$$ q + (z^5 - z) * q^7 - z^3 * q^13 + (z^4 + z^2) * q^19 + (z^4 + z^2) * q^31 + 2*z^3 * q^37 + (-z^5 + z) * q^43 + (-z^4 + z^2 + 1) * q^49 - q^61 + (z^5 - z) * q^67 + 2*z^3 * q^73 + (z^4 + z^2) * q^91 + z^3 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 8 q^{49} - 4 q^{61}+O(q^{100})$$ 4 * q + 8 * q^49 - 4 * q^61

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$2801$$ $$3151$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1999.1
 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i
0 0 0 0 0 −1.73205 0 0 0
1999.2 0 0 0 0 0 −1.73205 0 0 0
1999.3 0 0 0 0 0 1.73205 0 0 0
1999.4 0 0 0 0 0 1.73205 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.1.j.b 4
3.b odd 2 1 CM 3600.1.j.b 4
4.b odd 2 1 inner 3600.1.j.b 4
5.b even 2 1 inner 3600.1.j.b 4
5.c odd 4 1 3600.1.e.c 2
5.c odd 4 1 3600.1.e.d yes 2
12.b even 2 1 inner 3600.1.j.b 4
15.d odd 2 1 inner 3600.1.j.b 4
15.e even 4 1 3600.1.e.c 2
15.e even 4 1 3600.1.e.d yes 2
20.d odd 2 1 inner 3600.1.j.b 4
20.e even 4 1 3600.1.e.c 2
20.e even 4 1 3600.1.e.d yes 2
60.h even 2 1 inner 3600.1.j.b 4
60.l odd 4 1 3600.1.e.c 2
60.l odd 4 1 3600.1.e.d yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3600.1.e.c 2 5.c odd 4 1
3600.1.e.c 2 15.e even 4 1
3600.1.e.c 2 20.e even 4 1
3600.1.e.c 2 60.l odd 4 1
3600.1.e.d yes 2 5.c odd 4 1
3600.1.e.d yes 2 15.e even 4 1
3600.1.e.d yes 2 20.e even 4 1
3600.1.e.d yes 2 60.l odd 4 1
3600.1.j.b 4 1.a even 1 1 trivial
3600.1.j.b 4 3.b odd 2 1 CM
3600.1.j.b 4 4.b odd 2 1 inner
3600.1.j.b 4 5.b even 2 1 inner
3600.1.j.b 4 12.b even 2 1 inner
3600.1.j.b 4 15.d odd 2 1 inner
3600.1.j.b 4 20.d odd 2 1 inner
3600.1.j.b 4 60.h even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} - 3$$ acting on $$S_{1}^{\mathrm{new}}(3600, [\chi])$$.

## Hecke characteristic polynomials

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