# Properties

 Label 3600.1.e.c Level $3600$ Weight $1$ Character orbit 3600.e Analytic conductor $1.797$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,1,Mod(3151,3600)]

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, 0]))

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

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

chi := DirichletCharacter("3600.3151");

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.e (of order $$2$$, degree $$1$$, 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: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6}^{2} + \zeta_{6}) q^{7}+O(q^{10})$$ q + (z^2 + z) * q^7 $$q + (\zeta_{6}^{2} + \zeta_{6}) q^{7} - q^{13} + (\zeta_{6}^{2} + \zeta_{6}) q^{19} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{31} - 2 q^{37} + (\zeta_{6}^{2} + \zeta_{6}) q^{43} + (\zeta_{6}^{2} - \zeta_{6} - 1) q^{49} - q^{61} + (\zeta_{6}^{2} + \zeta_{6}) q^{67} + 2 q^{73} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{91} - q^{97}+O(q^{100})$$ q + (z^2 + z) * q^7 - q^13 + (z^2 + z) * q^19 + (-z^2 - z) * q^31 - 2 * q^37 + (z^2 + z) * q^43 + (z^2 - z - 1) * q^49 - q^61 + (z^2 + z) * q^67 + 2 * q^73 + (-z^2 - z) * q^91 - q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 2 q^{13} - 4 q^{37} - 4 q^{49} - 2 q^{61} + 4 q^{73} - 2 q^{97}+O(q^{100})$$ 2 * q - 2 * q^13 - 4 * q^37 - 4 * q^49 - 2 * q^61 + 4 * q^73 - 2 * q^97

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3151.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 0 0 1.73205i 0 0 0
3151.2 0 0 0 0 0 1.73205i 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
12.b 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.e.c 2
3.b odd 2 1 CM 3600.1.e.c 2
4.b odd 2 1 inner 3600.1.e.c 2
5.b even 2 1 3600.1.e.d yes 2
5.c odd 4 2 3600.1.j.b 4
12.b even 2 1 inner 3600.1.e.c 2
15.d odd 2 1 3600.1.e.d yes 2
15.e even 4 2 3600.1.j.b 4
20.d odd 2 1 3600.1.e.d yes 2
20.e even 4 2 3600.1.j.b 4
60.h even 2 1 3600.1.e.d yes 2
60.l odd 4 2 3600.1.j.b 4

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3600, [\chi])$$:

 $$T_{7}^{2} + 3$$ T7^2 + 3 $$T_{13} + 1$$ T13 + 1

## Hecke characteristic polynomials

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