# Properties

 Label 3600.1.bf.b Level $3600$ Weight $1$ Character orbit 3600.bf Analytic conductor $1.797$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -15 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,1,Mod(2557,3600)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3600, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 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("3600.2557");

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.bf (of order $$4$$, degree $$2$$, 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: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.153600.2

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + q^8 $$q + q^{2} + q^{4} + q^{8} + q^{16} + (i - 1) q^{17} + (i + 1) q^{19} + ( - i + 1) q^{23} + q^{32} + (i - 1) q^{34} + (i + 1) q^{38} + ( - i + 1) q^{46} + (i - 1) q^{47} - i q^{49} - i q^{53} + ( - i + 1) q^{61} + q^{64} + (i - 1) q^{68} + (i + 1) q^{76} + i q^{79} - q^{83} + ( - i + 1) q^{92} + (i - 1) q^{94} - i q^{98} +O(q^{100})$$ q + q^2 + q^4 + q^8 + q^16 + (z - 1) * q^17 + (z + 1) * q^19 + (-z + 1) * q^23 + q^32 + (z - 1) * q^34 + (z + 1) * q^38 + (-z + 1) * q^46 + (z - 1) * q^47 - z * q^49 - z * q^53 + (-z + 1) * q^61 + q^64 + (z - 1) * q^68 + (z + 1) * q^76 + z * q^79 - q^83 + (-z + 1) * q^92 + (z - 1) * q^94 - z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 2 q^{16} - 2 q^{17} + 2 q^{19} + 2 q^{23} + 2 q^{32} - 2 q^{34} + 2 q^{38} + 2 q^{46} - 2 q^{47} + 2 q^{61} + 2 q^{64} - 2 q^{68} + 2 q^{76} - 4 q^{83} + 2 q^{92} - 2 q^{94}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 2 * q^16 - 2 * q^17 + 2 * q^19 + 2 * q^23 + 2 * q^32 - 2 * q^34 + 2 * q^38 + 2 * q^46 - 2 * q^47 + 2 * q^61 + 2 * q^64 - 2 * q^68 + 2 * q^76 - 4 * q^83 + 2 * q^92 - 2 * q^94

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2557.1
 − 1.00000i 1.00000i
1.00000 0 1.00000 0 0 0 1.00000 0 0
3493.1 1.00000 0 1.00000 0 0 0 1.00000 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
80.t odd 4 1 inner
240.bb even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.1.bf.b yes 2
3.b odd 2 1 3600.1.bf.a yes 2
5.b even 2 1 3600.1.bf.a yes 2
5.c odd 4 1 3600.1.bb.a 2
5.c odd 4 1 3600.1.bb.b yes 2
15.d odd 2 1 CM 3600.1.bf.b yes 2
15.e even 4 1 3600.1.bb.a 2
15.e even 4 1 3600.1.bb.b yes 2
16.e even 4 1 3600.1.bb.a 2
48.i odd 4 1 3600.1.bb.b yes 2
80.i odd 4 1 3600.1.bf.a yes 2
80.q even 4 1 3600.1.bb.b yes 2
80.t odd 4 1 inner 3600.1.bf.b yes 2
240.bb even 4 1 inner 3600.1.bf.b yes 2
240.bf even 4 1 3600.1.bf.a yes 2
240.bm odd 4 1 3600.1.bb.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3600.1.bb.a 2 5.c odd 4 1
3600.1.bb.a 2 15.e even 4 1
3600.1.bb.a 2 16.e even 4 1
3600.1.bb.a 2 240.bm odd 4 1
3600.1.bb.b yes 2 5.c odd 4 1
3600.1.bb.b yes 2 15.e even 4 1
3600.1.bb.b yes 2 48.i odd 4 1
3600.1.bb.b yes 2 80.q even 4 1
3600.1.bf.a yes 2 3.b odd 2 1
3600.1.bf.a yes 2 5.b even 2 1
3600.1.bf.a yes 2 80.i odd 4 1
3600.1.bf.a yes 2 240.bf even 4 1
3600.1.bf.b yes 2 1.a even 1 1 trivial
3600.1.bf.b yes 2 15.d odd 2 1 CM
3600.1.bf.b yes 2 80.t odd 4 1 inner
3600.1.bf.b yes 2 240.bb even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{17}^{2} + 2T_{17} + 2$$ acting on $$S_{1}^{\mathrm{new}}(3600, [\chi])$$.

## Hecke characteristic polynomials

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