# Properties

 Label 3600.1.bo.a Level $3600$ Weight $1$ Character orbit 3600.bo Analytic conductor $1.797$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -15 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,1,Mod(451,3600)]

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([2, 3, 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("3600.451");

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.bo (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: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 720) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.0.92160.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} - \zeta_{8}^{3} q^{8} +O(q^{10})$$ q - z * q^2 + z^2 * q^4 - z^3 * q^8 $$q - \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} - \zeta_{8}^{3} q^{8} - q^{16} + (\zeta_{8}^{3} - \zeta_{8}) q^{17} + (\zeta_{8}^{2} - 1) q^{19} + (\zeta_{8}^{3} - \zeta_{8}) q^{23} + 2 \zeta_{8}^{2} q^{31} + \zeta_{8} q^{32} + (\zeta_{8}^{2} + 1) q^{34} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{38} + (\zeta_{8}^{2} + 1) q^{46} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{47} - q^{49} + ( - \zeta_{8}^{2} - 1) q^{61} - 2 \zeta_{8}^{3} q^{62} - \zeta_{8}^{2} q^{64} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{68} + ( - \zeta_{8}^{2} - 1) q^{76} + 2 \zeta_{8}^{3} q^{83} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{92} + (\zeta_{8}^{2} - 1) q^{94} + \zeta_{8} q^{98} +O(q^{100})$$ q - z * q^2 + z^2 * q^4 - z^3 * q^8 - q^16 + (z^3 - z) * q^17 + (z^2 - 1) * q^19 + (z^3 - z) * q^23 + 2*z^2 * q^31 + z * q^32 + (z^2 + 1) * q^34 + (-z^3 + z) * q^38 + (z^2 + 1) * q^46 + (-z^3 - z) * q^47 - q^49 + (-z^2 - 1) * q^61 - 2*z^3 * q^62 - z^2 * q^64 + (-z^3 - z) * q^68 + (-z^2 - 1) * q^76 + 2*z^3 * q^83 + (-z^3 - z) * q^92 + (z^2 - 1) * q^94 + z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 4 q^{16} - 4 q^{19} + 4 q^{34} + 4 q^{46} - 4 q^{49} - 4 q^{61} - 4 q^{76} - 4 q^{94}+O(q^{100})$$ 4 * q - 4 * q^16 - 4 * q^19 + 4 * q^34 + 4 * q^46 - 4 * q^49 - 4 * q^61 - 4 * q^76 - 4 * q^94

## Character values

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

 $$n$$ $$577$$ $$901$$ $$2801$$ $$3151$$ $$\chi(n)$$ $$1$$ $$-\zeta_{8}^{2}$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
451.1
 0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i
−0.707107 0.707107i 0 1.00000i 0 0 0 0.707107 0.707107i 0 0
451.2 0.707107 + 0.707107i 0 1.00000i 0 0 0 −0.707107 + 0.707107i 0 0
2251.1 −0.707107 + 0.707107i 0 1.00000i 0 0 0 0.707107 + 0.707107i 0 0
2251.2 0.707107 0.707107i 0 1.00000i 0 0 0 −0.707107 0.707107i 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 $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
5.b even 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner
80.k odd 4 1 inner
240.t 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.bo.a 4
3.b odd 2 1 inner 3600.1.bo.a 4
5.b even 2 1 inner 3600.1.bo.a 4
5.c odd 4 2 720.1.r.a 4
15.d odd 2 1 CM 3600.1.bo.a 4
15.e even 4 2 720.1.r.a 4
16.f odd 4 1 inner 3600.1.bo.a 4
20.e even 4 2 2880.1.r.a 4
48.k even 4 1 inner 3600.1.bo.a 4
60.l odd 4 2 2880.1.r.a 4
80.i odd 4 1 2880.1.r.a 4
80.j even 4 1 720.1.r.a 4
80.k odd 4 1 inner 3600.1.bo.a 4
80.s even 4 1 720.1.r.a 4
80.t odd 4 1 2880.1.r.a 4
240.t even 4 1 inner 3600.1.bo.a 4
240.z odd 4 1 720.1.r.a 4
240.bb even 4 1 2880.1.r.a 4
240.bd odd 4 1 720.1.r.a 4
240.bf even 4 1 2880.1.r.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.1.r.a 4 5.c odd 4 2
720.1.r.a 4 15.e even 4 2
720.1.r.a 4 80.j even 4 1
720.1.r.a 4 80.s even 4 1
720.1.r.a 4 240.z odd 4 1
720.1.r.a 4 240.bd odd 4 1
2880.1.r.a 4 20.e even 4 2
2880.1.r.a 4 60.l odd 4 2
2880.1.r.a 4 80.i odd 4 1
2880.1.r.a 4 80.t odd 4 1
2880.1.r.a 4 240.bb even 4 1
2880.1.r.a 4 240.bf even 4 1
3600.1.bo.a 4 1.a even 1 1 trivial
3600.1.bo.a 4 3.b odd 2 1 inner
3600.1.bo.a 4 5.b even 2 1 inner
3600.1.bo.a 4 15.d odd 2 1 CM
3600.1.bo.a 4 16.f odd 4 1 inner
3600.1.bo.a 4 48.k even 4 1 inner
3600.1.bo.a 4 80.k odd 4 1 inner
3600.1.bo.a 4 240.t even 4 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(3600, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 2)^{2}$$
$19$ $$(T^{2} + 2 T + 2)^{2}$$
$23$ $$(T^{2} - 2)^{2}$$
$29$ $$T^{4}$$
$31$ $$(T^{2} + 4)^{2}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$(T^{2} + 2)^{2}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$(T^{2} + 2 T + 2)^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4} + 16$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$