# Properties

 Label 3600.1.dx.a Level $3600$ Weight $1$ Character orbit 3600.dx Analytic conductor $1.797$ Analytic rank $0$ Dimension $16$ Projective image $D_{20}$ CM discriminant -4 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,1,Mod(287,3600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3600, base_ring=CyclotomicField(20))

chi = DirichletCharacter(H, H._module([10, 0, 10, 9]))

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

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

chi := DirichletCharacter("3600.287");

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.dx (of order $$20$$, degree $$8$$, 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: $$16$$ Relative dimension: $$2$$ over $$\Q(\zeta_{20})$$ Coefficient field: $$\Q(\zeta_{40})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - x^{12} + x^{8} - x^{4} + 1$$ x^16 - x^12 + x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{20}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{20} + \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{40}^{7} q^{5} +O(q^{10})$$ q - z^7 * q^5 $$q - \zeta_{40}^{7} q^{5} + (\zeta_{40}^{18} - \zeta_{40}^{4}) q^{13} + ( - \zeta_{40}^{13} + \zeta_{40}) q^{17} + \zeta_{40}^{14} q^{25} + ( - \zeta_{40}^{9} - \zeta_{40}^{7}) q^{29} + (\zeta_{40}^{12} + \zeta_{40}^{10}) q^{37} + ( - \zeta_{40}^{9} - \zeta_{40}^{3}) q^{41} - \zeta_{40}^{10} q^{49} + (\zeta_{40}^{15} - \zeta_{40}^{11}) q^{53} + ( - \zeta_{40}^{6} - \zeta_{40}^{2}) q^{61} + (\zeta_{40}^{11} + \zeta_{40}^{5}) q^{65} + ( - \zeta_{40}^{16} - \zeta_{40}^{2}) q^{73} + ( - \zeta_{40}^{8} - 1) q^{85} + (\zeta_{40}^{15} - \zeta_{40}^{13}) q^{89} + ( - \zeta_{40}^{14} + \zeta_{40}^{12}) q^{97} +O(q^{100})$$ q - z^7 * q^5 + (z^18 - z^4) * q^13 + (-z^13 + z) * q^17 + z^14 * q^25 + (-z^9 - z^7) * q^29 + (z^12 + z^10) * q^37 + (-z^9 - z^3) * q^41 - z^10 * q^49 + (z^15 - z^11) * q^53 + (-z^6 - z^2) * q^61 + (z^11 + z^5) * q^65 + (-z^16 - z^2) * q^73 + (-z^8 - 1) * q^85 + (z^15 - z^13) * q^89 + (-z^14 + z^12) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q+O(q^{10})$$ 16 * q $$16 q - 4 q^{13} + 4 q^{37} + 4 q^{73} - 12 q^{85} + 4 q^{97}+O(q^{100})$$ 16 * q - 4 * q^13 + 4 * q^37 + 4 * q^73 - 12 * q^85 + 4 * q^97

## Character values

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

 $$n$$ $$577$$ $$901$$ $$2801$$ $$3151$$ $$\chi(n)$$ $$-\zeta_{40}^{18}$$ $$1$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
287.1
 −0.156434 + 0.987688i 0.156434 − 0.987688i 0.987688 + 0.156434i −0.987688 − 0.156434i −0.891007 + 0.453990i 0.891007 − 0.453990i 0.987688 − 0.156434i −0.987688 + 0.156434i 0.453990 − 0.891007i −0.453990 + 0.891007i −0.891007 − 0.453990i 0.891007 + 0.453990i −0.156434 − 0.987688i 0.156434 + 0.987688i 0.453990 + 0.891007i −0.453990 − 0.891007i
0 0 0 −0.891007 + 0.453990i 0 0 0 0 0
287.2 0 0 0 0.891007 0.453990i 0 0 0 0 0
863.1 0 0 0 −0.453990 0.891007i 0 0 0 0 0
863.2 0 0 0 0.453990 + 0.891007i 0 0 0 0 0
1583.1 0 0 0 −0.987688 + 0.156434i 0 0 0 0 0
1583.2 0 0 0 0.987688 0.156434i 0 0 0 0 0
1727.1 0 0 0 −0.453990 + 0.891007i 0 0 0 0 0
1727.2 0 0 0 0.453990 0.891007i 0 0 0 0 0
2303.1 0 0 0 −0.156434 + 0.987688i 0 0 0 0 0
2303.2 0 0 0 0.156434 0.987688i 0 0 0 0 0
2447.1 0 0 0 −0.987688 0.156434i 0 0 0 0 0
2447.2 0 0 0 0.987688 + 0.156434i 0 0 0 0 0
3023.1 0 0 0 −0.891007 0.453990i 0 0 0 0 0
3023.2 0 0 0 0.891007 + 0.453990i 0 0 0 0 0
3167.1 0 0 0 −0.156434 0.987688i 0 0 0 0 0
3167.2 0 0 0 0.156434 + 0.987688i 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 287.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
3.b odd 2 1 inner
12.b even 2 1 inner
25.f odd 20 1 inner
75.l even 20 1 inner
100.l even 20 1 inner
300.u odd 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.1.dx.a 16
3.b odd 2 1 inner 3600.1.dx.a 16
4.b odd 2 1 CM 3600.1.dx.a 16
12.b even 2 1 inner 3600.1.dx.a 16
25.f odd 20 1 inner 3600.1.dx.a 16
75.l even 20 1 inner 3600.1.dx.a 16
100.l even 20 1 inner 3600.1.dx.a 16
300.u odd 20 1 inner 3600.1.dx.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3600.1.dx.a 16 1.a even 1 1 trivial
3600.1.dx.a 16 3.b odd 2 1 inner
3600.1.dx.a 16 4.b odd 2 1 CM
3600.1.dx.a 16 12.b even 2 1 inner
3600.1.dx.a 16 25.f odd 20 1 inner
3600.1.dx.a 16 75.l even 20 1 inner
3600.1.dx.a 16 100.l even 20 1 inner
3600.1.dx.a 16 300.u odd 20 1 inner

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$T^{16} - T^{12} + \cdots + 1$$
$7$ $$T^{16}$$
$11$ $$T^{16}$$
$13$ $$(T^{8} + 2 T^{7} + 2 T^{6} + \cdots + 1)^{2}$$
$17$ $$T^{16} + 4 T^{12} + \cdots + 1$$
$19$ $$T^{16}$$
$23$ $$T^{16}$$
$29$ $$T^{16} + 4 T^{14} + \cdots + 1$$
$31$ $$T^{16}$$
$37$ $$(T^{8} - 2 T^{7} + 7 T^{6} + \cdots + 1)^{2}$$
$41$ $$T^{16} - 4 T^{14} + \cdots + 1$$
$43$ $$T^{16}$$
$47$ $$T^{16}$$
$53$ $$T^{16} - 11 T^{12} + \cdots + 1$$
$59$ $$T^{16}$$
$61$ $$(T^{8} + 10 T^{4} + \cdots + 25)^{2}$$
$67$ $$T^{16}$$
$71$ $$T^{16}$$
$73$ $$(T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 1)^{2}$$
$79$ $$T^{16}$$
$83$ $$T^{16}$$
$89$ $$T^{16} + 4 T^{14} + \cdots + 1$$
$97$ $$(T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 1)^{2}$$