# Properties

 Label 3600.1.ct.a Level $3600$ Weight $1$ Character orbit 3600.ct Analytic conductor $1.797$ Analytic rank $0$ Dimension $4$ Projective image $D_{10}$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,1,Mod(559,3600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3600, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 0, 0, 7]))

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

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

chi := DirichletCharacter("3600.559");

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.ct (of order $$10$$, degree $$4$$, 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: $$\Q(\zeta_{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ x^4 - x^3 + x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 400) Projective image: $$D_{10}$$ Projective field: Galois closure of 10.2.195312500000000.4

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{10}^{4} q^{5}+O(q^{10})$$ q + z^4 * q^5 $$q + \zeta_{10}^{4} q^{5} + (\zeta_{10}^{3} - \zeta_{10}) q^{13} + ( - \zeta_{10}^{2} - \zeta_{10}) q^{17} - \zeta_{10}^{3} q^{25} + ( - \zeta_{10}^{4} + \zeta_{10}^{3}) q^{29} + (\zeta_{10}^{4} - 1) q^{37} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{41} - q^{49} + ( - \zeta_{10}^{2} + 1) q^{53} + ( - \zeta_{10}^{4} - \zeta_{10}^{2}) q^{61} + ( - \zeta_{10}^{2} + 1) q^{65} + (\zeta_{10}^{4} - \zeta_{10}^{2}) q^{73} + (\zeta_{10} + 1) q^{85} + (\zeta_{10} - 1) q^{89} + ( - \zeta_{10}^{4} - \zeta_{10}^{3}) q^{97} +O(q^{100})$$ q + z^4 * q^5 + (z^3 - z) * q^13 + (-z^2 - z) * q^17 - z^3 * q^25 + (-z^4 + z^3) * q^29 + (z^4 - 1) * q^37 + (-z^3 - z) * q^41 - q^49 + (-z^2 + 1) * q^53 + (-z^4 - z^2) * q^61 + (-z^2 + 1) * q^65 + (z^4 - z^2) * q^73 + (z + 1) * q^85 + (z - 1) * q^89 + (-z^4 - z^3) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{5}+O(q^{10})$$ 4 * q - q^5 $$4 q - q^{5} - q^{25} + 2 q^{29} - 5 q^{37} - 2 q^{41} - 4 q^{49} + 5 q^{53} + 2 q^{61} + 5 q^{65} + 5 q^{85} - 3 q^{89}+O(q^{100})$$ 4 * q - q^5 - q^25 + 2 * q^29 - 5 * q^37 - 2 * q^41 - 4 * q^49 + 5 * q^53 + 2 * q^61 + 5 * q^65 + 5 * q^85 - 3 * q^89

## Character values

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

 $$n$$ $$577$$ $$901$$ $$2801$$ $$3151$$ $$\chi(n)$$ $$\zeta_{10}$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
559.1
 −0.309017 − 0.951057i 0.809017 + 0.587785i 0.809017 − 0.587785i −0.309017 + 0.951057i
0 0 0 0.309017 0.951057i 0 0 0 0 0
1279.1 0 0 0 −0.809017 + 0.587785i 0 0 0 0 0
2719.1 0 0 0 −0.809017 0.587785i 0 0 0 0 0
3439.1 0 0 0 0.309017 + 0.951057i 0 0 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
25.e even 10 1 inner
100.h odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.1.ct.a 4
3.b odd 2 1 400.1.x.a 4
4.b odd 2 1 CM 3600.1.ct.a 4
12.b even 2 1 400.1.x.a 4
15.d odd 2 1 2000.1.x.a 4
15.e even 4 2 2000.1.z.a 8
24.f even 2 1 1600.1.bf.a 4
24.h odd 2 1 1600.1.bf.a 4
25.e even 10 1 inner 3600.1.ct.a 4
60.h even 2 1 2000.1.x.a 4
60.l odd 4 2 2000.1.z.a 8
75.h odd 10 1 400.1.x.a 4
75.j odd 10 1 2000.1.x.a 4
75.l even 20 2 2000.1.z.a 8
100.h odd 10 1 inner 3600.1.ct.a 4
300.n even 10 1 2000.1.x.a 4
300.r even 10 1 400.1.x.a 4
300.u odd 20 2 2000.1.z.a 8
600.z odd 10 1 1600.1.bf.a 4
600.bk even 10 1 1600.1.bf.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.1.x.a 4 3.b odd 2 1
400.1.x.a 4 12.b even 2 1
400.1.x.a 4 75.h odd 10 1
400.1.x.a 4 300.r even 10 1
1600.1.bf.a 4 24.f even 2 1
1600.1.bf.a 4 24.h odd 2 1
1600.1.bf.a 4 600.z odd 10 1
1600.1.bf.a 4 600.bk even 10 1
2000.1.x.a 4 15.d odd 2 1
2000.1.x.a 4 60.h even 2 1
2000.1.x.a 4 75.j odd 10 1
2000.1.x.a 4 300.n even 10 1
2000.1.z.a 8 15.e even 4 2
2000.1.z.a 8 60.l odd 4 2
2000.1.z.a 8 75.l even 20 2
2000.1.z.a 8 300.u odd 20 2
3600.1.ct.a 4 1.a even 1 1 trivial
3600.1.ct.a 4 4.b odd 2 1 CM
3600.1.ct.a 4 25.e even 10 1 inner
3600.1.ct.a 4 100.h odd 10 1 inner

## Hecke kernels

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

## Hecke characteristic polynomials

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