# Properties

 Label 3600.1.cj.a Level $3600$ Weight $1$ Character orbit 3600.cj Analytic conductor $1.797$ Analytic rank $0$ Dimension $8$ Projective image $D_{10}$ 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(271,3600)]

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, 6]))

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

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

chi := DirichletCharacter("3600.271");

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.cj (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: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{20})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ x^8 - x^6 + x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{10}$$ Projective field: Galois closure of 10.0.9492187500000000.9

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{20}^{9} q^{5}+O(q^{10})$$ q + z^9 * q^5 $$q + \zeta_{20}^{9} q^{5} + ( - \zeta_{20}^{8} + \zeta_{20}^{6}) q^{13} + ( - \zeta_{20}^{7} - \zeta_{20}) q^{17} - \zeta_{20}^{8} q^{25} + ( - \zeta_{20}^{9} - \zeta_{20}^{3}) q^{29} + (\zeta_{20}^{4} + 1) q^{37} + (\zeta_{20}^{3} + \zeta_{20}) q^{41} + q^{49} + (\zeta_{20}^{7} + \zeta_{20}^{5}) q^{53} + (\zeta_{20}^{4} - \zeta_{20}^{2}) q^{61} + (\zeta_{20}^{7} - \zeta_{20}^{5}) q^{65} + ( - \zeta_{20}^{4} + \zeta_{20}^{2}) q^{73} + (\zeta_{20}^{6} + 1) q^{85} + ( - \zeta_{20}^{5} + \zeta_{20}) q^{89} + ( - \zeta_{20}^{8} - \zeta_{20}^{4}) q^{97} +O(q^{100})$$ q + z^9 * q^5 + (-z^8 + z^6) * q^13 + (-z^7 - z) * q^17 - z^8 * q^25 + (-z^9 - z^3) * q^29 + (z^4 + 1) * q^37 + (z^3 + z) * q^41 + q^49 + (z^7 + z^5) * q^53 + (z^4 - z^2) * q^61 + (z^7 - z^5) * q^65 + (-z^4 + z^2) * q^73 + (z^6 + 1) * q^85 + (-z^5 + z) * q^89 + (-z^8 - z^4) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q + 4 q^{13} + 2 q^{25} + 6 q^{37} + 8 q^{49} - 4 q^{61} + 4 q^{73} + 10 q^{85} + 4 q^{97}+O(q^{100})$$ 8 * q + 4 * q^13 + 2 * q^25 + 6 * q^37 + 8 * q^49 - 4 * q^61 + 4 * q^73 + 10 * q^85 + 4 * q^97

## Character values

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

 $$n$$ $$577$$ $$901$$ $$2801$$ $$3151$$ $$\chi(n)$$ $$-\zeta_{20}^{6}$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
271.1
 0.587785 − 0.809017i −0.587785 + 0.809017i 0.951057 − 0.309017i −0.951057 + 0.309017i 0.951057 + 0.309017i −0.951057 − 0.309017i 0.587785 + 0.809017i −0.587785 − 0.809017i
0 0 0 −0.587785 0.809017i 0 0 0 0 0
271.2 0 0 0 0.587785 + 0.809017i 0 0 0 0 0
991.1 0 0 0 −0.951057 0.309017i 0 0 0 0 0
991.2 0 0 0 0.951057 + 0.309017i 0 0 0 0 0
1711.1 0 0 0 −0.951057 + 0.309017i 0 0 0 0 0
1711.2 0 0 0 0.951057 0.309017i 0 0 0 0 0
2431.1 0 0 0 −0.587785 + 0.809017i 0 0 0 0 0
2431.2 0 0 0 0.587785 0.809017i 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 271.2 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.d even 5 1 inner
75.j odd 10 1 inner
100.j odd 10 1 inner
300.n even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.1.cj.a 8
3.b odd 2 1 inner 3600.1.cj.a 8
4.b odd 2 1 CM 3600.1.cj.a 8
12.b even 2 1 inner 3600.1.cj.a 8
25.d even 5 1 inner 3600.1.cj.a 8
75.j odd 10 1 inner 3600.1.cj.a 8
100.j odd 10 1 inner 3600.1.cj.a 8
300.n even 10 1 inner 3600.1.cj.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3600.1.cj.a 8 1.a even 1 1 trivial
3600.1.cj.a 8 3.b odd 2 1 inner
3600.1.cj.a 8 4.b odd 2 1 CM
3600.1.cj.a 8 12.b even 2 1 inner
3600.1.cj.a 8 25.d even 5 1 inner
3600.1.cj.a 8 75.j odd 10 1 inner
3600.1.cj.a 8 100.j odd 10 1 inner
3600.1.cj.a 8 300.n even 10 1 inner

## Hecke kernels

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

## Hecke characteristic polynomials

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