# Properties

 Label 2000.1.z.a Level $2000$ Weight $1$ Character orbit 2000.z Analytic conductor $0.998$ 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] = [2000,1,Mod(351,2000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2000, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 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("2000.351");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2000 = 2^{4} \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2000.z (of order $$10$$, degree $$4$$, not 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: $$0.998130025266$$ 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$5$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{20}^{8} q^{9}+O(q^{10})$$ q + z^8 * q^9 $$q + \zeta_{20}^{8} q^{9} + (\zeta_{20}^{9} + \zeta_{20}^{7}) q^{13} + (\zeta_{20}^{9} + \zeta_{20}^{3}) q^{17} + (\zeta_{20}^{6} + \zeta_{20}^{2}) q^{29} + (\zeta_{20}^{5} - \zeta_{20}) q^{37} + ( - \zeta_{20}^{4} + \zeta_{20}^{2}) q^{41} + q^{49} + ( - \zeta_{20}^{5} - \zeta_{20}^{3}) q^{53} + ( - \zeta_{20}^{8} + \zeta_{20}^{6}) q^{61} + (\zeta_{20}^{3} + \zeta_{20}) q^{73} - \zeta_{20}^{6} q^{81} + ( - \zeta_{20}^{4} - 1) q^{89} + (\zeta_{20}^{7} + \zeta_{20}) q^{97} +O(q^{100})$$ q + z^8 * q^9 + (z^9 + z^7) * q^13 + (z^9 + z^3) * q^17 + (z^6 + z^2) * q^29 + (z^5 - z) * q^37 + (-z^4 + z^2) * q^41 + q^49 + (-z^5 - z^3) * q^53 + (-z^8 + z^6) * q^61 + (z^3 + z) * q^73 - z^6 * q^81 + (-z^4 - 1) * q^89 + (z^7 + z) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 2 q^{9}+O(q^{10})$$ 8 * q - 2 * q^9 $$8 q - 2 q^{9} + 4 q^{29} + 4 q^{41} + 8 q^{49} + 4 q^{61} - 2 q^{81} - 6 q^{89}+O(q^{100})$$ 8 * q - 2 * q^9 + 4 * q^29 + 4 * q^41 + 8 * q^49 + 4 * q^61 - 2 * q^81 - 6 * q^89

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2000\mathbb{Z}\right)^\times$$.

 $$n$$ $$501$$ $$751$$ $$1377$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\zeta_{20}^{4}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
351.1
 −0.587785 − 0.809017i 0.587785 + 0.809017i −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 0 0 0 0 0 0 0.309017 + 0.951057i 0
351.2 0 0 0 0 0 0 0 0.309017 + 0.951057i 0
1151.1 0 0 0 0 0 0 0 0.309017 0.951057i 0
1151.2 0 0 0 0 0 0 0 0.309017 0.951057i 0
1551.1 0 0 0 0 0 0 0 −0.809017 0.587785i 0
1551.2 0 0 0 0 0 0 0 −0.809017 0.587785i 0
1951.1 0 0 0 0 0 0 0 −0.809017 + 0.587785i 0
1951.2 0 0 0 0 0 0 0 −0.809017 + 0.587785i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 351.2 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 $$\Q(\sqrt{-1})$$
5.b even 2 1 inner
20.d odd 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner
100.h odd 10 1 inner
100.j odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2000.1.z.a 8
4.b odd 2 1 CM 2000.1.z.a 8
5.b even 2 1 inner 2000.1.z.a 8
5.c odd 4 1 400.1.x.a 4
5.c odd 4 1 2000.1.x.a 4
15.e even 4 1 3600.1.ct.a 4
20.d odd 2 1 inner 2000.1.z.a 8
20.e even 4 1 400.1.x.a 4
20.e even 4 1 2000.1.x.a 4
25.d even 5 1 inner 2000.1.z.a 8
25.e even 10 1 inner 2000.1.z.a 8
25.f odd 20 1 400.1.x.a 4
25.f odd 20 1 2000.1.x.a 4
40.i odd 4 1 1600.1.bf.a 4
40.k even 4 1 1600.1.bf.a 4
60.l odd 4 1 3600.1.ct.a 4
75.l even 20 1 3600.1.ct.a 4
100.h odd 10 1 inner 2000.1.z.a 8
100.j odd 10 1 inner 2000.1.z.a 8
100.l even 20 1 400.1.x.a 4
100.l even 20 1 2000.1.x.a 4
200.v even 20 1 1600.1.bf.a 4
200.x odd 20 1 1600.1.bf.a 4
300.u odd 20 1 3600.1.ct.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.1.x.a 4 5.c odd 4 1
400.1.x.a 4 20.e even 4 1
400.1.x.a 4 25.f odd 20 1
400.1.x.a 4 100.l even 20 1
1600.1.bf.a 4 40.i odd 4 1
1600.1.bf.a 4 40.k even 4 1
1600.1.bf.a 4 200.v even 20 1
1600.1.bf.a 4 200.x odd 20 1
2000.1.x.a 4 5.c odd 4 1
2000.1.x.a 4 20.e even 4 1
2000.1.x.a 4 25.f odd 20 1
2000.1.x.a 4 100.l even 20 1
2000.1.z.a 8 1.a even 1 1 trivial
2000.1.z.a 8 4.b odd 2 1 CM
2000.1.z.a 8 5.b even 2 1 inner
2000.1.z.a 8 20.d odd 2 1 inner
2000.1.z.a 8 25.d even 5 1 inner
2000.1.z.a 8 25.e even 10 1 inner
2000.1.z.a 8 100.h odd 10 1 inner
2000.1.z.a 8 100.j odd 10 1 inner
3600.1.ct.a 4 15.e even 4 1
3600.1.ct.a 4 60.l odd 4 1
3600.1.ct.a 4 75.l even 20 1
3600.1.ct.a 4 300.u odd 20 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(2000, [\chi])$$.

## Hecke characteristic polynomials

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