# Properties

 Label 500.1.h.a Level $500$ Weight $1$ Character orbit 500.h Analytic conductor $0.250$ Analytic rank $0$ Dimension $8$ Projective image $D_{5}$ CM discriminant -4 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [500,1,Mod(99,500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(500, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 9]))

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

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

chi := DirichletCharacter("500.99");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$500 = 2^{2} \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 500.h (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: $$0.249532506317$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 100) Projective image: $$D_{5}$$ Projective field: Galois closure of 5.1.6250000.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{20} q^{2} + \zeta_{20}^{2} q^{4} - \zeta_{20}^{3} q^{8} - \zeta_{20}^{4} q^{9} +O(q^{10})$$ q - z * q^2 + z^2 * q^4 - z^3 * q^8 - z^4 * q^9 $$q - \zeta_{20} q^{2} + \zeta_{20}^{2} q^{4} - \zeta_{20}^{3} q^{8} - \zeta_{20}^{4} q^{9} + ( - \zeta_{20}^{7} + \zeta_{20}) q^{13} + \zeta_{20}^{4} q^{16} + ( - \zeta_{20}^{9} + \zeta_{20}^{7}) q^{17} + \zeta_{20}^{5} q^{18} + (\zeta_{20}^{8} - \zeta_{20}^{2}) q^{26} + ( - \zeta_{20}^{8} + \zeta_{20}^{6}) q^{29} - \zeta_{20}^{5} q^{32} + ( - \zeta_{20}^{8} - 1) q^{34} - \zeta_{20}^{6} q^{36} + ( - \zeta_{20}^{5} + \zeta_{20}^{3}) q^{37} + ( - \zeta_{20}^{6} - \zeta_{20}^{2}) q^{41} - q^{49} + ( - \zeta_{20}^{9} + \zeta_{20}^{3}) q^{52} + (\zeta_{20}^{9} + \zeta_{20}^{5}) q^{53} + (\zeta_{20}^{9} - \zeta_{20}^{7}) q^{58} + (\zeta_{20}^{8} + \zeta_{20}^{4}) q^{61} + \zeta_{20}^{6} q^{64} + (\zeta_{20}^{9} + \zeta_{20}) q^{68} + \zeta_{20}^{7} q^{72} + (\zeta_{20}^{9} - \zeta_{20}^{3}) q^{73} + (\zeta_{20}^{6} - \zeta_{20}^{4}) q^{74} + \zeta_{20}^{8} q^{81} + (\zeta_{20}^{7} + \zeta_{20}^{3}) q^{82} + (\zeta_{20}^{2} - 1) q^{89} + (\zeta_{20}^{3} - \zeta_{20}) q^{97} + \zeta_{20} q^{98} +O(q^{100})$$ q - z * q^2 + z^2 * q^4 - z^3 * q^8 - z^4 * q^9 + (-z^7 + z) * q^13 + z^4 * q^16 + (-z^9 + z^7) * q^17 + z^5 * q^18 + (z^8 - z^2) * q^26 + (-z^8 + z^6) * q^29 - z^5 * q^32 + (-z^8 - 1) * q^34 - z^6 * q^36 + (-z^5 + z^3) * q^37 + (-z^6 - z^2) * q^41 - q^49 + (-z^9 + z^3) * q^52 + (z^9 + z^5) * q^53 + (z^9 - z^7) * q^58 + (z^8 + z^4) * q^61 + z^6 * q^64 + (z^9 + z) * q^68 + z^7 * q^72 + (z^9 - z^3) * q^73 + (z^6 - z^4) * q^74 + z^8 * q^81 + (z^7 + z^3) * q^82 + (z^2 - 1) * q^89 + (z^3 - z) * q^97 + z * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2 q^{4} + 2 q^{9}+O(q^{10})$$ 8 * q + 2 * q^4 + 2 * q^9 $$8 q + 2 q^{4} + 2 q^{9} - 2 q^{16} - 4 q^{26} + 4 q^{29} - 6 q^{34} - 2 q^{36} - 4 q^{41} - 8 q^{49} - 4 q^{61} + 2 q^{64} + 4 q^{74} - 2 q^{81} - 6 q^{89}+O(q^{100})$$ 8 * q + 2 * q^4 + 2 * q^9 - 2 * q^16 - 4 * q^26 + 4 * q^29 - 6 * q^34 - 2 * q^36 - 4 * q^41 - 8 * q^49 - 4 * q^61 + 2 * q^64 + 4 * q^74 - 2 * q^81 - 6 * q^89

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/500\mathbb{Z}\right)^\times$$.

 $$n$$ $$251$$ $$377$$ $$\chi(n)$$ $$-1$$ $$\zeta_{20}^{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 0.951057 − 0.309017i −0.951057 + 0.309017i 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 0.809017 0.587785i 0 0 0 −0.587785 + 0.809017i −0.309017 + 0.951057i 0
99.2 0.951057 0.309017i 0 0.809017 0.587785i 0 0 0 0.587785 0.809017i −0.309017 + 0.951057i 0
199.1 −0.587785 0.809017i 0 −0.309017 + 0.951057i 0 0 0 0.951057 0.309017i 0.809017 + 0.587785i 0
199.2 0.587785 + 0.809017i 0 −0.309017 + 0.951057i 0 0 0 −0.951057 + 0.309017i 0.809017 + 0.587785i 0
299.1 −0.587785 + 0.809017i 0 −0.309017 0.951057i 0 0 0 0.951057 + 0.309017i 0.809017 0.587785i 0
299.2 0.587785 0.809017i 0 −0.309017 0.951057i 0 0 0 −0.951057 0.309017i 0.809017 0.587785i 0
399.1 −0.951057 0.309017i 0 0.809017 + 0.587785i 0 0 0 −0.587785 0.809017i −0.309017 0.951057i 0
399.2 0.951057 + 0.309017i 0 0.809017 + 0.587785i 0 0 0 0.587785 + 0.809017i −0.309017 0.951057i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 99.2 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 500.1.h.a 8
4.b odd 2 1 CM 500.1.h.a 8
5.b even 2 1 inner 500.1.h.a 8
5.c odd 4 1 100.1.j.a 4
5.c odd 4 1 500.1.j.a 4
15.e even 4 1 900.1.x.a 4
20.d odd 2 1 inner 500.1.h.a 8
20.e even 4 1 100.1.j.a 4
20.e even 4 1 500.1.j.a 4
25.d even 5 1 inner 500.1.h.a 8
25.d even 5 1 2500.1.d.a 4
25.d even 5 2 2500.1.h.e 8
25.e even 10 1 inner 500.1.h.a 8
25.e even 10 1 2500.1.d.a 4
25.e even 10 2 2500.1.h.e 8
25.f odd 20 1 100.1.j.a 4
25.f odd 20 1 500.1.j.a 4
25.f odd 20 1 2500.1.b.a 2
25.f odd 20 1 2500.1.b.b 2
25.f odd 20 2 2500.1.j.a 4
25.f odd 20 2 2500.1.j.b 4
40.i odd 4 1 1600.1.bh.a 4
40.k even 4 1 1600.1.bh.a 4
60.l odd 4 1 900.1.x.a 4
75.l even 20 1 900.1.x.a 4
100.h odd 10 1 inner 500.1.h.a 8
100.h odd 10 1 2500.1.d.a 4
100.h odd 10 2 2500.1.h.e 8
100.j odd 10 1 inner 500.1.h.a 8
100.j odd 10 1 2500.1.d.a 4
100.j odd 10 2 2500.1.h.e 8
100.l even 20 1 100.1.j.a 4
100.l even 20 1 500.1.j.a 4
100.l even 20 1 2500.1.b.a 2
100.l even 20 1 2500.1.b.b 2
100.l even 20 2 2500.1.j.a 4
100.l even 20 2 2500.1.j.b 4
200.v even 20 1 1600.1.bh.a 4
200.x odd 20 1 1600.1.bh.a 4
300.u odd 20 1 900.1.x.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.1.j.a 4 5.c odd 4 1
100.1.j.a 4 20.e even 4 1
100.1.j.a 4 25.f odd 20 1
100.1.j.a 4 100.l even 20 1
500.1.h.a 8 1.a even 1 1 trivial
500.1.h.a 8 4.b odd 2 1 CM
500.1.h.a 8 5.b even 2 1 inner
500.1.h.a 8 20.d odd 2 1 inner
500.1.h.a 8 25.d even 5 1 inner
500.1.h.a 8 25.e even 10 1 inner
500.1.h.a 8 100.h odd 10 1 inner
500.1.h.a 8 100.j odd 10 1 inner
500.1.j.a 4 5.c odd 4 1
500.1.j.a 4 20.e even 4 1
500.1.j.a 4 25.f odd 20 1
500.1.j.a 4 100.l even 20 1
900.1.x.a 4 15.e even 4 1
900.1.x.a 4 60.l odd 4 1
900.1.x.a 4 75.l even 20 1
900.1.x.a 4 300.u odd 20 1
1600.1.bh.a 4 40.i odd 4 1
1600.1.bh.a 4 40.k even 4 1
1600.1.bh.a 4 200.v even 20 1
1600.1.bh.a 4 200.x odd 20 1
2500.1.b.a 2 25.f odd 20 1
2500.1.b.a 2 100.l even 20 1
2500.1.b.b 2 25.f odd 20 1
2500.1.b.b 2 100.l even 20 1
2500.1.d.a 4 25.d even 5 1
2500.1.d.a 4 25.e even 10 1
2500.1.d.a 4 100.h odd 10 1
2500.1.d.a 4 100.j odd 10 1
2500.1.h.e 8 25.d even 5 2
2500.1.h.e 8 25.e even 10 2
2500.1.h.e 8 100.h odd 10 2
2500.1.h.e 8 100.j odd 10 2
2500.1.j.a 4 25.f odd 20 2
2500.1.j.a 4 100.l even 20 2
2500.1.j.b 4 25.f odd 20 2
2500.1.j.b 4 100.l even 20 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(500, [\chi])$$.

## Hecke characteristic polynomials

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