# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [500,1,Mod(51,500)]

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, 8]))

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

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

chi := DirichletCharacter("500.51");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$500 = 2^{2} \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 500.j (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: $$4$$ 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: $$\Z[a_1, a_2]$$ 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.

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

## Character values

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

 $$n$$ $$251$$ $$377$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{10}^{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
51.1
 0.809017 + 0.587785i −0.309017 + 0.951057i −0.309017 − 0.951057i 0.809017 − 0.587785i
0.809017 0.587785i 0 0.309017 0.951057i 0 0 0 −0.309017 0.951057i −0.809017 0.587785i 0
151.1 −0.309017 0.951057i 0 −0.809017 + 0.587785i 0 0 0 0.809017 + 0.587785i 0.309017 0.951057i 0
351.1 −0.309017 + 0.951057i 0 −0.809017 0.587785i 0 0 0 0.809017 0.587785i 0.309017 + 0.951057i 0
451.1 0.809017 + 0.587785i 0 0.309017 + 0.951057i 0 0 0 −0.309017 + 0.951057i −0.809017 + 0.587785i 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.d even 5 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.j.a 4
4.b odd 2 1 CM 500.1.j.a 4
5.b even 2 1 100.1.j.a 4
5.c odd 4 2 500.1.h.a 8
15.d odd 2 1 900.1.x.a 4
20.d odd 2 1 100.1.j.a 4
20.e even 4 2 500.1.h.a 8
25.d even 5 1 inner 500.1.j.a 4
25.d even 5 1 2500.1.b.a 2
25.d even 5 2 2500.1.j.b 4
25.e even 10 1 100.1.j.a 4
25.e even 10 1 2500.1.b.b 2
25.e even 10 2 2500.1.j.a 4
25.f odd 20 2 500.1.h.a 8
25.f odd 20 2 2500.1.d.a 4
25.f odd 20 4 2500.1.h.e 8
40.e odd 2 1 1600.1.bh.a 4
40.f even 2 1 1600.1.bh.a 4
60.h even 2 1 900.1.x.a 4
75.h odd 10 1 900.1.x.a 4
100.h odd 10 1 100.1.j.a 4
100.h odd 10 1 2500.1.b.b 2
100.h odd 10 2 2500.1.j.a 4
100.j odd 10 1 inner 500.1.j.a 4
100.j odd 10 1 2500.1.b.a 2
100.j odd 10 2 2500.1.j.b 4
100.l even 20 2 500.1.h.a 8
100.l even 20 2 2500.1.d.a 4
100.l even 20 4 2500.1.h.e 8
200.o even 10 1 1600.1.bh.a 4
200.s odd 10 1 1600.1.bh.a 4
300.r even 10 1 900.1.x.a 4

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

## Hecke kernels

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

## Hecke characteristic polynomials

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