# Properties

 Label 2500.1.j.a Level $2500$ Weight $1$ Character orbit 2500.j Analytic conductor $1.248$ 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] = [2500,1,Mod(251,2500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2500, 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("2500.251");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2500 = 2^{2} \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2500.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: $$1.24766253158$$ 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2500\mathbb{Z}\right)^\times$$.

 $$n$$ $$1251$$ $$1877$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{10}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 −0.309017 + 0.951057i 0.809017 − 0.587785i 0.809017 + 0.587785i −0.309017 − 0.951057i
−0.809017 + 0.587785i 0 0.309017 0.951057i 0 0 0 0.309017 + 0.951057i −0.809017 0.587785i 0
751.1 0.309017 + 0.951057i 0 −0.809017 + 0.587785i 0 0 0 −0.809017 0.587785i 0.309017 0.951057i 0
1751.1 0.309017 0.951057i 0 −0.809017 0.587785i 0 0 0 −0.809017 + 0.587785i 0.309017 + 0.951057i 0
2251.1 −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 2500.1.j.a 4
4.b odd 2 1 CM 2500.1.j.a 4
5.b even 2 1 2500.1.j.b 4
5.c odd 4 2 2500.1.h.e 8
20.d odd 2 1 2500.1.j.b 4
20.e even 4 2 2500.1.h.e 8
25.d even 5 2 100.1.j.a 4
25.d even 5 1 2500.1.b.b 2
25.d even 5 1 inner 2500.1.j.a 4
25.e even 10 2 500.1.j.a 4
25.e even 10 1 2500.1.b.a 2
25.e even 10 1 2500.1.j.b 4
25.f odd 20 4 500.1.h.a 8
25.f odd 20 2 2500.1.d.a 4
25.f odd 20 2 2500.1.h.e 8
75.j odd 10 2 900.1.x.a 4
100.h odd 10 2 500.1.j.a 4
100.h odd 10 1 2500.1.b.a 2
100.h odd 10 1 2500.1.j.b 4
100.j odd 10 2 100.1.j.a 4
100.j odd 10 1 2500.1.b.b 2
100.j odd 10 1 inner 2500.1.j.a 4
100.l even 20 4 500.1.h.a 8
100.l even 20 2 2500.1.d.a 4
100.l even 20 2 2500.1.h.e 8
200.n odd 10 2 1600.1.bh.a 4
200.t even 10 2 1600.1.bh.a 4
300.n even 10 2 900.1.x.a 4

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2500, [\chi])$$:

 $$T_{3}$$ T3 $$T_{13}^{4} - 3T_{13}^{3} + 4T_{13}^{2} - 2T_{13} + 1$$ T13^4 - 3*T13^3 + 4*T13^2 - 2*T13 + 1

## Hecke characteristic polynomials

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