# Properties

 Label 2500.1.d.a Level $2500$ Weight $1$ Character orbit 2500.d 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(2499,2500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2500, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

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

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

chi := DirichletCharacter("2500.2499");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2500 = 2^{2} \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2500.d (of order $$2$$, degree $$1$$, 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: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} - q^{4} - \beta_{3} q^{8} - q^{9}+O(q^{10})$$ q + b3 * q^2 - q^4 - b3 * q^8 - q^9 $$q + \beta_{3} q^{2} - q^{4} - \beta_{3} q^{8} - q^{9} - \beta_1 q^{13} + q^{16} + ( - \beta_{3} - \beta_1) q^{17} - \beta_{3} q^{18} + \beta_{2} q^{26} + (\beta_{2} + 1) q^{29} + \beta_{3} q^{32} + (\beta_{2} + 1) q^{34} + q^{36} + ( - \beta_{3} - \beta_1) q^{37} + \beta_{2} q^{41} - q^{49} + \beta_1 q^{52} - \beta_1 q^{53} + (\beta_{3} + \beta_1) q^{58} + \beta_{2} q^{61} - q^{64} + (\beta_{3} + \beta_1) q^{68} + \beta_{3} q^{72} - \beta_1 q^{73} + (\beta_{2} + 1) q^{74} + q^{81} + \beta_1 q^{82} + (\beta_{2} + 1) q^{89} + ( - \beta_{3} - \beta_1) q^{97} - \beta_{3} q^{98}+O(q^{100})$$ q + b3 * q^2 - q^4 - b3 * q^8 - q^9 - b1 * q^13 + q^16 + (-b3 - b1) * q^17 - b3 * q^18 + b2 * q^26 + (b2 + 1) * q^29 + b3 * q^32 + (b2 + 1) * q^34 + q^36 + (-b3 - b1) * q^37 + b2 * q^41 - q^49 + b1 * q^52 - b1 * q^53 + (b3 + b1) * q^58 + b2 * q^61 - q^64 + (b3 + b1) * q^68 + b3 * q^72 - b1 * q^73 + (b2 + 1) * q^74 + q^81 + b1 * q^82 + (b2 + 1) * q^89 + (-b3 - b1) * q^97 - b3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 - 4 * q^9 $$4 q - 4 q^{4} - 4 q^{9} + 4 q^{16} - 2 q^{26} + 2 q^{29} + 2 q^{34} + 4 q^{36} - 2 q^{41} - 4 q^{49} - 2 q^{61} - 4 q^{64} + 2 q^{74} + 4 q^{81} + 2 q^{89}+O(q^{100})$$ 4 * q - 4 * q^4 - 4 * q^9 + 4 * q^16 - 2 * q^26 + 2 * q^29 + 2 * q^34 + 4 * q^36 - 2 * q^41 - 4 * q^49 - 2 * q^61 - 4 * q^64 + 2 * q^74 + 4 * q^81 + 2 * q^89

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ v^2 + 1 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2\nu$$ v^3 + 2*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ b2 - 1 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2\beta_1$$ b3 - 2*b1

## Character values

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

 $$n$$ $$1251$$ $$1877$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2499.1
 1.61803i − 0.618034i 0.618034i − 1.61803i
1.00000i 0 −1.00000 0 0 0 1.00000i −1.00000 0
2499.2 1.00000i 0 −1.00000 0 0 0 1.00000i −1.00000 0
2499.3 1.00000i 0 −1.00000 0 0 0 1.00000i −1.00000 0
2499.4 1.00000i 0 −1.00000 0 0 0 1.00000i −1.00000 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 $$\Q(\sqrt{-1})$$
5.b even 2 1 inner
20.d odd 2 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.1.j.a 4 25.f odd 20 2
100.1.j.a 4 100.l even 20 2
500.1.h.a 8 25.d even 5 2
500.1.h.a 8 25.e even 10 2
500.1.h.a 8 100.h odd 10 2
500.1.h.a 8 100.j odd 10 2
500.1.j.a 4 25.f odd 20 2
500.1.j.a 4 100.l even 20 2
900.1.x.a 4 75.l even 20 2
900.1.x.a 4 300.u odd 20 2
1600.1.bh.a 4 200.v even 20 2
1600.1.bh.a 4 200.x odd 20 2
2500.1.b.a 2 5.c odd 4 1
2500.1.b.a 2 20.e even 4 1
2500.1.b.b 2 5.c odd 4 1
2500.1.b.b 2 20.e even 4 1
2500.1.d.a 4 1.a even 1 1 trivial
2500.1.d.a 4 4.b odd 2 1 CM
2500.1.d.a 4 5.b even 2 1 inner
2500.1.d.a 4 20.d odd 2 1 inner
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 $$S_{1}^{\mathrm{new}}(2500, [\chi])$$.

## Hecke characteristic polynomials

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