# Properties

 Label 2523.1.b.a Level $2523$ Weight $1$ Character orbit 2523.b Self dual yes Analytic conductor $1.259$ Analytic rank $0$ Dimension $2$ Projective image $D_{5}$ CM discriminant -3 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2523,1,Mod(842,2523)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2523, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("2523.842");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2523 = 3 \cdot 29^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2523.b (of order $$2$$, degree $$1$$, 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: yes Analytic conductor: $$1.25914102687$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{5}$$ Projective field: Galois closure of 5.1.6365529.1

## $q$-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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{4} + (\beta - 1) q^{7} + q^{9}+O(q^{10})$$ q - q^3 + q^4 + (b - 1) * q^7 + q^9 $$q - q^{3} + q^{4} + (\beta - 1) q^{7} + q^{9} - q^{12} - \beta q^{13} + q^{16} + \beta q^{19} + ( - \beta + 1) q^{21} + q^{25} - q^{27} + (\beta - 1) q^{28} + ( - \beta + 1) q^{31} + q^{36} + \beta q^{37} + \beta q^{39} + ( - \beta + 1) q^{43} - q^{48} + ( - \beta + 1) q^{49} - \beta q^{52} - \beta q^{57} + ( - \beta + 1) q^{61} + (\beta - 1) q^{63} + q^{64} - \beta q^{67} + \beta q^{73} - q^{75} + \beta q^{76} + \beta q^{79} + q^{81} + ( - \beta + 1) q^{84} - q^{91} + (\beta - 1) q^{93} + ( - \beta + 1) q^{97} +O(q^{100})$$ q - q^3 + q^4 + (b - 1) * q^7 + q^9 - q^12 - b * q^13 + q^16 + b * q^19 + (-b + 1) * q^21 + q^25 - q^27 + (b - 1) * q^28 + (-b + 1) * q^31 + q^36 + b * q^37 + b * q^39 + (-b + 1) * q^43 - q^48 + (-b + 1) * q^49 - b * q^52 - b * q^57 + (-b + 1) * q^61 + (b - 1) * q^63 + q^64 - b * q^67 + b * q^73 - q^75 + b * q^76 + b * q^79 + q^81 + (-b + 1) * q^84 - q^91 + (b - 1) * q^93 + (-b + 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{4} - q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^4 - q^7 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{4} - q^{7} + 2 q^{9} - 2 q^{12} - q^{13} + 2 q^{16} + q^{19} + q^{21} + 2 q^{25} - 2 q^{27} - q^{28} + q^{31} + 2 q^{36} + q^{37} + q^{39} + q^{43} - 2 q^{48} + q^{49} - q^{52} - q^{57} + q^{61} - q^{63} + 2 q^{64} - q^{67} + q^{73} - 2 q^{75} + q^{76} + q^{79} + 2 q^{81} + q^{84} - 2 q^{91} - q^{93} + q^{97}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^4 - q^7 + 2 * q^9 - 2 * q^12 - q^13 + 2 * q^16 + q^19 + q^21 + 2 * q^25 - 2 * q^27 - q^28 + q^31 + 2 * q^36 + q^37 + q^39 + q^43 - 2 * q^48 + q^49 - q^52 - q^57 + q^61 - q^63 + 2 * q^64 - q^67 + q^73 - 2 * q^75 + q^76 + q^79 + 2 * q^81 + q^84 - 2 * q^91 - q^93 + q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2523\mathbb{Z}\right)^\times$$.

 $$n$$ $$842$$ $$1684$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
842.1
 −0.618034 1.61803
0 −1.00000 1.00000 0 0 −1.61803 0 1.00000 0
842.2 0 −1.00000 1.00000 0 0 0.618034 0 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2523.1.b.a 2
3.b odd 2 1 CM 2523.1.b.a 2
29.b even 2 1 2523.1.b.c yes 2
29.c odd 4 2 2523.1.d.a 4
29.d even 7 6 2523.1.j.c 12
29.e even 14 6 2523.1.j.a 12
29.f odd 28 12 2523.1.h.c 24
87.d odd 2 1 2523.1.b.c yes 2
87.f even 4 2 2523.1.d.a 4
87.h odd 14 6 2523.1.j.a 12
87.j odd 14 6 2523.1.j.c 12
87.k even 28 12 2523.1.h.c 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2523.1.b.a 2 1.a even 1 1 trivial
2523.1.b.a 2 3.b odd 2 1 CM
2523.1.b.c yes 2 29.b even 2 1
2523.1.b.c yes 2 87.d odd 2 1
2523.1.d.a 4 29.c odd 4 2
2523.1.d.a 4 87.f even 4 2
2523.1.h.c 24 29.f odd 28 12
2523.1.h.c 24 87.k even 28 12
2523.1.j.a 12 29.e even 14 6
2523.1.j.a 12 87.h odd 14 6
2523.1.j.c 12 29.d even 7 6
2523.1.j.c 12 87.j odd 14 6

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2523, [\chi])$$:

 $$T_{2}$$ T2 $$T_{19}^{2} - T_{19} - 1$$ T19^2 - T19 - 1

## Hecke characteristic polynomials

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