# Properties

 Label 1444.1.b.a Level $1444$ Weight $1$ Character orbit 1444.b Self dual yes Analytic conductor $0.721$ Analytic rank $0$ Dimension $2$ Projective image $D_{5}$ CM discriminant -4 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1444,1,Mod(723,1444)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1444, 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("1444.723");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1444 = 2^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1444.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: $$0.720649878242$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{5}$$ Projective field: Galois closure of 5.1.2085136.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + (\beta - 1) q^{5} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^4 + (b - 1) * q^5 - q^8 + q^9 $$q - q^{2} + q^{4} + (\beta - 1) q^{5} - q^{8} + q^{9} + ( - \beta + 1) q^{10} + \beta q^{13} + q^{16} - \beta q^{17} - q^{18} + (\beta - 1) q^{20} + ( - \beta + 1) q^{25} - \beta q^{26} + \beta q^{29} - q^{32} + \beta q^{34} + q^{36} + ( - \beta + 1) q^{37} + ( - \beta + 1) q^{40} + ( - \beta + 1) q^{41} + (\beta - 1) q^{45} + q^{49} + (\beta - 1) q^{50} + \beta q^{52} + ( - \beta + 1) q^{53} - \beta q^{58} - \beta q^{61} + q^{64} + q^{65} - \beta q^{68} - q^{72} + (\beta - 1) q^{73} + (\beta - 1) q^{74} + (\beta - 1) q^{80} + q^{81} + (\beta - 1) q^{82} - q^{85} + ( - \beta + 1) q^{89} + ( - \beta + 1) q^{90} + \beta q^{97} - q^{98} +O(q^{100})$$ q - q^2 + q^4 + (b - 1) * q^5 - q^8 + q^9 + (-b + 1) * q^10 + b * q^13 + q^16 - b * q^17 - q^18 + (b - 1) * q^20 + (-b + 1) * q^25 - b * q^26 + b * q^29 - q^32 + b * q^34 + q^36 + (-b + 1) * q^37 + (-b + 1) * q^40 + (-b + 1) * q^41 + (b - 1) * q^45 + q^49 + (b - 1) * q^50 + b * q^52 + (-b + 1) * q^53 - b * q^58 - b * q^61 + q^64 + q^65 - b * q^68 - q^72 + (b - 1) * q^73 + (b - 1) * q^74 + (b - 1) * q^80 + q^81 + (b - 1) * q^82 - q^85 + (-b + 1) * q^89 + (-b + 1) * q^90 + b * q^97 - q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - q^5 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{8} + 2 q^{9} + q^{10} + q^{13} + 2 q^{16} - q^{17} - 2 q^{18} - q^{20} + q^{25} - q^{26} + q^{29} - 2 q^{32} + q^{34} + 2 q^{36} + q^{37} + q^{40} + q^{41} - q^{45} + 2 q^{49} - q^{50} + q^{52} + q^{53} - q^{58} - q^{61} + 2 q^{64} + 2 q^{65} - q^{68} - 2 q^{72} - q^{73} - q^{74} - q^{80} + 2 q^{81} - q^{82} - 2 q^{85} + q^{89} + q^{90} + q^{97} - 2 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - q^5 - 2 * q^8 + 2 * q^9 + q^10 + q^13 + 2 * q^16 - q^17 - 2 * q^18 - q^20 + q^25 - q^26 + q^29 - 2 * q^32 + q^34 + 2 * q^36 + q^37 + q^40 + q^41 - q^45 + 2 * q^49 - q^50 + q^52 + q^53 - q^58 - q^61 + 2 * q^64 + 2 * q^65 - q^68 - 2 * q^72 - q^73 - q^74 - q^80 + 2 * q^81 - q^82 - 2 * q^85 + q^89 + q^90 + q^97 - 2 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1444\mathbb{Z}\right)^\times$$.

 $$n$$ $$723$$ $$1085$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
723.1
 −0.618034 1.61803
−1.00000 0 1.00000 −1.61803 0 0 −1.00000 1.00000 1.61803
723.2 −1.00000 0 1.00000 0.618034 0 0 −1.00000 1.00000 −0.618034
 $$n$$: 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 $$\Q(\sqrt{-1})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1444.1.b.a 2
4.b odd 2 1 CM 1444.1.b.a 2
19.b odd 2 1 1444.1.b.b yes 2
19.c even 3 2 1444.1.g.b 4
19.d odd 6 2 1444.1.g.a 4
19.e even 9 6 1444.1.l.b 12
19.f odd 18 6 1444.1.l.a 12
76.d even 2 1 1444.1.b.b yes 2
76.f even 6 2 1444.1.g.a 4
76.g odd 6 2 1444.1.g.b 4
76.k even 18 6 1444.1.l.a 12
76.l odd 18 6 1444.1.l.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1444.1.b.a 2 1.a even 1 1 trivial
1444.1.b.a 2 4.b odd 2 1 CM
1444.1.b.b yes 2 19.b odd 2 1
1444.1.b.b yes 2 76.d even 2 1
1444.1.g.a 4 19.d odd 6 2
1444.1.g.a 4 76.f even 6 2
1444.1.g.b 4 19.c even 3 2
1444.1.g.b 4 76.g odd 6 2
1444.1.l.a 12 19.f odd 18 6
1444.1.l.a 12 76.k even 18 6
1444.1.l.b 12 19.e even 9 6
1444.1.l.b 12 76.l odd 18 6

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{2} - T_{13} - 1$$ acting on $$S_{1}^{\mathrm{new}}(1444, [\chi])$$.

## Hecke characteristic polynomials

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