# Properties

 Label 1444.1.g.a Level $1444$ Weight $1$ Character orbit 1444.g Analytic conductor $0.721$ 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] = [1444,1,Mod(1151,1444)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1444, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 4]))

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

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

chi := DirichletCharacter("1444.1151");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1444 = 2^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1444.g (of order $$6$$, degree $$2$$, 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: $$0.720649878242$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 2x^{2} + x + 1$$ x^4 - x^3 + 2*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 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - 1) q^{2} + \beta_{3} q^{4} + (\beta_{3} - \beta_1 + 1) q^{5} + q^{8} + \beta_{3} q^{9}+O(q^{10})$$ q + (-b3 - 1) * q^2 + b3 * q^4 + (b3 - b1 + 1) * q^5 + q^8 + b3 * q^9 $$q + ( - \beta_{3} - 1) q^{2} + \beta_{3} q^{4} + (\beta_{3} - \beta_1 + 1) q^{5} + q^{8} + \beta_{3} q^{9} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{10} + ( - \beta_{2} - \beta_1) q^{13} + ( - \beta_{3} - 1) q^{16} + \beta_1 q^{17} + q^{18} + ( - \beta_{2} - 1) q^{20} + (\beta_{3} - \beta_{2} - \beta_1) q^{25} + \beta_{2} q^{26} + ( - \beta_{2} - \beta_1) q^{29} + \beta_{3} q^{32} + ( - \beta_{2} - \beta_1) q^{34} + ( - \beta_{3} - 1) q^{36} + ( - \beta_{2} - 1) q^{37} + (\beta_{3} - \beta_1 + 1) q^{40} + (\beta_{3} - \beta_1 + 1) q^{41} + ( - \beta_{2} - 1) q^{45} + q^{49} + (\beta_{2} + 1) q^{50} + \beta_1 q^{52} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{53} + \beta_{2} q^{58} + ( - \beta_{2} - \beta_1) q^{61} + q^{64} - q^{65} + \beta_{2} q^{68} + \beta_{3} q^{72} + (\beta_{3} - \beta_1 + 1) q^{73} + (\beta_{3} - \beta_1 + 1) q^{74} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{80} + ( - \beta_{3} - 1) q^{81} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{82} - \beta_{3} q^{85} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{89} + (\beta_{3} - \beta_1 + 1) q^{90} + \beta_1 q^{97} + ( - \beta_{3} - 1) q^{98}+O(q^{100})$$ q + (-b3 - 1) * q^2 + b3 * q^4 + (b3 - b1 + 1) * q^5 + q^8 + b3 * q^9 + (-b3 + b2 + b1) * q^10 + (-b2 - b1) * q^13 + (-b3 - 1) * q^16 + b1 * q^17 + q^18 + (-b2 - 1) * q^20 + (b3 - b2 - b1) * q^25 + b2 * q^26 + (-b2 - b1) * q^29 + b3 * q^32 + (-b2 - b1) * q^34 + (-b3 - 1) * q^36 + (-b2 - 1) * q^37 + (b3 - b1 + 1) * q^40 + (b3 - b1 + 1) * q^41 + (-b2 - 1) * q^45 + q^49 + (b2 + 1) * q^50 + b1 * q^52 + (-b3 + b2 + b1) * q^53 + b2 * q^58 + (-b2 - b1) * q^61 + q^64 - q^65 + b2 * q^68 + b3 * q^72 + (b3 - b1 + 1) * q^73 + (b3 - b1 + 1) * q^74 + (-b3 + b2 + b1) * q^80 + (-b3 - 1) * q^81 + (-b3 + b2 + b1) * q^82 - b3 * q^85 + (-b3 + b2 + b1) * q^89 + (b3 - b1 + 1) * q^90 + b1 * q^97 + (-b3 - 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 2 q^{4} + q^{5} + 4 q^{8} - 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 - 2 * q^4 + q^5 + 4 * q^8 - 2 * q^9 $$4 q - 2 q^{2} - 2 q^{4} + q^{5} + 4 q^{8} - 2 q^{9} + q^{10} + q^{13} - 2 q^{16} + q^{17} + 4 q^{18} - 2 q^{20} - q^{25} - 2 q^{26} + q^{29} - 2 q^{32} + q^{34} - 2 q^{36} - 2 q^{37} + q^{40} + q^{41} - 2 q^{45} + 4 q^{49} + 2 q^{50} + q^{52} + q^{53} - 2 q^{58} + q^{61} + 4 q^{64} - 4 q^{65} - 2 q^{68} - 2 q^{72} + q^{73} + q^{74} + q^{80} - 2 q^{81} + q^{82} + 2 q^{85} + q^{89} + q^{90} + q^{97} - 2 q^{98}+O(q^{100})$$ 4 * q - 2 * q^2 - 2 * q^4 + q^5 + 4 * q^8 - 2 * q^9 + q^10 + q^13 - 2 * q^16 + q^17 + 4 * q^18 - 2 * q^20 - q^25 - 2 * q^26 + q^29 - 2 * q^32 + q^34 - 2 * q^36 - 2 * q^37 + q^40 + q^41 - 2 * q^45 + 4 * q^49 + 2 * q^50 + q^52 + q^53 - 2 * q^58 + q^61 + 4 * q^64 - 4 * q^65 - 2 * 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

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

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

## Character values

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

 $$n$$ $$723$$ $$1085$$ $$\chi(n)$$ $$-1$$ $$\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1151.1
 0.809017 − 1.40126i −0.309017 + 0.535233i 0.809017 + 1.40126i −0.309017 − 0.535233i
−0.500000 + 0.866025i 0 −0.500000 0.866025i −0.309017 + 0.535233i 0 0 1.00000 −0.500000 0.866025i −0.309017 0.535233i
1151.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0.809017 1.40126i 0 0 1.00000 −0.500000 0.866025i 0.809017 + 1.40126i
1375.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −0.309017 0.535233i 0 0 1.00000 −0.500000 + 0.866025i −0.309017 + 0.535233i
1375.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0.809017 + 1.40126i 0 0 1.00000 −0.500000 + 0.866025i 0.809017 1.40126i
 $$n$$: 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 $$\Q(\sqrt{-1})$$
19.c even 3 1 inner
76.g odd 6 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{4} - T_{13}^{3} + 2T_{13}^{2} + T_{13} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1444, [\chi])$$.

## Hecke characteristic polynomials

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