Properties

 Label 1444.1.h.a Level $1444$ Weight $1$ Character orbit 1444.h Analytic conductor $0.721$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -19 Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1444,1,Mod(69,1444)]

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([0, 5]))

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

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

chi := DirichletCharacter("1444.69");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1444 = 2^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1444.h (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: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ 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: no (minimal twist has level 76) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.76.1 Artin image: $C_3\times S_3$ Artin field: Galois closure of 6.0.39617584.1

$q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6}^{2} q^{5} - q^{7} - \zeta_{6} q^{9} +O(q^{10})$$ q - z^2 * q^5 - q^7 - z * q^9 $$q - \zeta_{6}^{2} q^{5} - q^{7} - \zeta_{6} q^{9} - q^{11} - \zeta_{6}^{2} q^{17} - 2 \zeta_{6} q^{23} + \zeta_{6}^{2} q^{35} - \zeta_{6}^{2} q^{43} - q^{45} + \zeta_{6} q^{47} + \zeta_{6}^{2} q^{55} + \zeta_{6} q^{61} + \zeta_{6} q^{63} - \zeta_{6}^{2} q^{73} + q^{77} + \zeta_{6}^{2} q^{81} + 2 q^{83} - \zeta_{6} q^{85} + \zeta_{6} q^{99} +O(q^{100})$$ q - z^2 * q^5 - q^7 - z * q^9 - q^11 - z^2 * q^17 - 2*z * q^23 + z^2 * q^35 - z^2 * q^43 - q^45 + z * q^47 + z^2 * q^55 + z * q^61 + z * q^63 - z^2 * q^73 + q^77 + z^2 * q^81 + 2 * q^83 - z * q^85 + z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{5} - 2 q^{7} - q^{9}+O(q^{10})$$ 2 * q + q^5 - 2 * q^7 - q^9 $$2 q + q^{5} - 2 q^{7} - q^{9} - 2 q^{11} + q^{17} - 2 q^{23} - q^{35} + q^{43} - 2 q^{45} + q^{47} - q^{55} + q^{61} + q^{63} + q^{73} + 2 q^{77} - q^{81} + 4 q^{83} - q^{85} + q^{99}+O(q^{100})$$ 2 * q + q^5 - 2 * q^7 - q^9 - 2 * q^11 + q^17 - 2 * q^23 - q^35 + q^43 - 2 * q^45 + q^47 - q^55 + q^61 + q^63 + q^73 + 2 * q^77 - q^81 + 4 * q^83 - q^85 + q^99

Character values

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

 $$n$$ $$723$$ $$1085$$ $$\chi(n)$$ $$1$$ $$\zeta_{6}$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
69.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 0.500000 + 0.866025i 0 −1.00000 0 −0.500000 + 0.866025i 0
293.1 0 0 0 0.500000 0.866025i 0 −1.00000 0 −0.500000 0.866025i 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
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
19.c even 3 1 inner
19.d 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.h.a 2
19.b odd 2 1 CM 1444.1.h.a 2
19.c even 3 1 76.1.c.a 1
19.c even 3 1 inner 1444.1.h.a 2
19.d odd 6 1 76.1.c.a 1
19.d odd 6 1 inner 1444.1.h.a 2
19.e even 9 6 1444.1.j.a 6
19.f odd 18 6 1444.1.j.a 6
57.f even 6 1 684.1.h.a 1
57.h odd 6 1 684.1.h.a 1
76.f even 6 1 304.1.e.a 1
76.g odd 6 1 304.1.e.a 1
95.h odd 6 1 1900.1.e.a 1
95.i even 6 1 1900.1.e.a 1
95.l even 12 2 1900.1.g.a 2
95.m odd 12 2 1900.1.g.a 2
133.g even 3 1 3724.1.bc.c 2
133.h even 3 1 3724.1.bc.c 2
133.i even 6 1 3724.1.bc.b 2
133.j odd 6 1 3724.1.bc.c 2
133.k odd 6 1 3724.1.bc.b 2
133.m odd 6 1 3724.1.e.c 1
133.n odd 6 1 3724.1.bc.c 2
133.p even 6 1 3724.1.e.c 1
133.s even 6 1 3724.1.bc.b 2
133.t odd 6 1 3724.1.bc.b 2
152.k odd 6 1 1216.1.e.b 1
152.l odd 6 1 1216.1.e.a 1
152.o even 6 1 1216.1.e.b 1
152.p even 6 1 1216.1.e.a 1
228.m even 6 1 2736.1.o.b 1
228.n odd 6 1 2736.1.o.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.1.c.a 1 19.c even 3 1
76.1.c.a 1 19.d odd 6 1
304.1.e.a 1 76.f even 6 1
304.1.e.a 1 76.g odd 6 1
684.1.h.a 1 57.f even 6 1
684.1.h.a 1 57.h odd 6 1
1216.1.e.a 1 152.l odd 6 1
1216.1.e.a 1 152.p even 6 1
1216.1.e.b 1 152.k odd 6 1
1216.1.e.b 1 152.o even 6 1
1444.1.h.a 2 1.a even 1 1 trivial
1444.1.h.a 2 19.b odd 2 1 CM
1444.1.h.a 2 19.c even 3 1 inner
1444.1.h.a 2 19.d odd 6 1 inner
1444.1.j.a 6 19.e even 9 6
1444.1.j.a 6 19.f odd 18 6
1900.1.e.a 1 95.h odd 6 1
1900.1.e.a 1 95.i even 6 1
1900.1.g.a 2 95.l even 12 2
1900.1.g.a 2 95.m odd 12 2
2736.1.o.b 1 228.m even 6 1
2736.1.o.b 1 228.n odd 6 1
3724.1.e.c 1 133.m odd 6 1
3724.1.e.c 1 133.p even 6 1
3724.1.bc.b 2 133.i even 6 1
3724.1.bc.b 2 133.k odd 6 1
3724.1.bc.b 2 133.s even 6 1
3724.1.bc.b 2 133.t odd 6 1
3724.1.bc.c 2 133.g even 3 1
3724.1.bc.c 2 133.h even 3 1
3724.1.bc.c 2 133.j odd 6 1
3724.1.bc.c 2 133.n odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

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