# Properties

 Label 1444.1.j.a Level $1444$ Weight $1$ Character orbit 1444.j Analytic conductor $0.721$ Analytic rank $0$ Dimension $6$ Projective image $D_{3}$ CM discriminant -19 Inner twists $12$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1444,1,Mod(333,1444)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1444, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 17]))

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

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

chi := DirichletCharacter("1444.333");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1444 = 2^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1444.j (of order $$18$$, degree $$6$$, 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: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 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: $S_3\times C_9$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{18} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{18}^{8} q^{5} + \zeta_{18}^{3} q^{7} + \zeta_{18}^{4} q^{9} +O(q^{10})$$ q - z^8 * q^5 + z^3 * q^7 + z^4 * q^9 $$q - \zeta_{18}^{8} q^{5} + \zeta_{18}^{3} q^{7} + \zeta_{18}^{4} q^{9} - \zeta_{18}^{6} q^{11} + \zeta_{18}^{5} q^{17} - 2 \zeta_{18} q^{23} + \zeta_{18}^{2} q^{35} - \zeta_{18}^{8} q^{43} + \zeta_{18}^{3} q^{45} - \zeta_{18}^{4} q^{47} - \zeta_{18}^{5} q^{55} + \zeta_{18} q^{61} + \zeta_{18}^{7} q^{63} - \zeta_{18}^{2} q^{73} + q^{77} + \zeta_{18}^{8} q^{81} - 2 \zeta_{18}^{3} q^{83} + \zeta_{18}^{4} q^{85} + \zeta_{18} q^{99} +O(q^{100})$$ q - z^8 * q^5 + z^3 * q^7 + z^4 * q^9 - z^6 * q^11 + z^5 * q^17 - 2*z * q^23 + z^2 * q^35 - z^8 * q^43 + z^3 * q^45 - z^4 * q^47 - z^5 * q^55 + z * q^61 + z^7 * q^63 - z^2 * q^73 + q^77 + z^8 * q^81 - 2*z^3 * q^83 + z^4 * q^85 + z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{7}+O(q^{10})$$ 6 * q + 3 * q^7 $$6 q + 3 q^{7} + 3 q^{11} + 3 q^{45} + 6 q^{77} - 6 q^{83}+O(q^{100})$$ 6 * q + 3 * q^7 + 3 * q^11 + 3 * q^45 + 6 * q^77 - 6 * q^83

## Character values

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

 $$n$$ $$723$$ $$1085$$ $$\chi(n)$$ $$1$$ $$\zeta_{18}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
333.1
 0.939693 − 0.342020i 0.939693 + 0.342020i −0.173648 + 0.984808i −0.766044 + 0.642788i −0.173648 − 0.984808i −0.766044 − 0.642788i
0 0 0 0.939693 + 0.342020i 0 0.500000 0.866025i 0 0.173648 0.984808i 0
477.1 0 0 0 0.939693 0.342020i 0 0.500000 + 0.866025i 0 0.173648 + 0.984808i 0
849.1 0 0 0 −0.173648 0.984808i 0 0.500000 0.866025i 0 0.766044 + 0.642788i 0
1021.1 0 0 0 −0.766044 0.642788i 0 0.500000 + 0.866025i 0 −0.939693 0.342020i 0
1029.1 0 0 0 −0.173648 + 0.984808i 0 0.500000 + 0.866025i 0 0.766044 0.642788i 0
1345.1 0 0 0 −0.766044 + 0.642788i 0 0.500000 0.866025i 0 −0.939693 + 0.342020i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 333.1 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 2 inner
19.d odd 6 2 inner
19.e even 9 3 inner
19.f odd 18 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1444.1.j.a 6
19.b odd 2 1 CM 1444.1.j.a 6
19.c even 3 2 inner 1444.1.j.a 6
19.d odd 6 2 inner 1444.1.j.a 6
19.e even 9 1 76.1.c.a 1
19.e even 9 2 1444.1.h.a 2
19.e even 9 3 inner 1444.1.j.a 6
19.f odd 18 1 76.1.c.a 1
19.f odd 18 2 1444.1.h.a 2
19.f odd 18 3 inner 1444.1.j.a 6
57.j even 18 1 684.1.h.a 1
57.l odd 18 1 684.1.h.a 1
76.k even 18 1 304.1.e.a 1
76.l odd 18 1 304.1.e.a 1
95.o odd 18 1 1900.1.e.a 1
95.p even 18 1 1900.1.e.a 1
95.q odd 36 2 1900.1.g.a 2
95.r even 36 2 1900.1.g.a 2
133.u even 9 1 3724.1.bc.c 2
133.w even 9 1 3724.1.bc.c 2
133.x odd 18 1 3724.1.bc.b 2
133.y odd 18 1 3724.1.e.c 1
133.z odd 18 1 3724.1.bc.b 2
133.ba even 18 1 3724.1.e.c 1
133.bb even 18 1 3724.1.bc.b 2
133.bd odd 18 1 3724.1.bc.c 2
133.be odd 18 1 3724.1.bc.c 2
133.bf even 18 1 3724.1.bc.b 2
152.s odd 18 1 1216.1.e.a 1
152.t even 18 1 1216.1.e.a 1
152.u odd 18 1 1216.1.e.b 1
152.v even 18 1 1216.1.e.b 1
228.u odd 18 1 2736.1.o.b 1
228.v even 18 1 2736.1.o.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.1.c.a 1 19.e even 9 1
76.1.c.a 1 19.f odd 18 1
304.1.e.a 1 76.k even 18 1
304.1.e.a 1 76.l odd 18 1
684.1.h.a 1 57.j even 18 1
684.1.h.a 1 57.l odd 18 1
1216.1.e.a 1 152.s odd 18 1
1216.1.e.a 1 152.t even 18 1
1216.1.e.b 1 152.u odd 18 1
1216.1.e.b 1 152.v even 18 1
1444.1.h.a 2 19.e even 9 2
1444.1.h.a 2 19.f odd 18 2
1444.1.j.a 6 1.a even 1 1 trivial
1444.1.j.a 6 19.b odd 2 1 CM
1444.1.j.a 6 19.c even 3 2 inner
1444.1.j.a 6 19.d odd 6 2 inner
1444.1.j.a 6 19.e even 9 3 inner
1444.1.j.a 6 19.f odd 18 3 inner
1900.1.e.a 1 95.o odd 18 1
1900.1.e.a 1 95.p even 18 1
1900.1.g.a 2 95.q odd 36 2
1900.1.g.a 2 95.r even 36 2
2736.1.o.b 1 228.u odd 18 1
2736.1.o.b 1 228.v even 18 1
3724.1.e.c 1 133.y odd 18 1
3724.1.e.c 1 133.ba even 18 1
3724.1.bc.b 2 133.x odd 18 1
3724.1.bc.b 2 133.z odd 18 1
3724.1.bc.b 2 133.bb even 18 1
3724.1.bc.b 2 133.bf even 18 1
3724.1.bc.c 2 133.u even 9 1
3724.1.bc.c 2 133.w even 9 1
3724.1.bc.c 2 133.bd odd 18 1
3724.1.bc.c 2 133.be odd 18 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

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6}$$
$5$ $$T^{6} - T^{3} + 1$$
$7$ $$(T^{2} - T + 1)^{3}$$
$11$ $$(T^{2} - T + 1)^{3}$$
$13$ $$T^{6}$$
$17$ $$T^{6} - T^{3} + 1$$
$19$ $$T^{6}$$
$23$ $$T^{6} + 8T^{3} + 64$$
$29$ $$T^{6}$$
$31$ $$T^{6}$$
$37$ $$T^{6}$$
$41$ $$T^{6}$$
$43$ $$T^{6} - T^{3} + 1$$
$47$ $$T^{6} - T^{3} + 1$$
$53$ $$T^{6}$$
$59$ $$T^{6}$$
$61$ $$T^{6} - T^{3} + 1$$
$67$ $$T^{6}$$
$71$ $$T^{6}$$
$73$ $$T^{6} - T^{3} + 1$$
$79$ $$T^{6}$$
$83$ $$(T^{2} + 2 T + 4)^{3}$$
$89$ $$T^{6}$$
$97$ $$T^{6}$$