# Properties

 Label 2888.1.s.a Level $2888$ Weight $1$ Character orbit 2888.s Analytic conductor $1.441$ Analytic rank $0$ Dimension $6$ Projective image $D_{3}$ CM discriminant -152 Inner twists $12$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2888,1,Mod(333,2888)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2888, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 9, 17]))

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

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

chi := DirichletCharacter("2888.333");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2888 = 2^{3} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2888.s (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: $$1.44129975648$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 152) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.152.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{18}^{5} q^{2} - \zeta_{18}^{2} q^{3} - \zeta_{18} q^{4} + \zeta_{18}^{7} q^{6} + \zeta_{18}^{3} q^{7} + \zeta_{18}^{6} q^{8} +O(q^{10})$$ q - z^5 * q^2 - z^2 * q^3 - z * q^4 + z^7 * q^6 + z^3 * q^7 + z^6 * q^8 $$q - \zeta_{18}^{5} q^{2} - \zeta_{18}^{2} q^{3} - \zeta_{18} q^{4} + \zeta_{18}^{7} q^{6} + \zeta_{18}^{3} q^{7} + \zeta_{18}^{6} q^{8} + \zeta_{18}^{3} q^{12} + \zeta_{18}^{7} q^{13} - \zeta_{18}^{8} q^{14} + \zeta_{18}^{2} q^{16} + \zeta_{18}^{5} q^{17} - \zeta_{18}^{5} q^{21} + \zeta_{18} q^{23} - \zeta_{18}^{8} q^{24} - \zeta_{18}^{7} q^{25} + \zeta_{18}^{3} q^{26} + \zeta_{18}^{6} q^{27} - \zeta_{18}^{4} q^{28} - \zeta_{18}^{4} q^{29} - \zeta_{18}^{7} q^{32} + \zeta_{18} q^{34} + q^{37} + q^{39} - \zeta_{18} q^{42} - \zeta_{18}^{6} q^{46} + \zeta_{18}^{4} q^{47} - \zeta_{18}^{4} q^{48} - \zeta_{18}^{3} q^{50} - \zeta_{18}^{7} q^{51} - \zeta_{18}^{8} q^{52} + \zeta_{18} q^{53} + \zeta_{18}^{2} q^{54} - q^{56} - q^{58} + \zeta_{18}^{5} q^{59} - \zeta_{18}^{3} q^{64} - \zeta_{18}^{4} q^{67} - \zeta_{18}^{6} q^{68} - \zeta_{18}^{3} q^{69} - \zeta_{18}^{2} q^{73} - 2 \zeta_{18}^{5} q^{74} - q^{75} - \zeta_{18}^{5} q^{78} - \zeta_{18}^{8} q^{81} + \zeta_{18}^{6} q^{84} + \zeta_{18}^{6} q^{87} - \zeta_{18} q^{91} - \zeta_{18}^{2} q^{92} + 2 q^{94} - q^{96} +O(q^{100})$$ q - z^5 * q^2 - z^2 * q^3 - z * q^4 + z^7 * q^6 + z^3 * q^7 + z^6 * q^8 + z^3 * q^12 + z^7 * q^13 - z^8 * q^14 + z^2 * q^16 + z^5 * q^17 - z^5 * q^21 + z * q^23 - z^8 * q^24 - z^7 * q^25 + z^3 * q^26 + z^6 * q^27 - z^4 * q^28 - z^4 * q^29 - z^7 * q^32 + z * q^34 + q^37 + q^39 - z * q^42 - z^6 * q^46 + z^4 * q^47 - z^4 * q^48 - z^3 * q^50 - z^7 * q^51 - z^8 * q^52 + z * q^53 + z^2 * q^54 - q^56 - q^58 + z^5 * q^59 - z^3 * q^64 - z^4 * q^67 - z^6 * q^68 - z^3 * q^69 - z^2 * q^73 - 2*z^5 * q^74 - q^75 - z^5 * q^78 - z^8 * q^81 + z^6 * q^84 + z^6 * q^87 - z * q^91 - z^2 * q^92 + 2 * q^94 - q^96 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{7} - 3 q^{8}+O(q^{10})$$ 6 * q + 3 * q^7 - 3 * q^8 $$6 q + 3 q^{7} - 3 q^{8} + 3 q^{12} + 3 q^{26} - 3 q^{27} + 12 q^{37} + 6 q^{39} + 3 q^{46} - 3 q^{50} - 6 q^{56} - 6 q^{58} - 3 q^{64} + 3 q^{68} - 3 q^{69} - 6 q^{75} - 3 q^{84} - 3 q^{87} + 12 q^{94} - 6 q^{96}+O(q^{100})$$ 6 * q + 3 * q^7 - 3 * q^8 + 3 * q^12 + 3 * q^26 - 3 * q^27 + 12 * q^37 + 6 * q^39 + 3 * q^46 - 3 * q^50 - 6 * q^56 - 6 * q^58 - 3 * q^64 + 3 * q^68 - 3 * q^69 - 6 * q^75 - 3 * q^84 - 3 * q^87 + 12 * q^94 - 6 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2888\mathbb{Z}\right)^\times$$.

 $$n$$ $$1445$$ $$2167$$ $$2529$$ $$\chi(n)$$ $$-1$$ $$1$$ $$\zeta_{18}$$

## 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 −0.766044 + 0.642788i −0.173648 − 0.984808i −0.173648 + 0.984808i −0.766044 − 0.642788i
0.173648 + 0.984808i −0.766044 + 0.642788i −0.939693 + 0.342020i 0 −0.766044 0.642788i 0.500000 0.866025i −0.500000 0.866025i 0 0
477.1 0.173648 0.984808i −0.766044 0.642788i −0.939693 0.342020i 0 −0.766044 + 0.642788i 0.500000 + 0.866025i −0.500000 + 0.866025i 0 0
1021.1 −0.939693 + 0.342020i −0.173648 + 0.984808i 0.766044 0.642788i 0 −0.173648 0.984808i 0.500000 + 0.866025i −0.500000 + 0.866025i 0 0
1029.1 0.766044 + 0.642788i 0.939693 0.342020i 0.173648 + 0.984808i 0 0.939693 + 0.342020i 0.500000 + 0.866025i −0.500000 + 0.866025i 0 0
2293.1 0.766044 0.642788i 0.939693 + 0.342020i 0.173648 0.984808i 0 0.939693 0.342020i 0.500000 0.866025i −0.500000 0.866025i 0 0
2789.1 −0.939693 0.342020i −0.173648 0.984808i 0.766044 + 0.642788i 0 −0.173648 + 0.984808i 0.500000 0.866025i −0.500000 0.866025i 0 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
152.g odd 2 1 CM by $$\Q(\sqrt{-38})$$
19.c even 3 2 inner
19.e even 9 3 inner
152.l odd 6 2 inner
152.s odd 18 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2888.1.s.a 6
8.b even 2 1 2888.1.s.b 6
19.b odd 2 1 2888.1.s.b 6
19.c even 3 2 inner 2888.1.s.a 6
19.d odd 6 2 2888.1.s.b 6
19.e even 9 1 152.1.g.b yes 1
19.e even 9 2 2888.1.l.a 2
19.e even 9 3 inner 2888.1.s.a 6
19.f odd 18 1 152.1.g.a 1
19.f odd 18 2 2888.1.l.b 2
19.f odd 18 3 2888.1.s.b 6
57.j even 18 1 1368.1.i.b 1
57.l odd 18 1 1368.1.i.a 1
76.k even 18 1 608.1.g.a 1
76.l odd 18 1 608.1.g.b 1
95.o odd 18 1 3800.1.o.b 1
95.p even 18 1 3800.1.o.a 1
95.q odd 36 2 3800.1.b.b 2
95.r even 36 2 3800.1.b.a 2
152.g odd 2 1 CM 2888.1.s.a 6
152.l odd 6 2 inner 2888.1.s.a 6
152.p even 6 2 2888.1.s.b 6
152.s odd 18 1 152.1.g.b yes 1
152.s odd 18 2 2888.1.l.a 2
152.s odd 18 3 inner 2888.1.s.a 6
152.t even 18 1 152.1.g.a 1
152.t even 18 2 2888.1.l.b 2
152.t even 18 3 2888.1.s.b 6
152.u odd 18 1 608.1.g.a 1
152.v even 18 1 608.1.g.b 1
456.bh odd 18 1 1368.1.i.b 1
456.bj even 18 1 1368.1.i.a 1
760.cj even 18 1 3800.1.o.b 1
760.ck odd 18 1 3800.1.o.a 1
760.cq odd 36 2 3800.1.b.a 2
760.cs even 36 2 3800.1.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.g.a 1 19.f odd 18 1
152.1.g.a 1 152.t even 18 1
152.1.g.b yes 1 19.e even 9 1
152.1.g.b yes 1 152.s odd 18 1
608.1.g.a 1 76.k even 18 1
608.1.g.a 1 152.u odd 18 1
608.1.g.b 1 76.l odd 18 1
608.1.g.b 1 152.v even 18 1
1368.1.i.a 1 57.l odd 18 1
1368.1.i.a 1 456.bj even 18 1
1368.1.i.b 1 57.j even 18 1
1368.1.i.b 1 456.bh odd 18 1
2888.1.l.a 2 19.e even 9 2
2888.1.l.a 2 152.s odd 18 2
2888.1.l.b 2 19.f odd 18 2
2888.1.l.b 2 152.t even 18 2
2888.1.s.a 6 1.a even 1 1 trivial
2888.1.s.a 6 19.c even 3 2 inner
2888.1.s.a 6 19.e even 9 3 inner
2888.1.s.a 6 152.g odd 2 1 CM
2888.1.s.a 6 152.l odd 6 2 inner
2888.1.s.a 6 152.s odd 18 3 inner
2888.1.s.b 6 8.b even 2 1
2888.1.s.b 6 19.b odd 2 1
2888.1.s.b 6 19.d odd 6 2
2888.1.s.b 6 19.f odd 18 3
2888.1.s.b 6 152.p even 6 2
2888.1.s.b 6 152.t even 18 3
3800.1.b.a 2 95.r even 36 2
3800.1.b.a 2 760.cq odd 36 2
3800.1.b.b 2 95.q odd 36 2
3800.1.b.b 2 760.cs even 36 2
3800.1.o.a 1 95.p even 18 1
3800.1.o.a 1 760.ck odd 18 1
3800.1.o.b 1 95.o odd 18 1
3800.1.o.b 1 760.cj even 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} - T_{3}^{3} + 1$$ acting on $$S_{1}^{\mathrm{new}}(2888, [\chi])$$.

## Hecke characteristic polynomials

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