# Properties

 Label 2888.1.l.b Level $2888$ Weight $1$ Character orbit 2888.l Analytic conductor $1.441$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -152 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2888,1,Mod(69,2888)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2888, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 5]))

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

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

chi := DirichletCharacter("2888.69");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2888 = 2^{3} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2888.l (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: $$1.44129975648$$ 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: $$\Z[a_1, a_2]$$ 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 Artin image: $C_6\times S_3$ Artin field: Galois closure of 12.0.1607222233084985344.6

## $q$-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.

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

## Character values

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

 $$n$$ $$1445$$ $$2167$$ $$2529$$ $$\chi(n)$$ $$-1$$ $$1$$ $$\zeta_{6}$$

## 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
0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 0 0.500000 0.866025i −1.00000 −1.00000 0 0
293.1 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0 0.500000 + 0.866025i −1.00000 −1.00000 0 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
152.g odd 2 1 CM by $$\Q(\sqrt{-38})$$
19.c even 3 1 inner
152.l odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2888.1.l.b 2
8.b even 2 1 2888.1.l.a 2
19.b odd 2 1 2888.1.l.a 2
19.c even 3 1 152.1.g.a 1
19.c even 3 1 inner 2888.1.l.b 2
19.d odd 6 1 152.1.g.b yes 1
19.d odd 6 1 2888.1.l.a 2
19.e even 9 6 2888.1.s.b 6
19.f odd 18 6 2888.1.s.a 6
57.f even 6 1 1368.1.i.a 1
57.h odd 6 1 1368.1.i.b 1
76.f even 6 1 608.1.g.b 1
76.g odd 6 1 608.1.g.a 1
95.h odd 6 1 3800.1.o.a 1
95.i even 6 1 3800.1.o.b 1
95.l even 12 2 3800.1.b.b 2
95.m odd 12 2 3800.1.b.a 2
152.g odd 2 1 CM 2888.1.l.b 2
152.k odd 6 1 608.1.g.b 1
152.l odd 6 1 152.1.g.a 1
152.l odd 6 1 inner 2888.1.l.b 2
152.o even 6 1 608.1.g.a 1
152.p even 6 1 152.1.g.b yes 1
152.p even 6 1 2888.1.l.a 2
152.s odd 18 6 2888.1.s.b 6
152.t even 18 6 2888.1.s.a 6
456.v even 6 1 1368.1.i.b 1
456.x odd 6 1 1368.1.i.a 1
760.z even 6 1 3800.1.o.a 1
760.bh odd 6 1 3800.1.o.b 1
760.bp even 12 2 3800.1.b.a 2
760.br odd 12 2 3800.1.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.g.a 1 19.c even 3 1
152.1.g.a 1 152.l odd 6 1
152.1.g.b yes 1 19.d odd 6 1
152.1.g.b yes 1 152.p even 6 1
608.1.g.a 1 76.g odd 6 1
608.1.g.a 1 152.o even 6 1
608.1.g.b 1 76.f even 6 1
608.1.g.b 1 152.k odd 6 1
1368.1.i.a 1 57.f even 6 1
1368.1.i.a 1 456.x odd 6 1
1368.1.i.b 1 57.h odd 6 1
1368.1.i.b 1 456.v even 6 1
2888.1.l.a 2 8.b even 2 1
2888.1.l.a 2 19.b odd 2 1
2888.1.l.a 2 19.d odd 6 1
2888.1.l.a 2 152.p even 6 1
2888.1.l.b 2 1.a even 1 1 trivial
2888.1.l.b 2 19.c even 3 1 inner
2888.1.l.b 2 152.g odd 2 1 CM
2888.1.l.b 2 152.l odd 6 1 inner
2888.1.s.a 6 19.f odd 18 6
2888.1.s.a 6 152.t even 18 6
2888.1.s.b 6 19.e even 9 6
2888.1.s.b 6 152.s odd 18 6
3800.1.b.a 2 95.m odd 12 2
3800.1.b.a 2 760.bp even 12 2
3800.1.b.b 2 95.l even 12 2
3800.1.b.b 2 760.br odd 12 2
3800.1.o.a 1 95.h odd 6 1
3800.1.o.a 1 760.z even 6 1
3800.1.o.b 1 95.i even 6 1
3800.1.o.b 1 760.bh odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + T_{3} + 1$$ acting on $$S_{1}^{\mathrm{new}}(2888, [\chi])$$.

## Hecke characteristic polynomials

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