# Properties

 Label 2888.1.f.b Level $2888$ Weight $1$ Character orbit 2888.f Self dual yes Analytic conductor $1.441$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -8 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2888,1,Mod(723,2888)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2888, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

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

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

chi := DirichletCharacter("2888.723");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2888 = 2^{3} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2888.f (of order $$2$$, degree $$1$$, 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: yes Analytic conductor: $$1.44129975648$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 152) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.2888.1 Artin image: $S_3$ Artin field: Galois closure of 3.1.2888.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - q^{6} + q^{8}+O(q^{10})$$ q + q^2 - q^3 + q^4 - q^6 + q^8 $$q + q^{2} - q^{3} + q^{4} - q^{6} + q^{8} - q^{11} - q^{12} + q^{16} + 2 q^{17} - q^{22} - q^{24} + q^{25} + q^{27} + q^{32} + q^{33} + 2 q^{34} - q^{41} + 2 q^{43} - q^{44} - q^{48} + q^{49} + q^{50} - 2 q^{51} + q^{54} - q^{59} + q^{64} + q^{66} - q^{67} + 2 q^{68} - q^{73} - q^{75} - q^{81} - q^{82} - q^{83} + 2 q^{86} - q^{88} + 2 q^{89} - q^{96} - q^{97} + q^{98}+O(q^{100})$$ q + q^2 - q^3 + q^4 - q^6 + q^8 - q^11 - q^12 + q^16 + 2 * q^17 - q^22 - q^24 + q^25 + q^27 + q^32 + q^33 + 2 * q^34 - q^41 + 2 * q^43 - q^44 - q^48 + q^49 + q^50 - 2 * q^51 + q^54 - q^59 + q^64 + q^66 - q^67 + 2 * q^68 - q^73 - q^75 - q^81 - q^82 - q^83 + 2 * q^86 - q^88 + 2 * q^89 - q^96 - q^97 + q^98

## Character values

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

 $$n$$ $$1445$$ $$2167$$ $$2529$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
723.1
 0
1.00000 −1.00000 1.00000 0 −1.00000 0 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2888.1.f.b 1
8.d odd 2 1 CM 2888.1.f.b 1
19.b odd 2 1 2888.1.f.a 1
19.c even 3 2 152.1.k.a 2
19.d odd 6 2 2888.1.k.a 2
19.e even 9 6 2888.1.u.c 6
19.f odd 18 6 2888.1.u.d 6
57.h odd 6 2 1368.1.bz.a 2
76.g odd 6 2 608.1.o.a 2
95.i even 6 2 3800.1.bd.c 2
95.m odd 12 4 3800.1.bn.b 4
152.b even 2 1 2888.1.f.a 1
152.k odd 6 2 152.1.k.a 2
152.o even 6 2 2888.1.k.a 2
152.p even 6 2 608.1.o.a 2
152.u odd 18 6 2888.1.u.c 6
152.v even 18 6 2888.1.u.d 6
456.u even 6 2 1368.1.bz.a 2
760.bm odd 6 2 3800.1.bd.c 2
760.bw even 12 4 3800.1.bn.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.k.a 2 19.c even 3 2
152.1.k.a 2 152.k odd 6 2
608.1.o.a 2 76.g odd 6 2
608.1.o.a 2 152.p even 6 2
1368.1.bz.a 2 57.h odd 6 2
1368.1.bz.a 2 456.u even 6 2
2888.1.f.a 1 19.b odd 2 1
2888.1.f.a 1 152.b even 2 1
2888.1.f.b 1 1.a even 1 1 trivial
2888.1.f.b 1 8.d odd 2 1 CM
2888.1.k.a 2 19.d odd 6 2
2888.1.k.a 2 152.o even 6 2
2888.1.u.c 6 19.e even 9 6
2888.1.u.c 6 152.u odd 18 6
2888.1.u.d 6 19.f odd 18 6
2888.1.u.d 6 152.v even 18 6
3800.1.bd.c 2 95.i even 6 2
3800.1.bd.c 2 760.bm odd 6 2
3800.1.bn.b 4 95.m odd 12 4
3800.1.bn.b 4 760.bw even 12 4

## Hecke kernels

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

## Hecke characteristic polynomials

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