# Properties

 Label 2888.1.f.d Level $2888$ Weight $1$ Character orbit 2888.f Self dual yes Analytic conductor $1.441$ Analytic rank $0$ Dimension $3$ Projective image $D_{9}$ 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: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 3x - 1$$ x^3 - 3*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 152) Projective image: $$D_{9}$$ Projective field: Galois closure of 9.1.69564674215936.1 Artin image: $D_9$ Artin field: Galois closure of 9.1.69564674215936.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} - \beta_1 q^{3} + q^{4} - \beta_1 q^{6} + q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10})$$ q + q^2 - b1 * q^3 + q^4 - b1 * q^6 + q^8 + (b2 + 1) * q^9 $$q + q^{2} - \beta_1 q^{3} + q^{4} - \beta_1 q^{6} + q^{8} + (\beta_{2} + 1) q^{9} + ( - \beta_{2} + \beta_1) q^{11} - \beta_1 q^{12} + q^{16} - q^{17} + (\beta_{2} + 1) q^{18} + ( - \beta_{2} + \beta_1) q^{22} - \beta_1 q^{24} + q^{25} + ( - \beta_1 - 1) q^{27} + q^{32} + ( - \beta_{2} + \beta_1 - 1) q^{33} - q^{34} + (\beta_{2} + 1) q^{36} + \beta_{2} q^{41} - q^{43} + ( - \beta_{2} + \beta_1) q^{44} - \beta_1 q^{48} + q^{49} + q^{50} + \beta_1 q^{51} + ( - \beta_1 - 1) q^{54} + ( - \beta_{2} + \beta_1) q^{59} + q^{64} + ( - \beta_{2} + \beta_1 - 1) q^{66} + \beta_{2} q^{67} - q^{68} + (\beta_{2} + 1) q^{72} + ( - \beta_{2} + \beta_1) q^{73} - \beta_1 q^{75} + (\beta_1 + 1) q^{81} + \beta_{2} q^{82} + \beta_{2} q^{83} - q^{86} + ( - \beta_{2} + \beta_1) q^{88} - q^{89} - \beta_1 q^{96} + \beta_{2} q^{97} + q^{98} + (\beta_1 - 1) q^{99}+O(q^{100})$$ q + q^2 - b1 * q^3 + q^4 - b1 * q^6 + q^8 + (b2 + 1) * q^9 + (-b2 + b1) * q^11 - b1 * q^12 + q^16 - q^17 + (b2 + 1) * q^18 + (-b2 + b1) * q^22 - b1 * q^24 + q^25 + (-b1 - 1) * q^27 + q^32 + (-b2 + b1 - 1) * q^33 - q^34 + (b2 + 1) * q^36 + b2 * q^41 - q^43 + (-b2 + b1) * q^44 - b1 * q^48 + q^49 + q^50 + b1 * q^51 + (-b1 - 1) * q^54 + (-b2 + b1) * q^59 + q^64 + (-b2 + b1 - 1) * q^66 + b2 * q^67 - q^68 + (b2 + 1) * q^72 + (-b2 + b1) * q^73 - b1 * q^75 + (b1 + 1) * q^81 + b2 * q^82 + b2 * q^83 - q^86 + (-b2 + b1) * q^88 - q^89 - b1 * q^96 + b2 * q^97 + q^98 + (b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 3 q^{4} + 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8 + 3 * q^9 $$3 q + 3 q^{2} + 3 q^{4} + 3 q^{8} + 3 q^{9} + 3 q^{16} - 3 q^{17} + 3 q^{18} + 3 q^{25} - 3 q^{27} + 3 q^{32} - 3 q^{33} - 3 q^{34} + 3 q^{36} - 3 q^{43} + 3 q^{49} + 3 q^{50} - 3 q^{54} + 3 q^{64} - 3 q^{66} - 3 q^{68} + 3 q^{72} + 3 q^{81} - 3 q^{86} - 3 q^{89} + 3 q^{98} - 3 q^{99}+O(q^{100})$$ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^8 + 3 * q^9 + 3 * q^16 - 3 * q^17 + 3 * q^18 + 3 * q^25 - 3 * q^27 + 3 * q^32 - 3 * q^33 - 3 * q^34 + 3 * q^36 - 3 * q^43 + 3 * q^49 + 3 * q^50 - 3 * q^54 + 3 * q^64 - 3 * q^66 - 3 * q^68 + 3 * q^72 + 3 * q^81 - 3 * q^86 - 3 * q^89 + 3 * q^98 - 3 * q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{18} + \zeta_{18}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2

## 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
 1.87939 −0.347296 −1.53209
1.00000 −1.87939 1.00000 0 −1.87939 0 1.00000 2.53209 0
723.2 1.00000 0.347296 1.00000 0 0.347296 0 1.00000 −0.879385 0
723.3 1.00000 1.53209 1.00000 0 1.53209 0 1.00000 1.34730 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.d 3
8.d odd 2 1 CM 2888.1.f.d 3
19.b odd 2 1 2888.1.f.c 3
19.c even 3 2 2888.1.k.b 6
19.d odd 6 2 2888.1.k.c 6
19.e even 9 2 152.1.u.a 6
19.e even 9 2 2888.1.u.b 6
19.e even 9 2 2888.1.u.g 6
19.f odd 18 2 2888.1.u.a 6
19.f odd 18 2 2888.1.u.e 6
19.f odd 18 2 2888.1.u.f 6
57.l odd 18 2 1368.1.eh.a 6
76.l odd 18 2 608.1.bg.a 6
95.p even 18 2 3800.1.cv.c 6
95.q odd 36 4 3800.1.cq.b 12
152.b even 2 1 2888.1.f.c 3
152.k odd 6 2 2888.1.k.b 6
152.o even 6 2 2888.1.k.c 6
152.t even 18 2 608.1.bg.a 6
152.u odd 18 2 152.1.u.a 6
152.u odd 18 2 2888.1.u.b 6
152.u odd 18 2 2888.1.u.g 6
152.v even 18 2 2888.1.u.a 6
152.v even 18 2 2888.1.u.e 6
152.v even 18 2 2888.1.u.f 6
456.bu even 18 2 1368.1.eh.a 6
760.bz odd 18 2 3800.1.cv.c 6
760.cp even 36 4 3800.1.cq.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.u.a 6 19.e even 9 2
152.1.u.a 6 152.u odd 18 2
608.1.bg.a 6 76.l odd 18 2
608.1.bg.a 6 152.t even 18 2
1368.1.eh.a 6 57.l odd 18 2
1368.1.eh.a 6 456.bu even 18 2
2888.1.f.c 3 19.b odd 2 1
2888.1.f.c 3 152.b even 2 1
2888.1.f.d 3 1.a even 1 1 trivial
2888.1.f.d 3 8.d odd 2 1 CM
2888.1.k.b 6 19.c even 3 2
2888.1.k.b 6 152.k odd 6 2
2888.1.k.c 6 19.d odd 6 2
2888.1.k.c 6 152.o even 6 2
2888.1.u.a 6 19.f odd 18 2
2888.1.u.a 6 152.v even 18 2
2888.1.u.b 6 19.e even 9 2
2888.1.u.b 6 152.u odd 18 2
2888.1.u.e 6 19.f odd 18 2
2888.1.u.e 6 152.v even 18 2
2888.1.u.f 6 19.f odd 18 2
2888.1.u.f 6 152.v even 18 2
2888.1.u.g 6 19.e even 9 2
2888.1.u.g 6 152.u odd 18 2
3800.1.cq.b 12 95.q odd 36 4
3800.1.cq.b 12 760.cp even 36 4
3800.1.cv.c 6 95.p even 18 2
3800.1.cv.c 6 760.bz odd 18 2

## Hecke kernels

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

## Hecke characteristic polynomials

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