Properties

 Label 2888.1.k.a Level $2888$ Weight $1$ Character orbit 2888.k Analytic conductor $1.441$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -8 Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2888,1,Mod(2595,2888)]

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([3, 3, 4]))

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

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

chi := DirichletCharacter("2888.2595");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2888 = 2^{3} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2888.k (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 3.1.2888.1 Artin image: $C_6\times S_3$ Artin field: Galois closure of 12.0.4452139149819904.6

$q$-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.

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

Character values

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

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

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2595.1
 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 0 −1.00000 0 0
2819.1 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 0 0.500000 0.866025i 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 $$\Q(\sqrt{-2})$$
19.c even 3 1 inner
152.k 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.k.a 2
8.d odd 2 1 CM 2888.1.k.a 2
19.b odd 2 1 152.1.k.a 2
19.c even 3 1 2888.1.f.a 1
19.c even 3 1 inner 2888.1.k.a 2
19.d odd 6 1 152.1.k.a 2
19.d odd 6 1 2888.1.f.b 1
19.e even 9 6 2888.1.u.d 6
19.f odd 18 6 2888.1.u.c 6
57.d even 2 1 1368.1.bz.a 2
57.f even 6 1 1368.1.bz.a 2
76.d even 2 1 608.1.o.a 2
76.f even 6 1 608.1.o.a 2
95.d odd 2 1 3800.1.bd.c 2
95.g even 4 2 3800.1.bn.b 4
95.h odd 6 1 3800.1.bd.c 2
95.l even 12 2 3800.1.bn.b 4
152.b even 2 1 152.1.k.a 2
152.g odd 2 1 608.1.o.a 2
152.k odd 6 1 2888.1.f.a 1
152.k odd 6 1 inner 2888.1.k.a 2
152.l odd 6 1 608.1.o.a 2
152.o even 6 1 152.1.k.a 2
152.o even 6 1 2888.1.f.b 1
152.u odd 18 6 2888.1.u.d 6
152.v even 18 6 2888.1.u.c 6
456.l odd 2 1 1368.1.bz.a 2
456.s odd 6 1 1368.1.bz.a 2
760.p even 2 1 3800.1.bd.c 2
760.y odd 4 2 3800.1.bn.b 4
760.bf even 6 1 3800.1.bd.c 2
760.bu odd 12 2 3800.1.bn.b 4

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

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

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