LMFDB - Newform orbit 2888.1.l.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2888,1,Mod(69,2888)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2888, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 5])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

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:traces = [2,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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:	$C_3\times S_3$
Artin field:	Galois closure of 6.0.1267762688.3

$q$-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} - 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} + \cdots - 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 + 2 * q^37 + q^39 - z * q^42 - q^46 - 2z 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 + 2z^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} - 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}+ \cdots - 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

0.500000	−	0.866025i
0.500000	+	0.866025i

−0.500000

−

0.866025i

0.500000

0.866025i

−0.500000

0.866025i

0.500000

−

0.866025i

−1.00000

1.00000

293.1

−0.500000

0.866025i

0.500000

−

0.866025i

−0.500000

−

0.866025i

0.500000

0.866025i

−1.00000

1.00000

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.a		2
8.b	even	2	1		2888.1.l.b		2
19.b	odd	2	1		2888.1.l.b		2
19.c	even	3	1		152.1.g.b	yes	1
19.c	even	3	1	inner	2888.1.l.a		2
19.d	odd	6	1		152.1.g.a	✓	1
19.d	odd	6	1		2888.1.l.b		2
19.e	even	9	6		2888.1.s.a		6
19.f	odd	18	6		2888.1.s.b		6
57.f	even	6	1		1368.1.i.b		1
57.h	odd	6	1		1368.1.i.a		1
76.f	even	6	1		608.1.g.a		1
76.g	odd	6	1		608.1.g.b		1
95.h	odd	6	1		3800.1.o.b		1
95.i	even	6	1		3800.1.o.a		1
95.l	even	12	2		3800.1.b.a		2
95.m	odd	12	2		3800.1.b.b		2
152.g	odd	2	1	CM	2888.1.l.a		2
152.k	odd	6	1		608.1.g.a		1
152.l	odd	6	1		152.1.g.b	yes	1
152.l	odd	6	1	inner	2888.1.l.a		2
152.o	even	6	1		608.1.g.b		1
152.p	even	6	1		152.1.g.a	✓	1
152.p	even	6	1		2888.1.l.b		2
152.s	odd	18	6		2888.1.s.a		6
152.t	even	18	6		2888.1.s.b		6
456.v	even	6	1		1368.1.i.a		1
456.x	odd	6	1		1368.1.i.b		1
760.z	even	6	1		3800.1.o.b		1
760.bh	odd	6	1		3800.1.o.a		1
760.bp	even	12	2		3800.1.b.b		2
760.br	odd	12	2		3800.1.b.a		2

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{2} - T_{3} + 1 $ T3^2 - T3 + 1 acting on $S_{1}^{\mathrm{new}}(2888, [\chi])$.

Hecke characteristic polynomials

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

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2888 = 2^{3} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2888.l (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{6}^{2} q^{2} - \zeta_{6}^{2} q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{6} - q^{7} + q^{8} - q^{12} + \zeta_{6} q^{13} - \zeta_{6}^{2} q^{14} + \zeta_{6}^{2} q^{16} - \zeta_{6}^{2} q^{17} + \zeta_{6}^{2} q^{21} + \cdots - q^{96} +O(q^{100}) \) 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 + 2 * q^37 + q^39 - z * q^42 - q^46 - 2z 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 + 2z^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
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + q^{3} - q^{4} + q^{6} - 2 q^{7} + 2 q^{8} - 2 q^{12} + q^{13} + q^{14} - q^{16} + q^{17} - q^{21} + q^{23} + q^{24} - q^{25} - 2 q^{26} + 2 q^{27} + q^{28} + q^{29} - q^{32}+ \cdots - 2 q^{96}+O(q^{100}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2888.1.l.a

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	$4$

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:	$C_3\times S_3$
Artin field:	Galois closure of 6.0.1267762688.3

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

$p$	$F_p(T)$
$2$	\( T^{2} + T + 1 \) T^2 + T + 1
$3$	\( T^{2} - T + 1 \) T^2 - T + 1
$5$	\( T^{2} \) T^2
$7$	\( (T + 1)^{2} \) (T + 1)^2
$11$	\( T^{2} \) T^2
$13$	\( T^{2} - T + 1 \) T^2 - T + 1
$17$	\( T^{2} - T + 1 \) T^2 - T + 1
$19$	\( T^{2} \) T^2
$23$	\( T^{2} - T + 1 \) T^2 - T + 1
$29$	\( T^{2} - T + 1 \) T^2 - T + 1
$31$	\( T^{2} \) T^2
$37$	\( (T - 2)^{2} \) (T - 2)^2
$41$	\( T^{2} \) T^2
$43$	\( T^{2} \) T^2
$47$	\( T^{2} + 2T + 4 \) T^2 + 2*T + 4
$53$	\( T^{2} - T + 1 \) T^2 - T + 1
$59$	\( T^{2} - T + 1 \) T^2 - T + 1
$61$	\( T^{2} \) T^2
$67$	\( T^{2} - T + 1 \) T^2 - T + 1
$71$	\( T^{2} \) T^2
$73$	\( T^{2} - T + 1 \) T^2 - T + 1
$79$	\( T^{2} \) T^2
$83$	\( T^{2} \) T^2
$89$	\( T^{2} \) T^2
$97$	\( T^{2} \) T^2
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\(n\):		e.g. 2-40 or 80-90
Significant digits:
Format: