LMFDB - Newform orbit 3864.1.dl.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3864,1,Mod(587,3864)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3864, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([11, 11, 11, 11, 20])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3864.587"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$N$	$=$	$3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	3864.dl (of order $22$ , degree $10$ , minimal)

Newform invariants

comment:select newform

sage:traces = [10,1,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.92838720881$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\Q(\zeta_{22})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{11}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{11} - \cdots)$

$q$ -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.

$f(q)$	$=$	$q + \zeta_{22}^{5} q^{2} + \zeta_{22}^{7} q^{3} + \zeta_{22}^{10} q^{4} - \zeta_{22} q^{6} - \zeta_{22}^{9} q^{7} - \zeta_{22}^{4} q^{8} - \zeta_{22}^{3} q^{9} - \zeta_{22}^{6} q^{12} + (\zeta_{22}^{8} - \zeta_{22}^{7}) q^{13} + \cdots + \zeta_{22} q^{98} +O(q^{100})$ q + z^5 * q^2 + z^7 * q^3 + z^10 * q^4 - z * q^6 - z^9 * q^7 - z^4 * q^8 - z^3 * q^9 - z^6 * q^12 + (z^8 - z^7) * q^13 + z^3 * q^14 - z^9 * q^16 + (z^9 - z^4) * q^17 - z^8 * q^18 + z^5 * q^21 - z^4 * q^23 + q^24 - z^5 * q^25 + (-z^2 + z) * q^26 - z^10 * q^27 + z^8 * q^28 + (-z^8 + z^5) * q^29 + (-z^5 - z^3) * q^31 + z^3 * q^32 + (-z^9 - z^3) * q^34 + z^2 * q^36 + (-z^4 + z^3) * q^39 + (z^5 - 1) * q^41 + z^10 * q^42 + (-z^3 + 1) * q^43 - z^9 * q^46 + z^5 * q^48 - z^7 * q^49 - z^10 * q^50 + (-z^5 + 1) * q^51 + (-z^7 + z^6) * q^52 + (-z^10 - z^8) * q^53 + z^4 * q^54 - z^2 * q^56 + (z^10 + z^2) * q^58 + (-z^4 - 1) * q^59 + (z^6 + z^2) * q^61 + (-z^10 - z^8) * q^62 - z * q^63 + z^8 * q^64 + (-z^7 - z^3) * q^67 + (-z^8 + z^3) * q^68 + q^69 + (z^7 + z^3) * q^71 + z^7 * q^72 + z * q^75 + (-z^9 + z^8) * q^78 + z^6 * q^81 + (z^10 - z^5) * q^82 + (z^9 - z^8) * q^83 - z^4 * q^84 + (-z^8 + z^5) * q^86 + (z^4 - z) * q^87 + (-z^8 - z^6) * q^89 + (z^6 - z^5) * q^91 + z^3 * q^92 + (-z^10 + z) * q^93 + z^10 * q^96 + z * q^98
$\operatorname{Tr}(f)(q)$	$=$	$10 q + q^{2} + q^{3} - q^{4} - q^{6} - q^{7} + q^{8} - q^{9} + q^{12} - 2 q^{13} + q^{14} - q^{16} + 2 q^{17} + q^{18} + q^{21} + q^{23} + 10 q^{24} - q^{25} + 2 q^{26} + q^{27} - q^{28}+ \cdots + q^{98}+O(q^{100})$ 10 * q + q^2 + q^3 - q^4 - q^6 - q^7 + q^8 - q^9 + q^12 - 2 * q^13 + q^14 - q^16 + 2 * q^17 + q^18 + q^21 + q^23 + 10 * q^24 - q^25 + 2 * q^26 + q^27 - q^28 + 2 * q^29 - 2 * q^31 + q^32 - 2 * q^34 - q^36 + 2 * q^39 - 9 * q^41 - q^42 + 9 * q^43 - q^46 + q^48 - q^49 + q^50 + 9 * q^51 - 2 * q^52 + 2 * q^53 - q^54 + q^56 - 2 * q^58 - 9 * q^59 - 2 * q^61 + 2 * q^62 - q^63 - q^64 - 2 * q^67 + 2 * q^68 + 10 * q^69 + 2 * q^71 + q^72 + q^75 - 2 * q^78 - q^81 - 2 * q^82 + 2 * q^83 + q^84 + 2 * q^86 - 2 * q^87 + 2 * q^89 - 2 * q^91 + q^92 + 2 * q^93 - q^96 + q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3864\mathbb{Z}\right)^\times$ .

$n$	$967$	$1289$	$1933$	$2761$	$2857$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$	$-\zeta_{22}^{5}$

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}

587.1

−0.415415	−	0.909632i
0.959493	+	0.281733i
0.959493	−	0.281733i
0.654861	−	0.755750i
0.142315	+	0.989821i
0.654861	+	0.755750i
−0.841254	+	0.540641i
−0.415415	+	0.909632i
−0.841254	−	0.540641i
0.142315	−	0.989821i

−0.841254

0.540641i

0.142315

−

0.989821i

0.415415

−

0.909632i

0.415415

0.909632i

−0.654861

−

0.755750i

0.142315

0.989821i

−0.959493

−

0.281733i

923.1

0.142315

0.989821i

−0.415415

0.909632i

−0.959493

0.281733i

−0.959493

−

0.281733i

0.841254

−

0.540641i

−0.415415

−

0.909632i

−0.654861

−

0.755750i

1595.1

0.142315

−

0.989821i

−0.415415

−

0.909632i

−0.959493

−

0.281733i

−0.959493

0.281733i

0.841254

0.540641i

−0.415415

0.909632i

−0.654861

0.755750i

2099.1

−0.415415

0.909632i

0.959493

0.281733i

−0.654861

−

0.755750i

−0.654861

0.755750i

−0.142315

0.989821i

0.959493

−

0.281733i

0.841254

0.540641i

2267.1

0.654861

0.755750i

−0.841254

−

0.540641i

−0.142315

0.989821i

−0.142315

−

0.989821i

−0.959493

−

0.281733i

−0.841254

0.540641i

0.415415

0.909632i

2603.1

−0.415415

−

0.909632i

0.959493

−

0.281733i

−0.654861

0.755750i

−0.654861

−

0.755750i

−0.142315

−

0.989821i

0.959493

0.281733i

0.841254

−

0.540641i

2939.1

0.959493

0.281733i

0.654861

−

0.755750i

0.841254

0.540641i

0.841254

−

0.540641i

0.415415

0.909632i

0.654861

0.755750i

−0.142315

−

0.989821i

3107.1

−0.841254

−

0.540641i

0.142315

0.989821i

0.415415

0.909632i

0.415415

−

0.909632i

−0.654861

0.755750i

0.142315

−

0.989821i

−0.959493

0.281733i

3275.1

0.959493

−

0.281733i

0.654861

0.755750i

0.841254

−

0.540641i

0.841254

0.540641i

0.415415

−

0.909632i

0.654861

−

0.755750i

−0.142315

0.989821i

3443.1

0.654861

−

0.755750i

−0.841254

0.540641i

−0.142315

−

0.989821i

−0.142315

0.989821i

−0.959493

0.281733i

−0.841254

−

0.540641i

0.415415

−

0.909632i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
168.e	odd	2	1	CM by $\Q(\sqrt{-42})$
23.c	even	11	1	inner
3864.dl	odd	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3864.1.dl.d	yes	10
3.b	odd	2	1		3864.1.dl.a	✓	10
7.b	odd	2	1		3864.1.dl.c	yes	10
8.d	odd	2	1		3864.1.dl.b	yes	10
21.c	even	2	1		3864.1.dl.b	yes	10
23.c	even	11	1	inner	3864.1.dl.d	yes	10
24.f	even	2	1		3864.1.dl.c	yes	10
56.e	even	2	1		3864.1.dl.a	✓	10
69.h	odd	22	1		3864.1.dl.a	✓	10
161.l	odd	22	1		3864.1.dl.c	yes	10
168.e	odd	2	1	CM	3864.1.dl.d	yes	10
184.k	odd	22	1		3864.1.dl.b	yes	10
483.v	even	22	1		3864.1.dl.b	yes	10
552.x	even	22	1		3864.1.dl.c	yes	10
1288.bo	even	22	1		3864.1.dl.a	✓	10
3864.dl	odd	22	1	inner	3864.1.dl.d	yes	10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3864.1.dl.a	✓	10	3.b	odd	2	1
3864.1.dl.a	✓	10	56.e	even	2	1
3864.1.dl.a	✓	10	69.h	odd	22	1
3864.1.dl.a	✓	10	1288.bo	even	22	1
3864.1.dl.b	yes	10	8.d	odd	2	1
3864.1.dl.b	yes	10	21.c	even	2	1
3864.1.dl.b	yes	10	184.k	odd	22	1
3864.1.dl.b	yes	10	483.v	even	22	1
3864.1.dl.c	yes	10	7.b	odd	2	1
3864.1.dl.c	yes	10	24.f	even	2	1
3864.1.dl.c	yes	10	161.l	odd	22	1
3864.1.dl.c	yes	10	552.x	even	22	1
3864.1.dl.d	yes	10	1.a	even	1	1	trivial
3864.1.dl.d	yes	10	23.c	even	11	1	inner
3864.1.dl.d	yes	10	168.e	odd	2	1	CM
3864.1.dl.d	yes	10	3864.dl	odd	22	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(3864, [\chi])$ :

T_{13}^{10} + 2T_{13}^{9} + 4T_{13}^{8} - 3T_{13}^{7} - 6T_{13}^{6} - 12T_{13}^{5} + 9T_{13}^{4} + 7T_{13}^{3} + 14T_{13}^{2} + 6T_{13} + 1

T13^10 + 2*T13^9 + 4*T13^8 - 3*T13^7 - 6*T13^6 - 12*T13^5 + 9*T13^4 + 7*T13^3 + 14*T13^2 + 6*T13 + 1

T_{17}^{10} - 2T_{17}^{9} + 4T_{17}^{8} + 3T_{17}^{7} - 6T_{17}^{6} + 12T_{17}^{5} + 9T_{17}^{4} - 7T_{17}^{3} + 14T_{17}^{2} - 6T_{17} + 1

T17^10 - 2*T17^9 + 4*T17^8 + 3*T17^7 - 6*T17^6 + 12*T17^5 + 9*T17^4 - 7*T17^3 + 14*T17^2 - 6*T17 + 1

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{10} - T^{9} + \cdots + 1$ T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$3$	$T^{10} - T^{9} + \cdots + 1$ T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$5$	$T^{10}$ T^10
$7$	$T^{10} + T^{9} + \cdots + 1$ T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$11$	$T^{10}$ T^10
$13$	$T^{10} + 2 T^{9} + \cdots + 1$ T^10 + 2T^9 + 4T^8 - 3T^7 - 6T^6 - 12T^5 + 9T^4 + 7T^3 + 14T^2 + 6*T + 1
$17$	$T^{10} - 2 T^{9} + \cdots + 1$ T^10 - 2T^9 + 4T^8 + 3T^7 - 6T^6 + 12T^5 + 9T^4 - 7T^3 + 14T^2 - 6*T + 1
$19$	$T^{10}$ T^10
$23$	$T^{10} - T^{9} + \cdots + 1$ T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$29$	$T^{10} - 2 T^{9} + \cdots + 1$ T^10 - 2T^9 + 4T^8 - 8T^7 + 16T^6 - 32T^5 + 53T^4 - 51T^3 + 25T^2 - 6*T + 1
$31$	$T^{10} + 2 T^{9} + \cdots + 1$ T^10 + 2T^9 + 4T^8 - 3T^7 - 6T^6 - 12T^5 + 9T^4 + 7T^3 + 14T^2 + 6*T + 1
$37$	$T^{10}$ T^10
$41$	$T^{10} + 9 T^{9} + \cdots + 1$ T^10 + 9T^9 + 37T^8 + 91T^7 + 148T^6 + 166T^5 + 130T^4 + 70T^3 + 25T^2 + 5*T + 1
$43$	$T^{10} - 9 T^{9} + \cdots + 1$ T^10 - 9T^9 + 37T^8 - 91T^7 + 148T^6 - 166T^5 + 130T^4 - 70T^3 + 25T^2 - 5*T + 1
$47$	$T^{10}$ T^10
$53$	$T^{10} - 2 T^{9} + \cdots + 1$ T^10 - 2T^9 + 4T^8 - 8T^7 + 16T^6 - 10T^5 + 20T^4 - 7T^3 + 3T^2 + 5*T + 1
$59$	$T^{10} + 9 T^{9} + \cdots + 1$ T^10 + 9T^9 + 37T^8 + 91T^7 + 148T^6 + 166T^5 + 130T^4 + 70T^3 + 25T^2 + 5*T + 1
$61$	$T^{10} + 2 T^{9} + \cdots + 1$ T^10 + 2T^9 + 4T^8 + 8T^7 + 16T^6 + 10T^5 + 20T^4 + 7T^3 + 3T^2 - 5*T + 1
$67$	$T^{10} + 2 T^{9} + \cdots + 1$ T^10 + 2T^9 + 4T^8 + 8T^7 + 16T^6 + 32T^5 + 53T^4 + 51T^3 + 25T^2 + 6*T + 1
$71$	$T^{10} - 2 T^{9} + \cdots + 1$ T^10 - 2T^9 + 4T^8 - 8T^7 + 16T^6 - 32T^5 + 53T^4 - 51T^3 + 25T^2 - 6*T + 1
$73$	$T^{10}$ T^10
$79$	$T^{10}$ T^10
$83$	$T^{10} - 2 T^{9} + \cdots + 1$ T^10 - 2T^9 + 4T^8 - 8T^7 + 5T^6 + T^5 - 2T^4 + 4T^3 + 14T^2 + 5T + 1
$89$	$T^{10} - 2 T^{9} + \cdots + 1$ T^10 - 2T^9 + 4T^8 + 3T^7 - 6T^6 + 12T^5 + 9T^4 - 7T^3 + 14T^2 - 6*T + 1
$97$	$T^{10}$ T^10
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 587.1
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	3864.1.dl.d

Level	$3864$
Weight	$1$
Character orbit	3864.dl
Analytic conductor	$1.928$
Analytic rank	$0$
Dimension	$10$
Projective image	$D_{11}$
CM discriminant	-168
Inner twists	$4$