LMFDB - Newform orbit 1587.1.h.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1587,1,Mod(170,1587)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1587, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 10]))

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

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

chi := DirichletCharacter("1587.170");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$1587 = 3 \cdot 23^{2}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1587.h (of order $22$ , degree $10$ , 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:	$0.792016175049$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$2$ over $\Q(\zeta_{22})$
Coefficient field:	20.0.5969915757478328440239161344.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{20} + 2x^{18} + 4x^{16} + 8x^{14} + 16x^{12} + 32x^{10} + 64x^{8} + 128x^{6} + 256x^{4} + 512x^{2} + 1024$ x^20 + 2x^18 + 4x^16 + 8x^14 + 16x^12 + 32x^10 + 64x^8 + 128x^6 + 256x^4 + 512*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.0.36501.1
Artin image:	$C_{11}\times D_8$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{88} - \cdots)$

$q$ -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 $1,\beta_1,\ldots,\beta_{19}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_{8} q^{3} + \beta_{2} q^{4} - \beta_{15} q^{7} + \beta_{16} q^{9} - \beta_{10} q^{12} + \beta_{4} q^{16} + \beta_{13} q^{19} + \beta_1 q^{21} + \beta_{12} q^{25} - \beta_{2} q^{27} - \beta_{17} q^{28}+ \cdots + \beta_{17} q^{97}+O(q^{100})$ q - b8 * q^3 + b2 * q^4 - b15 * q^7 + b16 * q^9 - b10 * q^12 + b4 * q^16 + b13 * q^19 + b1 * q^21 + b12 * q^25 - b2 * q^27 - b17 * q^28 + b18 * q^36 + b5 * q^37 + b19 * q^43 - b12 * q^48 + b8 * q^49 + (b19 + b17 + b15 + b13 + b11 + b9 + b7 + b5 + b3 + b1) * q^57 - b3 * q^61 - b9 * q^63 + b6 * q^64 - b1 * q^67 + (b18 + b16 + b14 + b12 + b10 + b8 + b6 + b4 + b2 + 1) * q^75 + b15 * q^76 - b7 * q^79 + b10 * q^81 + b3 * q^84 + b17 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$20 q + 2 q^{3} - 2 q^{4} - 2 q^{9} + 2 q^{12} - 2 q^{16} - 2 q^{25} + 2 q^{27} - 2 q^{36} + 2 q^{48} - 2 q^{49} - 2 q^{64} + 2 q^{75} - 2 q^{81}+O(q^{100})$ 20 * q + 2 * q^3 - 2 * q^4 - 2 * q^9 + 2 * q^12 - 2 * q^16 - 2 * q^25 + 2 * q^27 - 2 * q^36 + 2 * q^48 - 2 * q^49 - 2 * q^64 + 2 * q^75 - 2 * q^81

Basis of coefficient ring in terms of a root $\nu$ of $x^{20} + 2x^{18} + 4x^{16} + 8x^{14} + 16x^{12} + 32x^{10} + 64x^{8} + 128x^{6} + 256x^{4} + 512x^{2} + 1024$ x^20 + 2*x^18 + 4*x^16 + 8*x^14 + 16*x^12 + 32*x^10 + 64*x^8 + 128*x^6 + 256*x^4 + 512*x^2 + 1024 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( \nu^{2} ) / 2$ (v^2) / 2
$\beta_{3}$	$=$	$( \nu^{3} ) / 2$ (v^3) / 2
$\beta_{4}$	$=$	$( \nu^{4} ) / 4$ (v^4) / 4
$\beta_{5}$	$=$	$( \nu^{5} ) / 4$ (v^5) / 4
$\beta_{6}$	$=$	$( \nu^{6} ) / 8$ (v^6) / 8
$\beta_{7}$	$=$	$( \nu^{7} ) / 8$ (v^7) / 8
$\beta_{8}$	$=$	$( \nu^{8} ) / 16$ (v^8) / 16
$\beta_{9}$	$=$	$( \nu^{9} ) / 16$ (v^9) / 16
$\beta_{10}$	$=$	$( \nu^{10} ) / 32$ (v^10) / 32
$\beta_{11}$	$=$	$( \nu^{11} ) / 32$ (v^11) / 32
$\beta_{12}$	$=$	$( \nu^{12} ) / 64$ (v^12) / 64
$\beta_{13}$	$=$	$( \nu^{13} ) / 64$ (v^13) / 64
$\beta_{14}$	$=$	$( \nu^{14} ) / 128$ (v^14) / 128
$\beta_{15}$	$=$	$( \nu^{15} ) / 128$ (v^15) / 128
$\beta_{16}$	$=$	$( \nu^{16} ) / 256$ (v^16) / 256
$\beta_{17}$	$=$	$( \nu^{17} ) / 256$ (v^17) / 256
$\beta_{18}$	$=$	$( \nu^{18} ) / 512$ (v^18) / 512
$\beta_{19}$	$=$	$( \nu^{19} ) / 512$ (v^19) / 512

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$2\beta_{2}$ 2*b2
$\nu^{3}$	$=$	$2\beta_{3}$ 2*b3
$\nu^{4}$	$=$	$4\beta_{4}$ 4*b4
$\nu^{5}$	$=$	$4\beta_{5}$ 4*b5
$\nu^{6}$	$=$	$8\beta_{6}$ 8*b6
$\nu^{7}$	$=$	$8\beta_{7}$ 8*b7
$\nu^{8}$	$=$	$16\beta_{8}$ 16*b8
$\nu^{9}$	$=$	$16\beta_{9}$ 16*b9
$\nu^{10}$	$=$	$32\beta_{10}$ 32*b10
$\nu^{11}$	$=$	$32\beta_{11}$ 32*b11
$\nu^{12}$	$=$	$64\beta_{12}$ 64*b12
$\nu^{13}$	$=$	$64\beta_{13}$ 64*b13
$\nu^{14}$	$=$	$128\beta_{14}$ 128*b14
$\nu^{15}$	$=$	$128\beta_{15}$ 128*b15
$\nu^{16}$	$=$	$256\beta_{16}$ 256*b16
$\nu^{17}$	$=$	$256\beta_{17}$ 256*b17
$\nu^{18}$	$=$	$512\beta_{18}$ 512*b18
$\nu^{19}$	$=$	$512\beta_{19}$ 512*b19

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1587\mathbb{Z}\right)^\times$ .

$n$	$530$	$1063$
$\chi(n)$	$-1$	$\beta_{12}$

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}

170.1

−1.35693	+	0.398430i
1.35693	−	0.398430i
0.926113	+	1.06879i
−0.926113	−	1.06879i
−0.201264	−	1.39982i
0.201264	+	1.39982i
−1.18971	+	0.764582i
1.18971	−	0.764582i
−0.587486	+	1.28641i
0.587486	−	1.28641i
−0.587486	−	1.28641i
0.587486	+	1.28641i
0.926113	−	1.06879i
−0.926113	+	1.06879i
−1.18971	−	0.764582i
1.18971	+	0.764582i
−0.201264	+	1.39982i
0.201264	−	1.39982i
−1.35693	−	0.398430i
1.35693	+	0.398430i

0.654861

0.755750i

0.841254

−

0.540641i

−0.587486

1.28641i

−0.142315

0.989821i

170.2

0.654861

0.755750i

0.841254

−

0.540641i

0.587486

−

1.28641i

−0.142315

0.989821i

266.1

−0.841254

−

0.540641i

−0.142315

0.989821i

−1.35693

−

0.398430i

0.415415

0.909632i

266.2

−0.841254

−

0.540641i

−0.142315

0.989821i

1.35693

0.398430i

0.415415

0.909632i

647.1

−0.415415

0.909632i

−0.959493

0.281733i

−1.18971

0.764582i

−0.654861

−

0.755750i

647.2

−0.415415

0.909632i

−0.959493

0.281733i

1.18971

−

0.764582i

−0.654861

−

0.755750i

863.1

0.142315

−

0.989821i

0.415415

−

0.909632i

−0.926113

−

1.06879i

−0.959493

−

0.281733i

863.2

0.142315

−

0.989821i

0.415415

−

0.909632i

0.926113

1.06879i

−0.959493

−

0.281733i

995.1

0.959493

0.281733i

−0.654861

−

0.755750i

−0.201264

1.39982i

0.841254

0.540641i

995.2

0.959493

0.281733i

−0.654861

−

0.755750i

0.201264

−

1.39982i

0.841254

0.540641i

1016.1

0.959493

−

0.281733i

−0.654861

0.755750i

−0.201264

−

1.39982i

0.841254

−

0.540641i

1016.2

0.959493

−

0.281733i

−0.654861

0.755750i

0.201264

1.39982i

0.841254

−

0.540641i

1235.1

−0.841254

0.540641i

−0.142315

−

0.989821i

−1.35693

0.398430i

0.415415

−

0.909632i

1235.2

−0.841254

0.540641i

−0.142315

−

0.989821i

1.35693

−

0.398430i

0.415415

−

0.909632i

1313.1

0.142315

0.989821i

0.415415

0.909632i

−0.926113

1.06879i

−0.959493

0.281733i

1313.2

0.142315

0.989821i

0.415415

0.909632i

0.926113

−

1.06879i

−0.959493

0.281733i

1457.1

−0.415415

−

0.909632i

−0.959493

−

0.281733i

−1.18971

−

0.764582i

−0.654861

0.755750i

1457.2

−0.415415

−

0.909632i

−0.959493

−

0.281733i

1.18971

0.764582i

−0.654861

0.755750i

1559.1

0.654861

−

0.755750i

0.841254

0.540641i

−0.587486

−

1.28641i

−0.142315

−

0.989821i

1559.2

0.654861

−

0.755750i

0.841254

0.540641i

0.587486

1.28641i

−0.142315

−

0.989821i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3})$
23.b	odd	2	1	inner
23.c	even	11	9	inner
23.d	odd	22	9	inner
69.c	even	2	1	inner
69.g	even	22	9	inner
69.h	odd	22	9	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1587.1.h.c		20
3.b	odd	2	1	CM	1587.1.h.c		20
23.b	odd	2	1	inner	1587.1.h.c		20
23.c	even	11	1		1587.1.b.b	✓	2
23.c	even	11	9	inner	1587.1.h.c		20
23.d	odd	22	1		1587.1.b.b	✓	2
23.d	odd	22	9	inner	1587.1.h.c		20
69.c	even	2	1	inner	1587.1.h.c		20
69.g	even	22	1		1587.1.b.b	✓	2
69.g	even	22	9	inner	1587.1.h.c		20
69.h	odd	22	1		1587.1.b.b	✓	2
69.h	odd	22	9	inner	1587.1.h.c		20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1587.1.b.b	✓	2	23.c	even	11	1
1587.1.b.b	✓	2	23.d	odd	22	1
1587.1.b.b	✓	2	69.g	even	22	1
1587.1.b.b	✓	2	69.h	odd	22	1
1587.1.h.c		20	1.a	even	1	1	trivial
1587.1.h.c		20	3.b	odd	2	1	CM
1587.1.h.c		20	23.b	odd	2	1	inner
1587.1.h.c		20	23.c	even	11	9	inner
1587.1.h.c		20	23.d	odd	22	9	inner
1587.1.h.c		20	69.c	even	2	1	inner
1587.1.h.c		20	69.g	even	22	9	inner
1587.1.h.c		20	69.h	odd	22	9	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(1587, [\chi])$ :

T_{2}

T_{7}^{20} + 2 T_{7}^{18} + 4 T_{7}^{16} + 8 T_{7}^{14} + 16 T_{7}^{12} + 32 T_{7}^{10} + 64 T_{7}^{8} + \cdots + 1024

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{20}$ T^20
$3$	$(T^{10} - T^{9} + T^{8} + \cdots + 1)^{2}$ (T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1)^2
$5$	$T^{20}$ T^20
$7$	$T^{20} + 2 T^{18} + \cdots + 1024$ T^20 + 2T^18 + 4T^16 + 8T^14 + 16T^12 + 32T^10 + 64T^8 + 128T^6 + 256T^4 + 512*T^2 + 1024
$11$	$T^{20}$ T^20
$13$	$T^{20}$ T^20
$17$	$T^{20}$ T^20
$19$	$T^{20} + 2 T^{18} + \cdots + 1024$ T^20 + 2T^18 + 4T^16 + 8T^14 + 16T^12 + 32T^10 + 64T^8 + 128T^6 + 256T^4 + 512*T^2 + 1024
$23$	$T^{20}$ T^20
$29$	$T^{20}$ T^20
$31$	$T^{20}$ T^20
$37$	$T^{20} + 2 T^{18} + \cdots + 1024$ T^20 + 2T^18 + 4T^16 + 8T^14 + 16T^12 + 32T^10 + 64T^8 + 128T^6 + 256T^4 + 512*T^2 + 1024
$41$	$T^{20}$ T^20
$43$	$T^{20} + 2 T^{18} + \cdots + 1024$ T^20 + 2T^18 + 4T^16 + 8T^14 + 16T^12 + 32T^10 + 64T^8 + 128T^6 + 256T^4 + 512*T^2 + 1024
$47$	$T^{20}$ T^20
$53$	$T^{20}$ T^20
$59$	$T^{20}$ T^20
$61$	$T^{20} + 2 T^{18} + \cdots + 1024$ T^20 + 2T^18 + 4T^16 + 8T^14 + 16T^12 + 32T^10 + 64T^8 + 128T^6 + 256T^4 + 512*T^2 + 1024
$67$	$T^{20} + 2 T^{18} + \cdots + 1024$ T^20 + 2T^18 + 4T^16 + 8T^14 + 16T^12 + 32T^10 + 64T^8 + 128T^6 + 256T^4 + 512*T^2 + 1024
$71$	$T^{20}$ T^20
$73$	$T^{20}$ T^20
$79$	$T^{20} + 2 T^{18} + \cdots + 1024$ T^20 + 2T^18 + 4T^16 + 8T^14 + 16T^12 + 32T^10 + 64T^8 + 128T^6 + 256T^4 + 512*T^2 + 1024
$83$	$T^{20}$ T^20
$89$	$T^{20}$ T^20
$97$	$T^{20} + 2 T^{18} + \cdots + 1024$ T^20 + 2T^18 + 4T^16 + 8T^14 + 16T^12 + 32T^10 + 64T^8 + 128T^6 + 256T^4 + 512*T^2 + 1024
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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

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 170.2
Significant digits:
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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	1587.1.h.c

Level	$1587$
Weight	$1$
Character orbit	1587.h
Analytic conductor	$0.792$
Analytic rank	$0$
Dimension	$20$
Projective image	$D_{4}$
CM discriminant	-3
Inner twists	$40$