LMFDB - Newform orbit 1449.1.bq.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1449,1,Mod(55,1449)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1449, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 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("1449.55");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1449 = 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1449.bq (of order $22$, degree $10$, 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:	$0.723145203305$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 161)
Projective image:	$D_{11}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{11} - \cdots)$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1449\mathbb{Z}\right)^\times$.

$n$	$442$	$829$	$1289$
$\chi(n)$	$-\zeta_{22}$	$-1$	$1$

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} $

55.1

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

1.61435

−

0.474017i

1.54019

−

0.989821i

0.415415

−

0.909632i

0.915415

−

1.05645i

118.1

−0.273100

−

1.89945i

−2.57385

0.755750i

0.841254

−

0.540641i

1.34125

2.93694i

307.1

−0.273100

1.89945i

−2.57385

−

0.755750i

0.841254

0.540641i

1.34125

−

2.93694i

370.1

−0.698939

−

0.449181i

−0.128663

−

0.281733i

−0.654861

0.755750i

−0.154861

1.07708i

496.1

−0.186393

−

0.215109i

0.130785

−

0.909632i

−0.959493

−

0.281733i

−0.459493

0.295298i

685.1

1.61435

0.474017i

1.54019

0.989821i

0.415415

0.909632i

0.915415

1.05645i

748.1

−0.698939

0.449181i

−0.128663

0.281733i

−0.654861

−

0.755750i

−0.154861

−

1.07708i

811.1

0.544078

−

1.19136i

−0.468468

−

0.540641i

−0.142315

0.989821i

0.357685

−

0.105026i

1189.1

−0.186393

0.215109i

0.130785

0.909632i

−0.959493

0.281733i

−0.459493

−

0.295298i

1315.1

0.544078

1.19136i

−0.468468

0.540641i

−0.142315

−

0.989821i

0.357685

0.105026i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by $\Q(\sqrt{-7}) $
23.c	even	11	1	inner
161.l	odd	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1449.1.bq.a		10
3.b	odd	2	1		161.1.l.a	✓	10
7.b	odd	2	1	CM	1449.1.bq.a		10
12.b	even	2	1		2576.1.cj.a		10
21.c	even	2	1		161.1.l.a	✓	10
21.g	even	6	2		1127.1.v.a		20
21.h	odd	6	2		1127.1.v.a		20
23.c	even	11	1	inner	1449.1.bq.a		10
69.c	even	2	1		3703.1.l.f		10
69.g	even	22	1		3703.1.b.b		5
69.g	even	22	2		3703.1.l.d		10
69.g	even	22	2		3703.1.l.e		10
69.g	even	22	1		3703.1.l.f		10
69.g	even	22	2		3703.1.l.g		10
69.g	even	22	2		3703.1.l.i		10
69.h	odd	22	1		161.1.l.a	✓	10
69.h	odd	22	1		3703.1.b.c		5
69.h	odd	22	2		3703.1.l.a		10
69.h	odd	22	2		3703.1.l.b		10
69.h	odd	22	2		3703.1.l.c		10
69.h	odd	22	2		3703.1.l.h		10
84.h	odd	2	1		2576.1.cj.a		10
161.l	odd	22	1	inner	1449.1.bq.a		10
276.o	even	22	1		2576.1.cj.a		10
483.c	odd	2	1		3703.1.l.f		10
483.v	even	22	1		161.1.l.a	✓	10
483.v	even	22	1		3703.1.b.c		5
483.v	even	22	2		3703.1.l.a		10
483.v	even	22	2		3703.1.l.b		10
483.v	even	22	2		3703.1.l.c		10
483.v	even	22	2		3703.1.l.h		10
483.w	odd	22	1		3703.1.b.b		5
483.w	odd	22	2		3703.1.l.d		10
483.w	odd	22	2		3703.1.l.e		10
483.w	odd	22	1		3703.1.l.f		10
483.w	odd	22	2		3703.1.l.g		10
483.w	odd	22	2		3703.1.l.i		10
483.z	odd	66	2		1127.1.v.a		20
483.bb	even	66	2		1127.1.v.a		20
1932.bi	odd	22	1		2576.1.cj.a		10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
161.1.l.a	✓	10	3.b	odd	2	1
161.1.l.a	✓	10	21.c	even	2	1
161.1.l.a	✓	10	69.h	odd	22	1
161.1.l.a	✓	10	483.v	even	22	1
1127.1.v.a		20	21.g	even	6	2
1127.1.v.a		20	21.h	odd	6	2
1127.1.v.a		20	483.z	odd	66	2
1127.1.v.a		20	483.bb	even	66	2
1449.1.bq.a		10	1.a	even	1	1	trivial
1449.1.bq.a		10	7.b	odd	2	1	CM
1449.1.bq.a		10	23.c	even	11	1	inner
1449.1.bq.a		10	161.l	odd	22	1	inner
2576.1.cj.a		10	12.b	even	2	1
2576.1.cj.a		10	84.h	odd	2	1
2576.1.cj.a		10	276.o	even	22	1
2576.1.cj.a		10	1932.bi	odd	22	1
3703.1.b.b		5	69.g	even	22	1
3703.1.b.b		5	483.w	odd	22	1
3703.1.b.c		5	69.h	odd	22	1
3703.1.b.c		5	483.v	even	22	1
3703.1.l.a		10	69.h	odd	22	2
3703.1.l.a		10	483.v	even	22	2
3703.1.l.b		10	69.h	odd	22	2
3703.1.l.b		10	483.v	even	22	2
3703.1.l.c		10	69.h	odd	22	2
3703.1.l.c		10	483.v	even	22	2
3703.1.l.d		10	69.g	even	22	2
3703.1.l.d		10	483.w	odd	22	2
3703.1.l.e		10	69.g	even	22	2
3703.1.l.e		10	483.w	odd	22	2
3703.1.l.f		10	69.c	even	2	1
3703.1.l.f		10	69.g	even	22	1
3703.1.l.f		10	483.c	odd	2	1
3703.1.l.f		10	483.w	odd	22	1
3703.1.l.g		10	69.g	even	22	2
3703.1.l.g		10	483.w	odd	22	2
3703.1.l.h		10	69.h	odd	22	2
3703.1.l.h		10	483.v	even	22	2
3703.1.l.i		10	69.g	even	22	2
3703.1.l.i		10	483.w	odd	22	2

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{10} - 2T_{2}^{9} + 4T_{2}^{8} - 8T_{2}^{7} + 5T_{2}^{6} + T_{2}^{5} - 2T_{2}^{4} + 4T_{2}^{3} + 14T_{2}^{2} + 5T_{2} + 1 $ T2^10 - 2*T2^9 + 4*T2^8 - 8*T2^7 + 5*T2^6 + T2^5 - 2*T2^4 + 4*T2^3 + 14*T2^2 + 5*T2 + 1 acting on $S_{1}^{\mathrm{new}}(1449, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ 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
$3$	$ T^{10} $ T^10
$5$	$ 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} - 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
$13$	$ T^{10} $ T^10
$17$	$ T^{10} $ T^10
$19$	$ 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 - 10T^5 + 20T^4 - 7T^3 + 3T^2 + 5*T + 1
$31$	$ T^{10} $ T^10
$37$	$ 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
$41$	$ T^{10} $ T^10
$43$	$ 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
$47$	$ T^{10} $ T^10
$53$	$ 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
$59$	$ T^{10} $ T^10
$61$	$ T^{10} $ T^10
$67$	$ 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
$71$	$ 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
$73$	$ T^{10} $ T^10
$79$	$ 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
$83$	$ T^{10} $ T^10
$89$	$ T^{10} $ T^10
$97$	$ T^{10} $ T^10
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1449.bq (of order \(22\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{22}^{10} + \zeta_{22}^{3}) q^{2} + ( - \zeta_{22}^{9} + \cdots + \zeta_{22}^{2}) q^{4}+ \cdots + (\zeta_{22}^{9} - \zeta_{22}^{8} + \cdots - \zeta_{22}) q^{8}+O(q^{10}) \) q + (-z^10 + z^3) * q^2 + (-z^9 + z^6 + z^2) * q^4 + z^4 * q^7 + (z^9 - z^8 + z^5 - z) * q^8	\( q + ( - \zeta_{22}^{10} + \zeta_{22}^{3}) q^{2} + ( - \zeta_{22}^{9} + \cdots + \zeta_{22}^{2}) q^{4}+ \cdots + (\zeta_{22}^{7} - 1) q^{98}+O(q^{100}) \) q + (-z^10 + z^3) * q^2 + (-z^9 + z^6 + z^2) * q^4 + z^4 * q^7 + (z^9 - z^8 + z^5 - z) * q^8 + (z^7 - z^2) * q^11 + (z^7 + z^3) * q^14 + (z^8 - z^7 + z^4 - z - 1) * q^16 + (z^10 + z^6 - z^5 - z) * q^22 - z^6 * q^23 - z * q^25 + (z^10 + z^6 + z^2) * q^28 + (-z^10 - z^8) * q^29 + (-z^10 + z^7 - z^6 - z^4 + z^3 + 1) * q^32 + (-z^9 - z) * q^37 + (-z^3 + z^2) * q^43 + (z^9 - z^8 + z^5 - z^4 - z^2 - 1) * q^44 + (-z^9 - z^5) * q^46 + z^8 * q^49 + (-z^4 - 1) * q^50 + (z^5 + z^3) * q^53 + (z^9 + z^5 - z^2 + z) * q^56 + (-z^9 - z^7 + z^2 + 1) * q^58 + (z^10 - z^9 - z^7 + z^6 - z^5 - z^3 + z^2) * q^64 + (z^10 - z^3) * q^67 + (z^9 - z^4) * q^71 + (-z^8 - z^4 + z - 1) * q^74 + (-z^6 - 1) * q^77 + (z^8 + z^6) * q^79 + (-z^6 + z^5 - z^2 + z) * q^86 + (z^10 + z^8 - z^7 - z^5 + z^4 - z^3 - z + 1) * q^88 + (-z^8 - z^4 + z) * q^92 + (z^7 - 1) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{2} - 3 q^{4} - q^{7} + 4 q^{8}+O(q^{10}) \) 10 * q + 2 * q^2 - 3 * q^4 - q^7 + 4 * q^8	\( 10 q + 2 q^{2} - 3 q^{4} - q^{7} + 4 q^{8} + 2 q^{11} + 2 q^{14} + 6 q^{16} - 4 q^{22} + q^{23} - q^{25} - 3 q^{28} + 2 q^{29} - 5 q^{32} - 2 q^{37} - 2 q^{43} - 5 q^{44} - 2 q^{46} - q^{49} - 9 q^{50} + 2 q^{53} + 4 q^{56} + 7 q^{58} - 7 q^{64} - 2 q^{67} + 2 q^{71} - 7 q^{74} - 9 q^{77} - 2 q^{79} + 4 q^{86} + 3 q^{88} + 3 q^{92} - 9 q^{98}+O(q^{100}) \) 10 * q + 2 * q^2 - 3 * q^4 - q^7 + 4 * q^8 + 2 * q^11 + 2 * q^14 + 6 * q^16 - 4 * q^22 + q^23 - q^25 - 3 * q^28 + 2 * q^29 - 5 * q^32 - 2 * q^37 - 2 * q^43 - 5 * q^44 - 2 * q^46 - q^49 - 9 * q^50 + 2 * q^53 + 4 * q^56 + 7 * q^58 - 7 * q^64 - 2 * q^67 + 2 * q^71 - 7 * q^74 - 9 * q^77 - 2 * q^79 + 4 * q^86 + 3 * q^88 + 3 * q^92 - 9 * q^98

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	1449.1.bq.a

Level	$1449$
Weight	$1$
Character orbit	1449.bq
Analytic conductor	$0.723$
Analytic rank	$0$
Dimension	$10$
Projective image	$D_{11}$
CM discriminant	-7
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.723145203305\)
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:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 161)
Projective image:	\(D_{11}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{11} - \cdots)\)

\(n\)	\(442\)	\(829\)	\(1289\)
\(\chi(n)\)	\(-\zeta_{22}\)	\(-1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 55.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)
23.c	even	11	1	inner
161.l	odd	22	1	inner

$p$	$F_p(T)$
$2$	\( T^{10} - 2 T^{9} + \cdots + 1 \) T^10 - 2T^9 + 4T^8 - 8T^7 + 5T^6 + T^5 - 2T^4 + 4T^3 + 14T^2 + 5T + 1
$3$	\( T^{10} \) T^10
$5$	\( T^{10} \) 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
$11$	\( T^{10} - 2 T^{9} + \cdots + 1 \) T^10 - 2T^9 + 4T^8 + 3T^7 - 6T^6 + 12T^5 + 9T^4 - 7T^3 + 14T^2 - 6*T + 1
$13$	\( T^{10} \) T^10
$17$	\( T^{10} \) T^10
$19$	\( T^{10} \) 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 - 10T^5 + 20T^4 - 7T^3 + 3T^2 + 5*T + 1
$31$	\( T^{10} \) T^10
$37$	\( T^{10} + 2 T^{9} + \cdots + 1 \) T^10 + 2T^9 + 4T^8 - 3T^7 - 6T^6 - 12T^5 + 9T^4 + 7T^3 + 14T^2 + 6*T + 1
$41$	\( T^{10} \) T^10
$43$	\( T^{10} + 2 T^{9} + \cdots + 1 \) T^10 + 2T^9 + 4T^8 + 8T^7 + 5T^6 - T^5 - 2T^4 - 4T^3 + 14T^2 - 5T + 1
$47$	\( T^{10} \) T^10
$53$	\( T^{10} - 2 T^{9} + \cdots + 1 \) T^10 - 2T^9 + 4T^8 + 3T^7 - 6T^6 + 12T^5 + 9T^4 - 7T^3 + 14T^2 - 6*T + 1
$59$	\( T^{10} \) T^10
$61$	\( T^{10} \) T^10
$67$	\( T^{10} + 2 T^{9} + \cdots + 1 \) T^10 + 2T^9 + 4T^8 + 8T^7 + 5T^6 - T^5 - 2T^4 - 4T^3 + 14T^2 - 5T + 1
$71$	\( T^{10} - 2 T^{9} + \cdots + 1 \) T^10 - 2T^9 + 4T^8 + 3T^7 - 6T^6 + 12T^5 + 9T^4 - 7T^3 + 14T^2 - 6*T + 1
$73$	\( T^{10} \) T^10
$79$	\( T^{10} + 2 T^{9} + \cdots + 1 \) T^10 + 2T^9 + 4T^8 - 3T^7 - 6T^6 - 12T^5 + 9T^4 + 7T^3 + 14T^2 + 6*T + 1
$83$	\( T^{10} \) T^10
$89$	\( T^{10} \) T^10
$97$	\( T^{10} \) T^10
show more
show less