LMFDB - Newform orbit 759.1.r.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [759,1,Mod(65,759)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(759, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 11, 15]))

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

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

chi := DirichletCharacter("759.65");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 759 = 3 \cdot 11 \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	759.r (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:	$0.378790344588$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{22}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{22} - \cdots)$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/759\mathbb{Z}\right)^\times$.

$n$	$166$	$254$	$277$
$\chi(n)$	$\zeta_{22}$	$-1$	$-1$

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} $

65.1

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

−0.415415

−

0.909632i

0.654861

−

0.755750i

−0.698939

−

0.449181i

−0.654861

0.755750i

263.1

0.654861

0.755750i

0.142315

−

0.989821i

0.544078

−

1.19136i

−0.142315

0.989821i

296.1

0.142315

0.989821i

0.959493

−

0.281733i

−0.186393

0.215109i

−0.959493

0.281733i

329.1

0.654861

−

0.755750i

0.142315

0.989821i

0.544078

1.19136i

−0.142315

−

0.989821i

362.1

−0.415415

0.909632i

0.654861

0.755750i

−0.698939

0.449181i

−0.654861

−

0.755750i

428.1

0.959493

−

0.281733i

−0.841254

0.540641i

−0.273100

−

1.89945i

0.841254

−

0.540641i

494.1

−0.841254

0.540641i

−0.415415

0.909632i

1.61435

−

0.474017i

0.415415

−

0.909632i

527.1

−0.841254

−

0.540641i

−0.415415

−

0.909632i

1.61435

0.474017i

0.415415

0.909632i

626.1

0.959493

0.281733i

−0.841254

−

0.540641i

−0.273100

1.89945i

0.841254

0.540641i

659.1

0.142315

−

0.989821i

0.959493

0.281733i

−0.186393

−

0.215109i

−0.959493

−

0.281733i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	CM by $\Q(\sqrt{-11}) $
69.g	even	22	1	inner
759.r	odd	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	759.1.r.b	yes	10
3.b	odd	2	1		759.1.r.a	✓	10
11.b	odd	2	1	CM	759.1.r.b	yes	10
23.d	odd	22	1		759.1.r.a	✓	10
33.d	even	2	1		759.1.r.a	✓	10
69.g	even	22	1	inner	759.1.r.b	yes	10
253.l	even	22	1		759.1.r.a	✓	10
759.r	odd	22	1	inner	759.1.r.b	yes	10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
759.1.r.a	✓	10	3.b	odd	2	1
759.1.r.a	✓	10	23.d	odd	22	1
759.1.r.a	✓	10	33.d	even	2	1
759.1.r.a	✓	10	253.l	even	22	1
759.1.r.b	yes	10	1.a	even	1	1	trivial
759.1.r.b	yes	10	11.b	odd	2	1	CM
759.1.r.b	yes	10	69.g	even	22	1	inner
759.1.r.b	yes	10	759.r	odd	22	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{10} - 2T_{5}^{9} + 4T_{5}^{8} - 8T_{5}^{7} + 5T_{5}^{6} + T_{5}^{5} - 2T_{5}^{4} + 4T_{5}^{3} + 14T_{5}^{2} + 5T_{5} + 1 $ T5^10 - 2*T5^9 + 4*T5^8 - 8*T5^7 + 5*T5^6 + T5^5 - 2*T5^4 + 4*T5^3 + 14*T5^2 + 5*T5 + 1 acting on $S_{1}^{\mathrm{new}}(759, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{10} $ T^10
$3$	$ 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} - 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
$7$	$ T^{10} $ T^10
$11$	$ 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
$13$	$ 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} $ T^10
$31$	$ 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
$37$	$ T^{10} + 11 T^{4} + \cdots + 11 $ T^10 + 11T^4 - 55T^3 + 77T^2 - 44T + 11
$41$	$ T^{10} $ T^10
$43$	$ T^{10} $ T^10
$47$	$ T^{10} + 11 T^{8} + \cdots + 11 $ T^10 + 11T^8 + 44T^6 + 77T^4 + 55T^2 + 11
$53$	$ 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} - 22 T^{5} + \cdots + 11 $ T^10 - 22T^5 + 33T^3 + 11T^2 - 11T + 11
$61$	$ T^{10} $ T^10
$67$	$ T^{10} + 11 T^{4} + \cdots + 11 $ T^10 + 11T^4 + 55T^3 + 77T^2 + 44T + 11
$71$	$ T^{10} - 11 T^{9} + \cdots + 11 $ T^10 - 11T^9 + 55T^8 - 165T^7 + 330T^6 - 462T^5 + 462T^4 - 330T^3 + 165T^2 - 55*T + 11
$73$	$ T^{10} $ T^10
$79$	$ T^{10} $ T^10
$83$	$ T^{10} $ T^10
$89$	$ 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
$97$	$ T^{10} + 11 T^{9} + \cdots + 11 $ T^10 + 11T^9 + 55T^8 + 165T^7 + 330T^6 + 462T^5 + 462T^4 + 330T^3 + 165T^2 + 55*T + 11
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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:	\( N \)	\(=\)	\( 759 = 3 \cdot 11 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	759.r (of order \(22\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{22} q^{3} - \zeta_{22}^{2} q^{4} + (\zeta_{22}^{7} + \zeta_{22}^{5}) q^{5} + \zeta_{22}^{2} q^{9} +O(q^{10}) \) q + z * q^3 - z^2 * q^4 + (z^7 + z^5) * q^5 + z^2 * q^9	\( q + \zeta_{22} q^{3} - \zeta_{22}^{2} q^{4} + (\zeta_{22}^{7} + \zeta_{22}^{5}) q^{5} + \zeta_{22}^{2} q^{9} + \zeta_{22}^{10} q^{11} - \zeta_{22}^{3} q^{12} + (\zeta_{22}^{8} + \zeta_{22}^{6}) q^{15} + \zeta_{22}^{4} q^{16} + ( - \zeta_{22}^{9} - \zeta_{22}^{7}) q^{20} - \zeta_{22}^{10} q^{23} + (\zeta_{22}^{10} - \zeta_{22}^{3} - \zeta_{22}) q^{25} + \zeta_{22}^{3} q^{27} + ( - \zeta_{22}^{9} + \zeta_{22}^{8}) q^{31} - q^{33} - \zeta_{22}^{4} q^{36} + ( - \zeta_{22}^{7} + \zeta_{22}^{3}) q^{37} + \zeta_{22} q^{44} + (\zeta_{22}^{9} + \zeta_{22}^{7}) q^{45} + ( - \zeta_{22}^{6} - \zeta_{22}^{5}) q^{47} + \zeta_{22}^{5} q^{48} - \zeta_{22}^{8} q^{49} + (\zeta_{22}^{6} + \zeta_{22}^{2}) q^{53} + ( - \zeta_{22}^{6} - \zeta_{22}^{4}) q^{55} + (\zeta_{22}^{9} - \zeta_{22}^{5}) q^{59} + ( - \zeta_{22}^{10} - \zeta_{22}^{8}) q^{60} - \zeta_{22}^{6} q^{64} + ( - \zeta_{22}^{8} - \zeta_{22}^{5}) q^{67} + q^{69} + ( - \zeta_{22}^{2} + 1) q^{71} + ( - \zeta_{22}^{4} - \zeta_{22}^{2} - 1) q^{75} + (\zeta_{22}^{9} - 1) q^{80} + \zeta_{22}^{4} q^{81} + (\zeta_{22}^{4} - \zeta_{22}) q^{89} - \zeta_{22} q^{92} + ( - \zeta_{22}^{10} + \zeta_{22}^{9}) q^{93} + ( - \zeta_{22} - 1) q^{97} - \zeta_{22} q^{99} +O(q^{100}) \) q + z * q^3 - z^2 * q^4 + (z^7 + z^5) * q^5 + z^2 * q^9 + z^10 * q^11 - z^3 * q^12 + (z^8 + z^6) * q^15 + z^4 * q^16 + (-z^9 - z^7) * q^20 - z^10 * q^23 + (z^10 - z^3 - z) * q^25 + z^3 * q^27 + (-z^9 + z^8) * q^31 - q^33 - z^4 * q^36 + (-z^7 + z^3) * q^37 + z * q^44 + (z^9 + z^7) * q^45 + (-z^6 - z^5) * q^47 + z^5 * q^48 - z^8 * q^49 + (z^6 + z^2) * q^53 + (-z^6 - z^4) * q^55 + (z^9 - z^5) * q^59 + (-z^10 - z^8) * q^60 - z^6 * q^64 + (-z^8 - z^5) * q^67 + q^69 + (-z^2 + 1) * q^71 + (-z^4 - z^2 - 1) * q^75 + (z^9 - 1) * q^80 + z^4 * q^81 + (z^4 - z) * q^89 - z * q^92 + (-z^10 + z^9) * q^93 + (-z - 1) * q^97 - z * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{3} + q^{4} + 2 q^{5} - q^{9}+O(q^{10}) \) 10 * q + q^3 + q^4 + 2 * q^5 - q^9	\( 10 q + q^{3} + q^{4} + 2 q^{5} - q^{9} - q^{11} - q^{12} - 2 q^{15} - q^{16} - 2 q^{20} + q^{23} - 3 q^{25} + q^{27} - 2 q^{31} - 10 q^{33} + q^{36} + q^{44} + 2 q^{45} + q^{48} + q^{49} - 2 q^{53} + 2 q^{55} + 2 q^{60} + q^{64} + 10 q^{69} + 11 q^{71} - 8 q^{75} - 9 q^{80} - q^{81} - 2 q^{89} - q^{92} + 2 q^{93} - 11 q^{97} - q^{99}+O(q^{100}) \) 10 * q + q^3 + q^4 + 2 * q^5 - q^9 - q^11 - q^12 - 2 * q^15 - q^16 - 2 * q^20 + q^23 - 3 * q^25 + q^27 - 2 * q^31 - 10 * q^33 + q^36 + q^44 + 2 * q^45 + q^48 + q^49 - 2 * q^53 + 2 * q^55 + 2 * q^60 + q^64 + 10 * q^69 + 11 * q^71 - 8 * q^75 - 9 * q^80 - q^81 - 2 * q^89 - q^92 + 2 * q^93 - 11 * q^97 - q^99

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	759.1.r.b

Level	$759$
Weight	$1$
Character orbit	759.r
Analytic conductor	$0.379$
Analytic rank	$0$
Dimension	$10$
Projective image	$D_{22}$
CM discriminant	-11
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.378790344588\)
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:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{22}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{22} - \cdots)\)

\(n\)	\(166\)	\(254\)	\(277\)
\(\chi(n)\)	\(\zeta_{22}\)	\(-1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 65.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{10} \) T^10
$3$	\( T^{10} - T^{9} + \cdots + 1 \) T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$5$	\( T^{10} - 2 T^{9} + \cdots + 1 \) T^10 - 2T^9 + 4T^8 - 8T^7 + 5T^6 + T^5 - 2T^4 + 4T^3 + 14T^2 + 5T + 1
$7$	\( T^{10} \) T^10
$11$	\( T^{10} + T^{9} + \cdots + 1 \) T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$13$	\( T^{10} \) 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
$29$	\( T^{10} \) T^10
$31$	\( T^{10} + 2 T^{9} + \cdots + 1 \) T^10 + 2T^9 + 4T^8 + 8T^7 + 5T^6 - T^5 - 2T^4 - 4T^3 + 14T^2 - 5T + 1
$37$	\( T^{10} + 11 T^{4} + \cdots + 11 \) T^10 + 11T^4 - 55T^3 + 77T^2 - 44T + 11
$41$	\( T^{10} \) T^10
$43$	\( T^{10} \) T^10
$47$	\( T^{10} + 11 T^{8} + \cdots + 11 \) T^10 + 11T^8 + 44T^6 + 77T^4 + 55T^2 + 11
$53$	\( T^{10} + 2 T^{9} + \cdots + 1 \) T^10 + 2T^9 + 4T^8 + 8T^7 + 16T^6 + 10T^5 + 20T^4 + 7T^3 + 3T^2 - 5*T + 1
$59$	\( T^{10} - 22 T^{5} + \cdots + 11 \) T^10 - 22T^5 + 33T^3 + 11T^2 - 11T + 11
$61$	\( T^{10} \) T^10
$67$	\( T^{10} + 11 T^{4} + \cdots + 11 \) T^10 + 11T^4 + 55T^3 + 77T^2 + 44T + 11
$71$	\( T^{10} - 11 T^{9} + \cdots + 11 \) T^10 - 11T^9 + 55T^8 - 165T^7 + 330T^6 - 462T^5 + 462T^4 - 330T^3 + 165T^2 - 55*T + 11
$73$	\( T^{10} \) T^10
$79$	\( T^{10} \) T^10
$83$	\( T^{10} \) T^10
$89$	\( T^{10} + 2 T^{9} + \cdots + 1 \) T^10 + 2T^9 + 4T^8 + 8T^7 + 16T^6 + 10T^5 + 20T^4 + 7T^3 + 3T^2 - 5*T + 1
$97$	\( T^{10} + 11 T^{9} + \cdots + 11 \) T^10 + 11T^9 + 55T^8 + 165T^7 + 330T^6 + 462T^5 + 462T^4 + 330T^3 + 165T^2 + 55*T + 11
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