LMFDB - Newform orbit 1573.1.s.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1573,1,Mod(148,1573)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1573, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([8, 15])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1573.148"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 1573 = 11^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1573.s (of order $20$, degree $8$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,2] 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:	$0.785029264872$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{20})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$S_{4}$
Projective field:	Galois closure of 4.0.265837.1

$q$-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_{20}^{7} + \zeta_{20}^{2}) q^{2} - \zeta_{20}^{6} q^{3} + \zeta_{20}^{9} q^{4} + ( - \zeta_{20}^{8} + \zeta_{20}^{3}) q^{6} + \zeta_{20}^{5} q^{12} + \zeta_{20}^{7} q^{13} + \zeta_{20}^{8} q^{16} + \cdots + ( - \zeta_{20}^{5} + 1) q^{98} +O(q^{100}) $ q + (z^7 + z^2) * q^2 - z^6 * q^3 + z^9 * q^4 + (-z^8 + z^3) * q^6 + z^5 * q^12 + z^7 * q^13 + z^8 * q^16 + z^3 * q^17 + (-z^6 + z) * q^19 - z^5 * q^23 - z * q^25 + (z^9 - z^4) * q^26 - z^8 * q^27 + z^4 * q^29 + (-z^5 - 1) * q^32 + (z^5 - 1) * q^34 + (-z^9 + z^4) * q^37 + 2z^3 q^38 + z^3 * q^39 - z^5 * q^43 + (-z^7 + z^2) * q^46 + (z^6 + z) * q^47 + z^4 * q^48 - z^3 * q^49 + (-z^8 - z^3) * q^50 - z^9 * q^51 - z^6 * q^52 - z^2 * q^53 + (z^5 + 1) * q^54 + (-z^7 - z^2) * q^57 + (z^6 - z) * q^58 + z^8 * q^61 - z^7 * q^64 - z^2 * q^68 - z * q^69 + (-z^8 + z^3) * q^71 + (z^9 - z^4) * q^73 + 2z^6 q^74 + z^7 * q^75 + (z^5 - 1) * q^76 + (z^5 - 1) * q^78 - z^2 * q^79 - z^4 * q^81 + (z^8 - z^3) * q^83 + (-z^7 + z^2) * q^86 + q^87 + z^4 * q^92 + 2z^8 q^94 + (z^6 - z) * q^96 + (-z^5 + 1) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} - 2 q^{3} + 2 q^{6} - 2 q^{16} - 2 q^{19} + 2 q^{26} + 2 q^{27} - 2 q^{29} - 8 q^{32} - 8 q^{34} - 2 q^{37} + 2 q^{46} + 2 q^{47} - 2 q^{48} + 2 q^{50} - 2 q^{52} - 2 q^{53} + 8 q^{54} - 2 q^{57}+ \cdots + 8 q^{98}+O(q^{100}) $ 8 * q + 2 * q^2 - 2 * q^3 + 2 * q^6 - 2 * q^16 - 2 * q^19 + 2 * q^26 + 2 * q^27 - 2 * q^29 - 8 * q^32 - 8 * q^34 - 2 * q^37 + 2 * q^46 + 2 * q^47 - 2 * q^48 + 2 * q^50 - 2 * q^52 - 2 * q^53 + 8 * q^54 - 2 * q^57 + 2 * q^58 - 2 * q^61 - 2 * q^68 + 2 * q^71 + 2 * q^73 + 4 * q^74 - 8 * q^76 - 8 * q^78 - 2 * q^79 + 2 * q^81 - 2 * q^83 + 2 * q^86 + 8 * q^87 - 2 * q^92 - 4 * q^94 + 2 * q^96 + 8 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1573\mathbb{Z}\right)^\times$.

$n$	$365$	$1211$
$\chi(n)$	$\zeta_{20}^{4}$	$\zeta_{20}^{5}$

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} $

148.1

−0.587785	+	0.809017i
−0.587785	−	0.809017i
0.587785	−	0.809017i
0.951057	−	0.309017i
−0.951057	−	0.309017i
−0.951057	+	0.309017i
0.951057	+	0.309017i
0.587785	+	0.809017i

−1.26007

−

0.642040i

−0.809017

−

0.587785i

0.587785

0.809017i

0.642040

1.26007i

372.1

−1.26007

0.642040i

−0.809017

0.587785i

0.587785

−

0.809017i

0.642040

−

1.26007i

632.1

0.642040

−

1.26007i

−0.809017

−

0.587785i

−0.587785

−

0.809017i

−1.26007

0.642040i

850.1

0.221232

−

1.39680i

0.309017

0.951057i

−0.951057

−

0.309017i

1.39680

−

0.221232i

928.1

1.39680

−

0.221232i

0.309017

−

0.951057i

0.951057

−

0.309017i

0.221232

−

1.39680i

1334.1

1.39680

0.221232i

0.309017

0.951057i

0.951057

0.309017i

0.221232

1.39680i

1412.1

0.221232

1.39680i

0.309017

−

0.951057i

−0.951057

0.309017i

1.39680

0.221232i

1461.1

0.642040

1.26007i

−0.809017

0.587785i

−0.587785

0.809017i

−1.26007

−

0.642040i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.c	even	5	3	inner
13.d	odd	4	1	inner
143.r	odd	20	3	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1573.1.s.b		8
11.b	odd	2	1		1573.1.s.a		8
11.c	even	5	1		1573.1.f.a	✓	2
11.c	even	5	3	inner	1573.1.s.b		8
11.d	odd	10	1		1573.1.f.b	yes	2
11.d	odd	10	3		1573.1.s.a		8
13.d	odd	4	1	inner	1573.1.s.b		8
143.g	even	4	1		1573.1.s.a		8
143.r	odd	20	1		1573.1.f.a	✓	2
143.r	odd	20	3	inner	1573.1.s.b		8
143.s	even	20	1		1573.1.f.b	yes	2
143.s	even	20	3		1573.1.s.a		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1573.1.f.a	✓	2	11.c	even	5	1
1573.1.f.a	✓	2	143.r	odd	20	1
1573.1.f.b	yes	2	11.d	odd	10	1
1573.1.f.b	yes	2	143.s	even	20	1
1573.1.s.a		8	11.b	odd	2	1
1573.1.s.a		8	11.d	odd	10	3
1573.1.s.a		8	143.g	even	4	1
1573.1.s.a		8	143.s	even	20	3
1573.1.s.b		8	1.a	even	1	1	trivial
1573.1.s.b		8	11.c	even	5	3	inner
1573.1.s.b		8	13.d	odd	4	1	inner
1573.1.s.b		8	143.r	odd	20	3	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{8} - 2T_{2}^{7} + 2T_{2}^{6} - 4T_{2}^{4} + 8T_{2}^{2} - 16T_{2} + 16 $ T2^8 - 2*T2^7 + 2*T2^6 - 4*T2^4 + 8*T2^2 - 16*T2 + 16 acting on $S_{1}^{\mathrm{new}}(1573, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} - 2 T^{7} + \cdots + 16 $ T^8 - 2T^7 + 2T^6 - 4T^4 + 8T^2 - 16*T + 16
$3$	$ (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} $ (T^4 + T^3 + T^2 + T + 1)^2
$5$	$ T^{8} $ T^8
$7$	$ T^{8} $ T^8
$11$	$ T^{8} $ T^8
$13$	$ T^{8} - T^{6} + T^{4} + \cdots + 1 $ T^8 - T^6 + T^4 - T^2 + 1
$17$	$ T^{8} - T^{6} + T^{4} + \cdots + 1 $ T^8 - T^6 + T^4 - T^2 + 1
$19$	$ T^{8} + 2 T^{7} + \cdots + 16 $ T^8 + 2T^7 + 2T^6 - 4T^4 + 8T^2 + 16*T + 16
$23$	$ (T^{2} + 1)^{4} $ (T^2 + 1)^4
$29$	$ (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} $ (T^4 + T^3 + T^2 + T + 1)^2
$31$	$ T^{8} $ T^8
$37$	$ T^{8} + 2 T^{7} + \cdots + 16 $ T^8 + 2T^7 + 2T^6 - 4T^4 + 8T^2 + 16*T + 16
$41$	$ T^{8} $ T^8
$43$	$ (T^{2} + 1)^{4} $ (T^2 + 1)^4
$47$	$ T^{8} - 2 T^{7} + \cdots + 16 $ T^8 - 2T^7 + 2T^6 - 4T^4 + 8T^2 - 16*T + 16
$53$	$ (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} $ (T^4 + T^3 + T^2 + T + 1)^2
$59$	$ T^{8} $ T^8
$61$	$ (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} $ (T^4 + T^3 + T^2 + T + 1)^2
$67$	$ T^{8} $ T^8
$71$	$ T^{8} - 2 T^{7} + \cdots + 16 $ T^8 - 2T^7 + 2T^6 - 4T^4 + 8T^2 - 16*T + 16
$73$	$ T^{8} - 2 T^{7} + \cdots + 16 $ T^8 - 2T^7 + 2T^6 - 4T^4 + 8T^2 - 16*T + 16
$79$	$ (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} $ (T^4 + T^3 + T^2 + T + 1)^2
$83$	$ T^{8} + 2 T^{7} + \cdots + 16 $ T^8 + 2T^7 + 2T^6 - 4T^4 + 8T^2 + 16*T + 16
$89$	$ T^{8} $ T^8
$97$	$ T^{8} $ T^8
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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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1573 = 11^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1573.s (of order \(20\), degree \(8\), not minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{20}^{7} + \zeta_{20}^{2}) q^{2} - \zeta_{20}^{6} q^{3} + \zeta_{20}^{9} q^{4} + ( - \zeta_{20}^{8} + \zeta_{20}^{3}) q^{6} + \zeta_{20}^{5} q^{12} + \zeta_{20}^{7} q^{13} + \zeta_{20}^{8} q^{16} + \cdots + ( - \zeta_{20}^{5} + 1) q^{98} +O(q^{100}) \) q + (z^7 + z^2) * q^2 - z^6 * q^3 + z^9 * q^4 + (-z^8 + z^3) * q^6 + z^5 * q^12 + z^7 * q^13 + z^8 * q^16 + z^3 * q^17 + (-z^6 + z) * q^19 - z^5 * q^23 - z * q^25 + (z^9 - z^4) * q^26 - z^8 * q^27 + z^4 * q^29 + (-z^5 - 1) * q^32 + (z^5 - 1) * q^34 + (-z^9 + z^4) * q^37 + 2z^3 q^38 + z^3 * q^39 - z^5 * q^43 + (-z^7 + z^2) * q^46 + (z^6 + z) * q^47 + z^4 * q^48 - z^3 * q^49 + (-z^8 - z^3) * q^50 - z^9 * q^51 - z^6 * q^52 - z^2 * q^53 + (z^5 + 1) * q^54 + (-z^7 - z^2) * q^57 + (z^6 - z) * q^58 + z^8 * q^61 - z^7 * q^64 - z^2 * q^68 - z * q^69 + (-z^8 + z^3) * q^71 + (z^9 - z^4) * q^73 + 2z^6 q^74 + z^7 * q^75 + (z^5 - 1) * q^76 + (z^5 - 1) * q^78 - z^2 * q^79 - z^4 * q^81 + (z^8 - z^3) * q^83 + (-z^7 + z^2) * q^86 + q^87 + z^4 * q^92 + 2z^8 q^94 + (z^6 - z) * q^96 + (-z^5 + 1) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} - 2 q^{3} + 2 q^{6} - 2 q^{16} - 2 q^{19} + 2 q^{26} + 2 q^{27} - 2 q^{29} - 8 q^{32} - 8 q^{34} - 2 q^{37} + 2 q^{46} + 2 q^{47} - 2 q^{48} + 2 q^{50} - 2 q^{52} - 2 q^{53} + 8 q^{54} - 2 q^{57}+ \cdots + 8 q^{98}+O(q^{100}) \) 8 * q + 2 * q^2 - 2 * q^3 + 2 * q^6 - 2 * q^16 - 2 * q^19 + 2 * q^26 + 2 * q^27 - 2 * q^29 - 8 * q^32 - 8 * q^34 - 2 * q^37 + 2 * q^46 + 2 * q^47 - 2 * q^48 + 2 * q^50 - 2 * q^52 - 2 * q^53 + 8 * q^54 - 2 * q^57 + 2 * q^58 - 2 * q^61 - 2 * q^68 + 2 * q^71 + 2 * q^73 + 4 * q^74 - 8 * q^76 - 8 * q^78 - 2 * q^79 + 2 * q^81 - 2 * q^83 + 2 * q^86 + 8 * q^87 - 2 * q^92 - 4 * q^94 + 2 * q^96 + 8 * q^98

Introduction

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Modular forms

Varieties

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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	1573.1.s.b

Level	$1573$
Weight	$1$
Character orbit	1573.s
Analytic conductor	$0.785$
Analytic rank	$0$
Dimension	$8$
Projective image	$S_{4}$
CM/RM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(0.785029264872\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{20})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(S_{4}\)
Projective field:	Galois closure of 4.0.265837.1

\(n\)	\(365\)	\(1211\)
\(\chi(n)\)	\(\zeta_{20}^{4}\)	\(\zeta_{20}^{5}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 148.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} - 2 T^{7} + \cdots + 16 \) T^8 - 2T^7 + 2T^6 - 4T^4 + 8T^2 - 16*T + 16
$3$	\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) (T^4 + T^3 + T^2 + T + 1)^2
$5$	\( T^{8} \) T^8
$7$	\( T^{8} \) T^8
$11$	\( T^{8} \) T^8
$13$	\( T^{8} - T^{6} + T^{4} + \cdots + 1 \) T^8 - T^6 + T^4 - T^2 + 1
$17$	\( T^{8} - T^{6} + T^{4} + \cdots + 1 \) T^8 - T^6 + T^4 - T^2 + 1
$19$	\( T^{8} + 2 T^{7} + \cdots + 16 \) T^8 + 2T^7 + 2T^6 - 4T^4 + 8T^2 + 16*T + 16
$23$	\( (T^{2} + 1)^{4} \) (T^2 + 1)^4
$29$	\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) (T^4 + T^3 + T^2 + T + 1)^2
$31$	\( T^{8} \) T^8
$37$	\( T^{8} + 2 T^{7} + \cdots + 16 \) T^8 + 2T^7 + 2T^6 - 4T^4 + 8T^2 + 16*T + 16
$41$	\( T^{8} \) T^8
$43$	\( (T^{2} + 1)^{4} \) (T^2 + 1)^4
$47$	\( T^{8} - 2 T^{7} + \cdots + 16 \) T^8 - 2T^7 + 2T^6 - 4T^4 + 8T^2 - 16*T + 16
$53$	\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) (T^4 + T^3 + T^2 + T + 1)^2
$59$	\( T^{8} \) T^8
$61$	\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) (T^4 + T^3 + T^2 + T + 1)^2
$67$	\( T^{8} \) T^8
$71$	\( T^{8} - 2 T^{7} + \cdots + 16 \) T^8 - 2T^7 + 2T^6 - 4T^4 + 8T^2 - 16*T + 16
$73$	\( T^{8} - 2 T^{7} + \cdots + 16 \) T^8 - 2T^7 + 2T^6 - 4T^4 + 8T^2 - 16*T + 16
$79$	\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) (T^4 + T^3 + T^2 + T + 1)^2
$83$	\( T^{8} + 2 T^{7} + \cdots + 16 \) T^8 + 2T^7 + 2T^6 - 4T^4 + 8T^2 + 16*T + 16
$89$	\( T^{8} \) T^8
$97$	\( T^{8} \) T^8
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