LMFDB - Newform orbit 1573.1.l.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1573,1,Mod(233,1573)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1573, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([1, 5])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1573.233"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 1573 = 11^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1573.l (of order $10$, degree $4$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,-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:	$4$
Coefficient field:	$\Q(\zeta_{10})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + x^{2} - x + 1 $ x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 143)
Projective image:	$D_{5}$
Projective field:	Galois closure of 5.1.20449.1
Artin image:	$C_5\times D_5$
Artin field:	Galois closure of 10.0.875489472034463.2

$q$-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_{10}^{2} - \zeta_{10}) q^{2} + (\zeta_{10}^{4} + 1) q^{3} + (\zeta_{10}^{4} + \cdots + \zeta_{10}^{2}) q^{4} + (\zeta_{10}^{2} - 2 \zeta_{10} + 1) q^{6} + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{7}+ \cdots + (\zeta_{10}^{4} - \zeta_{10}^{3} + \cdots + 2) q^{98}+O(q^{100}) $ q + (z^2 - z) * q^2 + (z^4 + 1) * q^3 + (z^4 - z^3 + z^2) * q^4 + (z^2 - 2z + 1) q^6 + (z^4 + z^2) * q^7 + (z^4 - z^3 - z + 1) * q^8 + (z^4 - z^3 + 1) * q^9 + (z^4 - 2z^3 + 2z^2 - z) * q^12 + z^4 * q^13 + (z^4 - z^3 - z + 1) * q^14 + (z^4 - z^3 + z^2 - z + 1) * q^16 + (z^4 + z^2 - 2z + 2) q^18 + (z^4 + 1) * q^19 + (z^4 - z^3 + z^2 - z) * q^21 + (z^4 - z) * q^23 + (2z^4 - 2z^3 + z^2 - z + 2) * q^24 + z^2 * q^25 + (-z + 1) * q^26 + (z^4 - z^3 + z^2 + 1) * q^27 + (z^4 - z^3 + z^2 - 2z + 1) q^28 + (z^4 - z^3 + z^2 - z + 2) * q^32 + (z^4 - 2z^3 + 3z^2 - 2z + 1) q^36 + (z^2 - 2z + 1) q^38 + (z^4 - z^3) * q^39 + (-z^3 - z) * q^41 + (2z^4 - 2z^3 + z^2 - z + 2) * q^42 + (-z^3 + z^2 - z + 1) * q^46 + (2z^4 - 2z^3 + 2z^2 - 2z + 2) * q^48 + (z^4 - z^3 - z) * q^49 + (z^4 - z^3) * q^50 + (-z^3 + z^2 - z) * q^52 + (z^2 - z) * q^53 + (2z^4 - z^3 + z^2 - 2z + 2) * q^54 + (z^4 - 2z^3 + 2z^2 - z + 2) * q^56 + (2z^4 - z^3 + 1) q^57 + (z^4 - z^3 + 2z^2 - z + 1) q^63 + (z^4 - z^3 + 2z^2 - 2z + 1) * q^64 + (z^4 - z^3 - z + 1) * q^69 + (3z^4 - 3z^3 + 2z^2 - 2z + 2) * q^72 + (-z + 1) * q^73 + (z^2 - z) * q^75 + (z^4 - 2z^3 + 2z^2 - z) * q^76 + (z^4 - z + 2) * q^78 + (z^4 - z^3 + z^2 - z + 1) * q^81 + (z^4 - z^3 + z^2 + 1) * q^82 + (z^2 + 1) * q^83 + (2z^4 - 2z^3 + 2z^2 - 3z + 3) * q^84 + (-z^3 - z) * q^91 + (z^4 - 2z^3 + z^2 - z + 1) q^92 + (3z^4 - 2z^3 + 2z^2 - 2z + 3) * q^96 + (z^4 - z^3 + z^2 - z + 2) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 3 q^{3} - 3 q^{4} + q^{6} - 2 q^{7} + q^{8} + 2 q^{9} - 6 q^{12} - q^{13} + q^{14} + 4 q^{18} + 3 q^{19} - 4 q^{21} - 2 q^{23} + 2 q^{24} - q^{25} + 3 q^{26} + q^{27} - q^{28} + 4 q^{32}+ \cdots + 4 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 3 * q^3 - 3 * q^4 + q^6 - 2 * q^7 + q^8 + 2 * q^9 - 6 * q^12 - q^13 + q^14 + 4 * q^18 + 3 * q^19 - 4 * q^21 - 2 * q^23 + 2 * q^24 - q^25 + 3 * q^26 + q^27 - q^28 + 4 * q^32 - 4 * q^36 + q^38 - 2 * q^39 - 2 * q^41 + 2 * q^42 + q^46 - 3 * q^49 - 2 * q^50 - 3 * q^52 - 2 * q^53 + 2 * q^54 + 2 * q^56 + q^57 - q^63 - 2 * q^64 + q^69 - 2 * q^72 + 3 * q^73 - 2 * q^75 - 6 * q^76 + 6 * q^78 + q^82 + 3 * q^83 + 3 * q^84 - 2 * q^91 - q^92 + 3 * q^96 + 4 * q^98

Character values

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

$n$	$365$	$1211$
$\chi(n)$	$\zeta_{10}^{3}$	$-1$

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} $

233.1

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

−0.500000

1.53884i

1.30902

−

0.951057i

−1.30902

−

0.951057i

0.809017

2.48990i

−0.500000

−

0.363271i

0.809017

−

0.587785i

0.500000

−

1.53884i

766.1

−0.500000

−

0.363271i

0.190983

−

0.587785i

−0.190983

−

0.587785i

−0.309017

0.224514i

−0.500000

−

1.53884i

−0.309017

0.951057i

0.500000

0.363271i

844.1

−0.500000

0.363271i

0.190983

0.587785i

−0.190983

0.587785i

−0.309017

−

0.224514i

−0.500000

1.53884i

−0.309017

−

0.951057i

0.500000

−

0.363271i

1546.1

−0.500000

−

1.53884i

1.30902

0.951057i

−1.30902

0.951057i

0.809017

−

2.48990i

−0.500000

0.363271i

0.809017

0.587785i

0.500000

1.53884i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
143.d	odd	2	1	CM by $\Q(\sqrt{-143}) $
11.c	even	5	1	inner
143.l	odd	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1573.1.l.b		4
11.b	odd	2	1		1573.1.l.c		4
11.c	even	5	1		143.1.d.a	✓	2
11.c	even	5	1	inner	1573.1.l.b		4
11.c	even	5	2		1573.1.l.d		4
11.d	odd	10	1		143.1.d.b	yes	2
11.d	odd	10	2		1573.1.l.a		4
11.d	odd	10	1		1573.1.l.c		4
13.b	even	2	1		1573.1.l.c		4
33.f	even	10	1		1287.1.g.a		2
33.h	odd	10	1		1287.1.g.b		2
44.g	even	10	1		2288.1.m.a		2
44.h	odd	10	1		2288.1.m.b		2
55.h	odd	10	1		3575.1.h.e		2
55.j	even	10	1		3575.1.h.f		2
55.k	odd	20	2		3575.1.c.d		4
55.l	even	20	2		3575.1.c.c		4
143.d	odd	2	1	CM	1573.1.l.b		4
143.l	odd	10	1		143.1.d.a	✓	2
143.l	odd	10	1	inner	1573.1.l.b		4
143.l	odd	10	2		1573.1.l.d		4
143.n	even	10	1		143.1.d.b	yes	2
143.n	even	10	2		1573.1.l.a		4
143.n	even	10	1		1573.1.l.c		4
143.q	even	15	2		1859.1.i.b		4
143.r	odd	20	2		1859.1.c.c		4
143.s	even	20	2		1859.1.c.c		4
143.t	odd	30	2		1859.1.i.a		4
143.u	even	30	2		1859.1.i.a		4
143.v	odd	30	2		1859.1.i.b		4
143.w	even	60	4		1859.1.k.c		8
143.x	odd	60	4		1859.1.k.c		8
429.v	odd	10	1		1287.1.g.a		2
429.y	even	10	1		1287.1.g.b		2
572.ba	odd	10	1		2288.1.m.a		2
572.bb	even	10	1		2288.1.m.b		2
715.bf	even	10	1		3575.1.h.e		2
715.bi	odd	10	1		3575.1.h.f		2
715.cb	even	20	2		3575.1.c.d		4
715.ce	odd	20	2		3575.1.c.c		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
143.1.d.a	✓	2	11.c	even	5	1
143.1.d.a	✓	2	143.l	odd	10	1
143.1.d.b	yes	2	11.d	odd	10	1
143.1.d.b	yes	2	143.n	even	10	1
1287.1.g.a		2	33.f	even	10	1
1287.1.g.a		2	429.v	odd	10	1
1287.1.g.b		2	33.h	odd	10	1
1287.1.g.b		2	429.y	even	10	1
1573.1.l.a		4	11.d	odd	10	2
1573.1.l.a		4	143.n	even	10	2
1573.1.l.b		4	1.a	even	1	1	trivial
1573.1.l.b		4	11.c	even	5	1	inner
1573.1.l.b		4	143.d	odd	2	1	CM
1573.1.l.b		4	143.l	odd	10	1	inner
1573.1.l.c		4	11.b	odd	2	1
1573.1.l.c		4	11.d	odd	10	1
1573.1.l.c		4	13.b	even	2	1
1573.1.l.c		4	143.n	even	10	1
1573.1.l.d		4	11.c	even	5	2
1573.1.l.d		4	143.l	odd	10	2
1859.1.c.c		4	143.r	odd	20	2
1859.1.c.c		4	143.s	even	20	2
1859.1.i.a		4	143.t	odd	30	2
1859.1.i.a		4	143.u	even	30	2
1859.1.i.b		4	143.q	even	15	2
1859.1.i.b		4	143.v	odd	30	2
1859.1.k.c		8	143.w	even	60	4
1859.1.k.c		8	143.x	odd	60	4
2288.1.m.a		2	44.g	even	10	1
2288.1.m.a		2	572.ba	odd	10	1
2288.1.m.b		2	44.h	odd	10	1
2288.1.m.b		2	572.bb	even	10	1
3575.1.c.c		4	55.l	even	20	2
3575.1.c.c		4	715.ce	odd	20	2
3575.1.c.d		4	55.k	odd	20	2
3575.1.c.d		4	715.cb	even	20	2
3575.1.h.e		2	55.h	odd	10	1
3575.1.h.e		2	715.bf	even	10	1
3575.1.h.f		2	55.j	even	10	1
3575.1.h.f		2	715.bi	odd	10	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{4} + 2T_{2}^{3} + 4T_{2}^{2} + 3T_{2} + 1 $ T2^4 + 2*T2^3 + 4*T2^2 + 3*T2 + 1 acting on $S_{1}^{\mathrm{new}}(1573, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} + 2 T^{3} + \cdots + 1 $ T^4 + 2T^3 + 4T^2 + 3*T + 1
$3$	$ T^{4} - 3 T^{3} + \cdots + 1 $ T^4 - 3T^3 + 4T^2 - 2*T + 1
$5$	$ T^{4} $ T^4
$7$	$ T^{4} + 2 T^{3} + \cdots + 1 $ T^4 + 2T^3 + 4T^2 + 3*T + 1
$11$	$ T^{4} $ T^4
$13$	$ T^{4} + T^{3} + T^{2} + \cdots + 1 $ T^4 + T^3 + T^2 + T + 1
$17$	$ T^{4} $ T^4
$19$	$ T^{4} - 3 T^{3} + \cdots + 1 $ T^4 - 3T^3 + 4T^2 - 2*T + 1
$23$	$ (T^{2} + T - 1)^{2} $ (T^2 + T - 1)^2
$29$	$ T^{4} $ T^4
$31$	$ T^{4} $ T^4
$37$	$ T^{4} $ T^4
$41$	$ T^{4} + 2 T^{3} + \cdots + 1 $ T^4 + 2T^3 + 4T^2 + 3*T + 1
$43$	$ T^{4} $ T^4
$47$	$ T^{4} $ T^4
$53$	$ T^{4} + 2 T^{3} + \cdots + 1 $ T^4 + 2T^3 + 4T^2 + 3*T + 1
$59$	$ T^{4} $ T^4
$61$	$ T^{4} $ T^4
$67$	$ T^{4} $ T^4
$71$	$ T^{4} $ T^4
$73$	$ T^{4} - 3 T^{3} + \cdots + 1 $ T^4 - 3T^3 + 4T^2 - 2*T + 1
$79$	$ T^{4} $ T^4
$83$	$ T^{4} - 3 T^{3} + \cdots + 1 $ T^4 - 3T^3 + 4T^2 - 2*T + 1
$89$	$ T^{4} $ T^4
$97$	$ T^{4} $ T^4
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Elliptic curves over $\Q$
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Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1573 = 11^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1573.l (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{10}^{2} - \zeta_{10}) q^{2} + (\zeta_{10}^{4} + 1) q^{3} + (\zeta_{10}^{4} + \cdots + \zeta_{10}^{2}) q^{4} + (\zeta_{10}^{2} - 2 \zeta_{10} + 1) q^{6} + (\zeta_{10}^{4} + \zeta_{10}^{2}) q^{7}+ \cdots + (\zeta_{10}^{4} - \zeta_{10}^{3} + \cdots + 2) q^{98}+O(q^{100}) \) q + (z^2 - z) * q^2 + (z^4 + 1) * q^3 + (z^4 - z^3 + z^2) * q^4 + (z^2 - 2z + 1) q^6 + (z^4 + z^2) * q^7 + (z^4 - z^3 - z + 1) * q^8 + (z^4 - z^3 + 1) * q^9 + (z^4 - 2z^3 + 2z^2 - z) * q^12 + z^4 * q^13 + (z^4 - z^3 - z + 1) * q^14 + (z^4 - z^3 + z^2 - z + 1) * q^16 + (z^4 + z^2 - 2z + 2) q^18 + (z^4 + 1) * q^19 + (z^4 - z^3 + z^2 - z) * q^21 + (z^4 - z) * q^23 + (2z^4 - 2z^3 + z^2 - z + 2) * q^24 + z^2 * q^25 + (-z + 1) * q^26 + (z^4 - z^3 + z^2 + 1) * q^27 + (z^4 - z^3 + z^2 - 2z + 1) q^28 + (z^4 - z^3 + z^2 - z + 2) * q^32 + (z^4 - 2z^3 + 3z^2 - 2z + 1) q^36 + (z^2 - 2z + 1) q^38 + (z^4 - z^3) * q^39 + (-z^3 - z) * q^41 + (2z^4 - 2z^3 + z^2 - z + 2) * q^42 + (-z^3 + z^2 - z + 1) * q^46 + (2z^4 - 2z^3 + 2z^2 - 2z + 2) * q^48 + (z^4 - z^3 - z) * q^49 + (z^4 - z^3) * q^50 + (-z^3 + z^2 - z) * q^52 + (z^2 - z) * q^53 + (2z^4 - z^3 + z^2 - 2z + 2) * q^54 + (z^4 - 2z^3 + 2z^2 - z + 2) * q^56 + (2z^4 - z^3 + 1) q^57 + (z^4 - z^3 + 2z^2 - z + 1) q^63 + (z^4 - z^3 + 2z^2 - 2z + 1) * q^64 + (z^4 - z^3 - z + 1) * q^69 + (3z^4 - 3z^3 + 2z^2 - 2z + 2) * q^72 + (-z + 1) * q^73 + (z^2 - z) * q^75 + (z^4 - 2z^3 + 2z^2 - z) * q^76 + (z^4 - z + 2) * q^78 + (z^4 - z^3 + z^2 - z + 1) * q^81 + (z^4 - z^3 + z^2 + 1) * q^82 + (z^2 + 1) * q^83 + (2z^4 - 2z^3 + 2z^2 - 3z + 3) * q^84 + (-z^3 - z) * q^91 + (z^4 - 2z^3 + z^2 - z + 1) q^92 + (3z^4 - 2z^3 + 2z^2 - 2z + 3) * q^96 + (z^4 - z^3 + z^2 - z + 2) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 3 q^{3} - 3 q^{4} + q^{6} - 2 q^{7} + q^{8} + 2 q^{9} - 6 q^{12} - q^{13} + q^{14} + 4 q^{18} + 3 q^{19} - 4 q^{21} - 2 q^{23} + 2 q^{24} - q^{25} + 3 q^{26} + q^{27} - q^{28} + 4 q^{32}+ \cdots + 4 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 3 * q^3 - 3 * q^4 + q^6 - 2 * q^7 + q^8 + 2 * q^9 - 6 * q^12 - q^13 + q^14 + 4 * q^18 + 3 * q^19 - 4 * q^21 - 2 * q^23 + 2 * q^24 - q^25 + 3 * q^26 + q^27 - q^28 + 4 * q^32 - 4 * q^36 + q^38 - 2 * q^39 - 2 * q^41 + 2 * q^42 + q^46 - 3 * q^49 - 2 * q^50 - 3 * q^52 - 2 * q^53 + 2 * q^54 + 2 * q^56 + q^57 - q^63 - 2 * q^64 + q^69 - 2 * q^72 + 3 * q^73 - 2 * q^75 - 6 * q^76 + 6 * q^78 + q^82 + 3 * q^83 + 3 * q^84 - 2 * q^91 - q^92 + 3 * q^96 + 4 * 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	1573.1.l.b

Level	$1573$
Weight	$1$
Character orbit	1573.l
Analytic conductor	$0.785$
Analytic rank	$0$
Dimension	$4$
Projective image	$D_{5}$
CM discriminant	-143
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.785029264872\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{10})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{4} - x^{3} + x^{2} - x + 1 \) x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 143)
Projective image:	\(D_{5}\)
Projective field:	Galois closure of 5.1.20449.1
Artin image:	$C_5\times D_5$
Artin field:	Galois closure of 10.0.875489472034463.2

$p$	$F_p(T)$
$2$	\( T^{4} + 2 T^{3} + \cdots + 1 \) T^4 + 2T^3 + 4T^2 + 3*T + 1
$3$	\( T^{4} - 3 T^{3} + \cdots + 1 \) T^4 - 3T^3 + 4T^2 - 2*T + 1
$5$	\( T^{4} \) T^4
$7$	\( T^{4} + 2 T^{3} + \cdots + 1 \) T^4 + 2T^3 + 4T^2 + 3*T + 1
$11$	\( T^{4} \) T^4
$13$	\( T^{4} + T^{3} + T^{2} + \cdots + 1 \) T^4 + T^3 + T^2 + T + 1
$17$	\( T^{4} \) T^4
$19$	\( T^{4} - 3 T^{3} + \cdots + 1 \) T^4 - 3T^3 + 4T^2 - 2*T + 1
$23$	\( (T^{2} + T - 1)^{2} \) (T^2 + T - 1)^2
$29$	\( T^{4} \) T^4
$31$	\( T^{4} \) T^4
$37$	\( T^{4} \) T^4
$41$	\( T^{4} + 2 T^{3} + \cdots + 1 \) T^4 + 2T^3 + 4T^2 + 3*T + 1
$43$	\( T^{4} \) T^4
$47$	\( T^{4} \) T^4
$53$	\( T^{4} + 2 T^{3} + \cdots + 1 \) T^4 + 2T^3 + 4T^2 + 3*T + 1
$59$	\( T^{4} \) T^4
$61$	\( T^{4} \) T^4
$67$	\( T^{4} \) T^4
$71$	\( T^{4} \) T^4
$73$	\( T^{4} - 3 T^{3} + \cdots + 1 \) T^4 - 3T^3 + 4T^2 - 2*T + 1
$79$	\( T^{4} \) T^4
$83$	\( T^{4} - 3 T^{3} + \cdots + 1 \) T^4 - 3T^3 + 4T^2 - 2*T + 1
$89$	\( T^{4} \) T^4
$97$	\( T^{4} \) T^4
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\(n\)	\(365\)	\(1211\)
\(\chi(n)\)	\(\zeta_{10}^{3}\)	\(-1\)

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