LMFDB - Newform orbit 1900.1.bf.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1900,1,Mod(657,1900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1900.657"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1900, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 3, 8])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1900.bf (of order $12$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.948223524003$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 380)
Projective image:	$S_{4}$
Projective field:	Galois closure of 4.0.180500.1

$q$-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_{12}^{4} + \zeta_{12}) q^{3} - \zeta_{12}^{5} q^{9} + q^{11} + (\zeta_{12}^{5} + \zeta_{12}^{2}) q^{13} + ( - \zeta_{12}^{4} - \zeta_{12}) q^{17} - \zeta_{12}^{5} q^{19} + \zeta_{12}^{5} q^{29} + \cdots - \zeta_{12}^{5} q^{99} +O(q^{100}) $ q + (-z^4 + z) * q^3 - z^5 * q^9 + q^11 + (z^5 + z^2) * q^13 + (-z^4 - z) * q^17 - z^5 * q^19 + z^5 * q^29 - q^31 + (-z^4 + z) * q^33 + 2z^3 q^39 + (-z^5 + z^2) * q^47 + z^3 * q^49 - 2z^2 q^51 + (-z^5 - z^2) * q^53 + (-z^3 + 1) * q^57 - z * q^59 - z^2 * q^61 + (z^5 - z^2) * q^67 + z^4 * q^71 + z * q^79 + z^4 * q^81 + (z^3 - 1) * q^87 - z^5 * q^89 + (z^4 - z) * q^93 - z^5 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 4 q^{11} + 2 q^{13} + 2 q^{17} - 4 q^{31} + 2 q^{33} + 2 q^{47} - 4 q^{51} - 2 q^{53} + 4 q^{57} - 2 q^{61} - 2 q^{67} - 2 q^{71} - 2 q^{81} - 4 q^{87} - 2 q^{93}+O(q^{100}) $ 4 * q + 2 * q^3 + 4 * q^11 + 2 * q^13 + 2 * q^17 - 4 * q^31 + 2 * q^33 + 2 * q^47 - 4 * q^51 - 2 * q^53 + 4 * q^57 - 2 * q^61 - 2 * q^67 - 2 * q^71 - 2 * q^81 - 4 * q^87 - 2 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$-\zeta_{12}^{3}$	$-\zeta_{12}^{2}$	$1$

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} $

657.1

−0.866025	−	0.500000i
0.866025	−	0.500000i
0.866025	+	0.500000i
−0.866025	+	0.500000i

−0.366025

−

1.36603i

−0.866025

0.500000i

957.1

1.36603

0.366025i

0.866025

0.500000i

1493.1

1.36603

−

0.366025i

0.866025

−

0.500000i

1793.1

−0.366025

1.36603i

−0.866025

−

0.500000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
19.c	even	3	1	inner
95.m	odd	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1900.1.bf.a		4
5.b	even	2	1		380.1.x.a	✓	4
5.c	odd	4	1		380.1.x.a	✓	4
5.c	odd	4	1	inner	1900.1.bf.a		4
15.d	odd	2	1		3420.1.es.b		4
15.e	even	4	1		3420.1.es.b		4
19.c	even	3	1	inner	1900.1.bf.a		4
20.d	odd	2	1		1520.1.cr.b		4
20.e	even	4	1		1520.1.cr.b		4
95.i	even	6	1		380.1.x.a	✓	4
95.m	odd	12	1		380.1.x.a	✓	4
95.m	odd	12	1	inner	1900.1.bf.a		4
285.n	odd	6	1		3420.1.es.b		4
285.v	even	12	1		3420.1.es.b		4
380.p	odd	6	1		1520.1.cr.b		4
380.v	even	12	1		1520.1.cr.b		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
380.1.x.a	✓	4	5.b	even	2	1
380.1.x.a	✓	4	5.c	odd	4	1
380.1.x.a	✓	4	95.i	even	6	1
380.1.x.a	✓	4	95.m	odd	12	1
1520.1.cr.b		4	20.d	odd	2	1
1520.1.cr.b		4	20.e	even	4	1
1520.1.cr.b		4	380.p	odd	6	1
1520.1.cr.b		4	380.v	even	12	1
1900.1.bf.a		4	1.a	even	1	1	trivial
1900.1.bf.a		4	5.c	odd	4	1	inner
1900.1.bf.a		4	19.c	even	3	1	inner
1900.1.bf.a		4	95.m	odd	12	1	inner
3420.1.es.b		4	15.d	odd	2	1
3420.1.es.b		4	15.e	even	4	1
3420.1.es.b		4	285.n	odd	6	1
3420.1.es.b		4	285.v	even	12	1

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(1900, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} - 2 T^{3} + \cdots + 4 $ T^4 - 2T^3 + 2T^2 - 4*T + 4
$5$	$ T^{4} $ T^4
$7$	$ T^{4} $ T^4
$11$	$ (T - 1)^{4} $ (T - 1)^4
$13$	$ T^{4} - 2 T^{3} + \cdots + 4 $ T^4 - 2T^3 + 2T^2 - 4*T + 4
$17$	$ T^{4} - 2 T^{3} + \cdots + 4 $ T^4 - 2T^3 + 2T^2 - 4*T + 4
$19$	$ T^{4} - T^{2} + 1 $ T^4 - T^2 + 1
$23$	$ T^{4} $ T^4
$29$	$ T^{4} - T^{2} + 1 $ T^4 - T^2 + 1
$31$	$ (T + 1)^{4} $ (T + 1)^4
$37$	$ T^{4} $ T^4
$41$	$ T^{4} $ T^4
$43$	$ T^{4} $ T^4
$47$	$ T^{4} - 2 T^{3} + \cdots + 4 $ T^4 - 2T^3 + 2T^2 - 4*T + 4
$53$	$ T^{4} + 2 T^{3} + \cdots + 4 $ T^4 + 2T^3 + 2T^2 + 4*T + 4
$59$	$ T^{4} - T^{2} + 1 $ T^4 - T^2 + 1
$61$	$ (T^{2} + T + 1)^{2} $ (T^2 + T + 1)^2
$67$	$ T^{4} + 2 T^{3} + \cdots + 4 $ T^4 + 2T^3 + 2T^2 + 4*T + 4
$71$	$ (T^{2} + T + 1)^{2} $ (T^2 + T + 1)^2
$73$	$ T^{4} $ T^4
$79$	$ T^{4} - T^{2} + 1 $ T^4 - T^2 + 1
$83$	$ T^{4} $ T^4
$89$	$ T^{4} - T^{2} + 1 $ T^4 - T^2 + 1
$97$	$ T^{4} $ T^4
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Level:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1900.bf (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{12}^{4} + \zeta_{12}) q^{3} - \zeta_{12}^{5} q^{9} + q^{11} + (\zeta_{12}^{5} + \zeta_{12}^{2}) q^{13} + ( - \zeta_{12}^{4} - \zeta_{12}) q^{17} - \zeta_{12}^{5} q^{19} + \zeta_{12}^{5} q^{29} + \cdots - \zeta_{12}^{5} q^{99} +O(q^{100}) \) q + (-z^4 + z) * q^3 - z^5 * q^9 + q^11 + (z^5 + z^2) * q^13 + (-z^4 - z) * q^17 - z^5 * q^19 + z^5 * q^29 - q^31 + (-z^4 + z) * q^33 + 2z^3 q^39 + (-z^5 + z^2) * q^47 + z^3 * q^49 - 2z^2 q^51 + (-z^5 - z^2) * q^53 + (-z^3 + 1) * q^57 - z * q^59 - z^2 * q^61 + (z^5 - z^2) * q^67 + z^4 * q^71 + z * q^79 + z^4 * q^81 + (z^3 - 1) * q^87 - z^5 * q^89 + (z^4 - z) * q^93 - z^5 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 4 q^{11} + 2 q^{13} + 2 q^{17} - 4 q^{31} + 2 q^{33} + 2 q^{47} - 4 q^{51} - 2 q^{53} + 4 q^{57} - 2 q^{61} - 2 q^{67} - 2 q^{71} - 2 q^{81} - 4 q^{87} - 2 q^{93}+O(q^{100}) \) 4 * q + 2 * q^3 + 4 * q^11 + 2 * q^13 + 2 * q^17 - 4 * q^31 + 2 * q^33 + 2 * q^47 - 4 * q^51 - 2 * q^53 + 4 * q^57 - 2 * q^61 - 2 * q^67 - 2 * q^71 - 2 * q^81 - 4 * q^87 - 2 * q^93

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	1900.1.bf.a

Level	$1900$
Weight	$1$
Character orbit	1900.bf
Analytic conductor	$0.948$
Analytic rank	$0$
Dimension	$4$
Projective image	$S_{4}$
CM/RM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.948223524003\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{12})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{4} - x^{2} + 1 \) x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 380)
Projective image:	\(S_{4}\)
Projective field:	Galois closure of 4.0.180500.1

\(n\)	\(77\)	\(401\)	\(951\)
\(\chi(n)\)	\(-\zeta_{12}^{3}\)	\(-\zeta_{12}^{2}\)	\(1\)

$p$	$F_p(T)$
$2$	\( T^{4} \) T^4
$3$	\( T^{4} - 2 T^{3} + \cdots + 4 \) T^4 - 2T^3 + 2T^2 - 4*T + 4
$5$	\( T^{4} \) T^4
$7$	\( T^{4} \) T^4
$11$	\( (T - 1)^{4} \) (T - 1)^4
$13$	\( T^{4} - 2 T^{3} + \cdots + 4 \) T^4 - 2T^3 + 2T^2 - 4*T + 4
$17$	\( T^{4} - 2 T^{3} + \cdots + 4 \) T^4 - 2T^3 + 2T^2 - 4*T + 4
$19$	\( T^{4} - T^{2} + 1 \) T^4 - T^2 + 1
$23$	\( T^{4} \) T^4
$29$	\( T^{4} - T^{2} + 1 \) T^4 - T^2 + 1
$31$	\( (T + 1)^{4} \) (T + 1)^4
$37$	\( T^{4} \) T^4
$41$	\( T^{4} \) T^4
$43$	\( T^{4} \) T^4
$47$	\( T^{4} - 2 T^{3} + \cdots + 4 \) T^4 - 2T^3 + 2T^2 - 4*T + 4
$53$	\( T^{4} + 2 T^{3} + \cdots + 4 \) T^4 + 2T^3 + 2T^2 + 4*T + 4
$59$	\( T^{4} - T^{2} + 1 \) T^4 - T^2 + 1
$61$	\( (T^{2} + T + 1)^{2} \) (T^2 + T + 1)^2
$67$	\( T^{4} + 2 T^{3} + \cdots + 4 \) T^4 + 2T^3 + 2T^2 + 4*T + 4
$71$	\( (T^{2} + T + 1)^{2} \) (T^2 + T + 1)^2
$73$	\( T^{4} \) T^4
$79$	\( T^{4} - T^{2} + 1 \) T^4 - T^2 + 1
$83$	\( T^{4} \) T^4
$89$	\( T^{4} - T^{2} + 1 \) T^4 - T^2 + 1
$97$	\( T^{4} \) T^4
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\(n\):		e.g. 2-40 or 990-1000
Significant digits:
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