LMFDB - Newform orbit 3700.1.ch.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3700,1,Mod(487,3700)] 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("3700.487"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3700, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 9, 15])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 3700 = 2^{2} \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3700.ch (of order $20$, degree $8$, 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:	$1.84654054674$
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:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{20}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{20} - \cdots)$

$q$-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}^{2} q^{2} + \zeta_{20}^{4} q^{4} + \zeta_{20}^{6} q^{5} + \zeta_{20}^{6} q^{8} + \zeta_{20}^{3} q^{9} + \zeta_{20}^{8} q^{10} + ( - \zeta_{20}^{9} - \zeta_{20}^{7}) q^{13} + \zeta_{20}^{8} q^{16}+ \cdots - \zeta_{20}^{7} q^{98} +O(q^{100}) $ q + z^2 * q^2 + z^4 * q^4 + z^6 * q^5 + z^6 * q^8 + z^3 * q^9 + z^8 * q^10 + (-z^9 - z^7) * q^13 + z^8 * q^16 + (-z^9 + z^3) * q^17 + z^5 * q^18 - q^20 - z^2 * q^25 + (-z^9 + z) * q^26 + (-z^2 - z) * q^29 - q^32 + (z^5 + z) * q^34 + z^7 * q^36 - q^37 - z^2 * q^40 + (z^4 + z^2) * q^41 + z^9 * q^45 - z^5 * q^49 - z^4 * q^50 + (z^3 + z) * q^52 + (-z^3 + 1) * q^53 + (-z^4 - z^3) * q^58 + (-z^8 - z) * q^61 - z^2 * q^64 + (z^5 + z^3) * q^65 + (z^7 + z^3) * q^68 + z^9 * q^72 + (z^8 - z) * q^73 - z^2 * q^74 - z^4 * q^80 + z^6 * q^81 + (z^6 + z^4) * q^82 + (z^9 + z^5) * q^85 + (z^9 - 1) * q^89 - z * q^90 + (z^6 - z^2) * q^97 - z^7 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{8} - 2 q^{10} - 2 q^{16} - 8 q^{20} - 2 q^{25} - 2 q^{29} - 8 q^{32} - 8 q^{37} - 2 q^{40} + 2 q^{50} + 8 q^{53} + 2 q^{58} + 2 q^{61} - 2 q^{64} - 2 q^{73} - 2 q^{74}+ \cdots - 8 q^{89}+O(q^{100}) $ 8 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 + 2 * q^8 - 2 * q^10 - 2 * q^16 - 8 * q^20 - 2 * q^25 - 2 * q^29 - 8 * q^32 - 8 * q^37 - 2 * q^40 + 2 * q^50 + 8 * q^53 + 2 * q^58 + 2 * q^61 - 2 * q^64 - 2 * q^73 - 2 * q^74 + 2 * q^80 + 2 * q^81 - 8 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1777$	$1851$
$\chi(n)$	$\zeta_{20}^{5}$	$-\zeta_{20}^{9}$	$-1$

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} $

487.1

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

0.809017

0.587785i

0.309017

0.951057i

−0.309017

0.951057i

−0.309017

0.951057i

−0.587785

−

0.809017i

−0.809017

0.587785i

623.1

0.809017

−

0.587785i

0.309017

−

0.951057i

−0.309017

−

0.951057i

−0.309017

−

0.951057i

−0.587785

0.809017i

−0.809017

−

0.587785i

1227.1

0.809017

−

0.587785i

0.309017

−

0.951057i

−0.309017

−

0.951057i

−0.309017

−

0.951057i

0.587785

−

0.809017i

−0.809017

−

0.587785i

1363.1

0.809017

0.587785i

0.309017

0.951057i

−0.309017

0.951057i

−0.309017

0.951057i

0.587785

0.809017i

−0.809017

0.587785i

1967.1

−0.309017

−

0.951057i

−0.809017

0.587785i

0.809017

0.587785i

0.809017

0.587785i

0.951057

0.309017i

0.309017

−

0.951057i

2103.1

−0.309017

0.951057i

−0.809017

−

0.587785i

0.809017

−

0.587785i

0.809017

−

0.587785i

0.951057

−

0.309017i

0.309017

0.951057i

3447.1

−0.309017

0.951057i

−0.809017

−

0.587785i

0.809017

−

0.587785i

0.809017

−

0.587785i

−0.951057

0.309017i

0.309017

0.951057i

3583.1

−0.309017

−

0.951057i

−0.809017

0.587785i

0.809017

0.587785i

0.809017

0.587785i

−0.951057

−

0.309017i

0.309017

−

0.951057i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
925.bd	even	20	1	inner
3700.ch	odd	20	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3700.1.ch.a	✓	8
4.b	odd	2	1	CM	3700.1.ch.a	✓	8
25.f	odd	20	1		3700.1.ck.a	yes	8
37.d	odd	4	1		3700.1.ck.a	yes	8
100.l	even	20	1		3700.1.ck.a	yes	8
148.g	even	4	1		3700.1.ck.a	yes	8
925.bd	even	20	1	inner	3700.1.ch.a	✓	8
3700.ch	odd	20	1	inner	3700.1.ch.a	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3700.1.ch.a	✓	8	1.a	even	1	1	trivial
3700.1.ch.a	✓	8	4.b	odd	2	1	CM
3700.1.ch.a	✓	8	925.bd	even	20	1	inner
3700.1.ch.a	✓	8	3700.ch	odd	20	1	inner
3700.1.ck.a	yes	8	25.f	odd	20	1
3700.1.ck.a	yes	8	37.d	odd	4	1
3700.1.ck.a	yes	8	100.l	even	20	1
3700.1.ck.a	yes	8	148.g	even	4	1

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(3700, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} $ (T^4 - T^3 + T^2 - T + 1)^2
$3$	$ T^{8} $ T^8
$5$	$ (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} $ (T^4 - T^3 + T^2 - T + 1)^2
$7$	$ T^{8} $ T^8
$11$	$ T^{8} $ T^8
$13$	$ T^{8} + 10 T^{4} + \cdots + 25 $ T^8 + 10T^4 + 25T^2 + 25
$17$	$ T^{8} - 4 T^{6} + \cdots + 1 $ T^8 - 4T^6 + 6T^4 + T^2 + 1
$19$	$ T^{8} $ T^8
$23$	$ T^{8} $ T^8
$29$	$ T^{8} + 2 T^{7} + \cdots + 1 $ T^8 + 2T^7 + 2T^6 - 4T^4 - 10T^3 + 13T^2 - 4T + 1
$31$	$ T^{8} $ T^8
$37$	$ (T + 1)^{8} $ (T + 1)^8
$41$	$ (T^{4} + 5 T + 5)^{2} $ (T^4 + 5*T + 5)^2
$43$	$ T^{8} $ T^8
$47$	$ T^{8} $ T^8
$53$	$ T^{8} - 8 T^{7} + \cdots + 1 $ T^8 - 8T^7 + 27T^6 - 50T^5 + 56T^4 - 40T^3 + 18T^2 - 4*T + 1
$59$	$ T^{8} $ T^8
$61$	$ T^{8} - 2 T^{7} + \cdots + 1 $ T^8 - 2T^7 + 2T^6 - 10T^5 + 16T^4 - 10T^3 + 13T^2 - 6*T + 1
$67$	$ T^{8} $ T^8
$71$	$ T^{8} $ T^8
$73$	$ T^{8} + 2 T^{7} + \cdots + 1 $ T^8 + 2T^7 + 2T^6 + 10T^5 + 16T^4 + 10T^3 + 13T^2 + 6*T + 1
$79$	$ T^{8} $ T^8
$83$	$ T^{8} $ T^8
$89$	$ T^{8} + 8 T^{7} + \cdots + 1 $ T^8 + 8T^7 + 27T^6 + 50T^5 + 56T^4 + 40T^3 + 18T^2 + 4*T + 1
$97$	$ (T^{4} + 5 T + 5)^{2} $ (T^4 + 5*T + 5)^2
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Level:	\( N \)	\(=\)	\( 3700 = 2^{2} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3700.ch (of order \(20\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{20}^{2} q^{2} + \zeta_{20}^{4} q^{4} + \zeta_{20}^{6} q^{5} + \zeta_{20}^{6} q^{8} + \zeta_{20}^{3} q^{9} + \zeta_{20}^{8} q^{10} + ( - \zeta_{20}^{9} - \zeta_{20}^{7}) q^{13} + \zeta_{20}^{8} q^{16}+ \cdots - \zeta_{20}^{7} q^{98} +O(q^{100}) \) q + z^2 * q^2 + z^4 * q^4 + z^6 * q^5 + z^6 * q^8 + z^3 * q^9 + z^8 * q^10 + (-z^9 - z^7) * q^13 + z^8 * q^16 + (-z^9 + z^3) * q^17 + z^5 * q^18 - q^20 - z^2 * q^25 + (-z^9 + z) * q^26 + (-z^2 - z) * q^29 - q^32 + (z^5 + z) * q^34 + z^7 * q^36 - q^37 - z^2 * q^40 + (z^4 + z^2) * q^41 + z^9 * q^45 - z^5 * q^49 - z^4 * q^50 + (z^3 + z) * q^52 + (-z^3 + 1) * q^53 + (-z^4 - z^3) * q^58 + (-z^8 - z) * q^61 - z^2 * q^64 + (z^5 + z^3) * q^65 + (z^7 + z^3) * q^68 + z^9 * q^72 + (z^8 - z) * q^73 - z^2 * q^74 - z^4 * q^80 + z^6 * q^81 + (z^6 + z^4) * q^82 + (z^9 + z^5) * q^85 + (z^9 - 1) * q^89 - z * q^90 + (z^6 - z^2) * q^97 - z^7 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{8} - 2 q^{10} - 2 q^{16} - 8 q^{20} - 2 q^{25} - 2 q^{29} - 8 q^{32} - 8 q^{37} - 2 q^{40} + 2 q^{50} + 8 q^{53} + 2 q^{58} + 2 q^{61} - 2 q^{64} - 2 q^{73} - 2 q^{74}+ \cdots - 8 q^{89}+O(q^{100}) \) 8 * q + 2 * q^2 - 2 * q^4 + 2 * q^5 + 2 * q^8 - 2 * q^10 - 2 * q^16 - 8 * q^20 - 2 * q^25 - 2 * q^29 - 8 * q^32 - 8 * q^37 - 2 * q^40 + 2 * q^50 + 8 * q^53 + 2 * q^58 + 2 * q^61 - 2 * q^64 - 2 * q^73 - 2 * q^74 + 2 * q^80 + 2 * q^81 - 8 * q^89

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	3700.1.ch.a

Level	$3700$
Weight	$1$
Character orbit	3700.ch
Analytic conductor	$1.847$
Analytic rank	$0$
Dimension	$8$
Projective image	$D_{20}$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.84654054674\)
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:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{20}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

\(n\)	\(1001\)	\(1777\)	\(1851\)
\(\chi(n)\)	\(\zeta_{20}^{5}\)	\(-\zeta_{20}^{9}\)	\(-1\)

\(n\):		e.g. 2-40 or 80-90
Embeddings:		e.g. 1-3 or 487.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
925.bd	even	20	1	inner
3700.ch	odd	20	1	inner

$p$	$F_p(T)$
$2$	\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) (T^4 - T^3 + T^2 - T + 1)^2
$3$	\( T^{8} \) T^8
$5$	\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) (T^4 - T^3 + T^2 - T + 1)^2
$7$	\( T^{8} \) T^8
$11$	\( T^{8} \) T^8
$13$	\( T^{8} + 10 T^{4} + \cdots + 25 \) T^8 + 10T^4 + 25T^2 + 25
$17$	\( T^{8} - 4 T^{6} + \cdots + 1 \) T^8 - 4T^6 + 6T^4 + T^2 + 1
$19$	\( T^{8} \) T^8
$23$	\( T^{8} \) T^8
$29$	\( T^{8} + 2 T^{7} + \cdots + 1 \) T^8 + 2T^7 + 2T^6 - 4T^4 - 10T^3 + 13T^2 - 4T + 1
$31$	\( T^{8} \) T^8
$37$	\( (T + 1)^{8} \) (T + 1)^8
$41$	\( (T^{4} + 5 T + 5)^{2} \) (T^4 + 5*T + 5)^2
$43$	\( T^{8} \) T^8
$47$	\( T^{8} \) T^8
$53$	\( T^{8} - 8 T^{7} + \cdots + 1 \) T^8 - 8T^7 + 27T^6 - 50T^5 + 56T^4 - 40T^3 + 18T^2 - 4*T + 1
$59$	\( T^{8} \) T^8
$61$	\( T^{8} - 2 T^{7} + \cdots + 1 \) T^8 - 2T^7 + 2T^6 - 10T^5 + 16T^4 - 10T^3 + 13T^2 - 6*T + 1
$67$	\( T^{8} \) T^8
$71$	\( T^{8} \) T^8
$73$	\( T^{8} + 2 T^{7} + \cdots + 1 \) T^8 + 2T^7 + 2T^6 + 10T^5 + 16T^4 + 10T^3 + 13T^2 + 6*T + 1
$79$	\( T^{8} \) T^8
$83$	\( T^{8} \) T^8
$89$	\( T^{8} + 8 T^{7} + \cdots + 1 \) T^8 + 8T^7 + 27T^6 + 50T^5 + 56T^4 + 40T^3 + 18T^2 + 4*T + 1
$97$	\( (T^{4} + 5 T + 5)^{2} \) (T^4 + 5*T + 5)^2
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