LMFDB - Newform orbit 3700.1.b.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3700,1,Mod(3551,3700)] 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("3700.3551"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3700, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) 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.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-4,0,0,0,0,-4,0,0,0,0,0,0,4,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$1.84654054674$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 740)
Projective image:	$D_{4}$
Projective field:	Galois closure of $\Q(\sqrt{26 +2 \sqrt{185}})$
Artin image:	$C_4\times D_8$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{32} + \cdots)$

$q$-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_{8}^{2} q^{2} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{3} - q^{4} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{6} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{7} - \zeta_{8}^{2} q^{8} - q^{9} + (\zeta_{8}^{3} + \zeta_{8}) q^{12} + \cdots - \zeta_{8}^{2} q^{98} +O(q^{100}) $ q + z^2 * q^2 + (-z^3 - z) * q^3 - q^4 + (-z^3 + z) * q^6 + (-z^3 - z) * q^7 - z^2 * q^8 - q^9 + (z^3 + z) * q^12 + (-z^3 + z) * q^14 + q^16 - z^2 * q^18 + (-z^3 + z) * q^19 - 2 * q^21 + (z^3 - z) * q^24 + (z^3 + z) * q^28 + (-z^3 + z) * q^31 + z^2 * q^32 + q^36 - z^2 * q^37 + (z^3 + z) * q^38 - 2 * q^41 - 2z^2 q^42 + (-z^3 - z) * q^47 + (-z^3 - z) * q^48 - q^49 + (z^3 - z) * q^56 - 2z^2 q^57 + (z^3 - z) * q^59 + (z^3 + z) * q^62 + (z^3 + z) * q^63 - q^64 + (z^3 + z) * q^67 + z^2 * q^72 + q^74 + (z^3 - z) * q^76 + (z^3 - z) * q^79 - q^81 - 2z^2 q^82 + (-z^3 - z) * q^83 + 2 * q^84 - 2z^2 q^93 + (-z^3 + z) * q^94 + (-z^3 + z) * q^96 + 2z^2 q^97 - z^2 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{9} + 4 q^{16} - 8 q^{21} + 4 q^{36} - 8 q^{41} - 4 q^{49} - 4 q^{64} + 4 q^{74} - 4 q^{81} + 8 q^{84}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^9 + 4 * q^16 - 8 * q^21 + 4 * q^36 - 8 * q^41 - 4 * q^49 - 4 * q^64 + 4 * q^74 - 4 * q^81 + 8 * q^84

Character values

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

$n$	$1001$	$1777$	$1851$
$\chi(n)$	$-1$	$1$	$-1$

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} $

3551.1

−0.707107	+	0.707107i
0.707107	−	0.707107i
0.707107	+	0.707107i
−0.707107	−	0.707107i

−

1.00000i

−

1.41421i

−1.00000

−1.41421

−

1.41421i

1.00000i

−1.00000

3551.2

−

1.00000i

1.41421i

−1.00000

1.41421

1.41421i

1.00000i

−1.00000

3551.3

1.00000i

−

1.41421i

−1.00000

1.41421

−

1.41421i

−

1.00000i

−1.00000

3551.4

1.00000i

1.41421i

−1.00000

−1.41421

1.41421i

−

1.00000i

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
740.g	odd	2	1	CM by $\Q(\sqrt{-185}) $
4.b	odd	2	1	inner
5.b	even	2	1	inner
20.d	odd	2	1	inner
37.b	even	2	1	inner
148.b	odd	2	1	inner
185.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3700.1.b.f		4
4.b	odd	2	1	inner	3700.1.b.f		4
5.b	even	2	1	inner	3700.1.b.f		4
5.c	odd	4	1		740.1.g.c	✓	2
5.c	odd	4	1		740.1.g.d	yes	2
20.d	odd	2	1	inner	3700.1.b.f		4
20.e	even	4	1		740.1.g.c	✓	2
20.e	even	4	1		740.1.g.d	yes	2
37.b	even	2	1	inner	3700.1.b.f		4
148.b	odd	2	1	inner	3700.1.b.f		4
185.d	even	2	1	inner	3700.1.b.f		4
185.h	odd	4	1		740.1.g.c	✓	2
185.h	odd	4	1		740.1.g.d	yes	2
740.g	odd	2	1	CM	3700.1.b.f		4
740.m	even	4	1		740.1.g.c	✓	2
740.m	even	4	1		740.1.g.d	yes	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
740.1.g.c	✓	2	5.c	odd	4	1
740.1.g.c	✓	2	20.e	even	4	1
740.1.g.c	✓	2	185.h	odd	4	1
740.1.g.c	✓	2	740.m	even	4	1
740.1.g.d	yes	2	5.c	odd	4	1
740.1.g.d	yes	2	20.e	even	4	1
740.1.g.d	yes	2	185.h	odd	4	1
740.1.g.d	yes	2	740.m	even	4	1
3700.1.b.f		4	1.a	even	1	1	trivial
3700.1.b.f		4	4.b	odd	2	1	inner
3700.1.b.f		4	5.b	even	2	1	inner
3700.1.b.f		4	20.d	odd	2	1	inner
3700.1.b.f		4	37.b	even	2	1	inner
3700.1.b.f		4	148.b	odd	2	1	inner
3700.1.b.f		4	185.d	even	2	1	inner
3700.1.b.f		4	740.g	odd	2	1	CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(3700, [\chi])$:

$ T_{3}^{2} + 2 $

T3^2 + 2

$ T_{13} $

T13

$ T_{19}^{2} - 2 $

T19^2 - 2

$ T_{23} $

T23

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} + 1)^{2} $ (T^2 + 1)^2
$3$	$ (T^{2} + 2)^{2} $ (T^2 + 2)^2
$5$	$ T^{4} $ T^4
$7$	$ (T^{2} + 2)^{2} $ (T^2 + 2)^2
$11$	$ T^{4} $ T^4
$13$	$ T^{4} $ T^4
$17$	$ T^{4} $ T^4
$19$	$ (T^{2} - 2)^{2} $ (T^2 - 2)^2
$23$	$ T^{4} $ T^4
$29$	$ T^{4} $ T^4
$31$	$ (T^{2} - 2)^{2} $ (T^2 - 2)^2
$37$	$ (T^{2} + 1)^{2} $ (T^2 + 1)^2
$41$	$ (T + 2)^{4} $ (T + 2)^4
$43$	$ T^{4} $ T^4
$47$	$ (T^{2} + 2)^{2} $ (T^2 + 2)^2
$53$	$ T^{4} $ T^4
$59$	$ (T^{2} - 2)^{2} $ (T^2 - 2)^2
$61$	$ T^{4} $ T^4
$67$	$ (T^{2} + 2)^{2} $ (T^2 + 2)^2
$71$	$ T^{4} $ T^4
$73$	$ T^{4} $ T^4
$79$	$ (T^{2} - 2)^{2} $ (T^2 - 2)^2
$83$	$ (T^{2} + 2)^{2} $ (T^2 + 2)^2
$89$	$ T^{4} $ T^4
$97$	$ (T^{2} + 4)^{2} $ (T^2 + 4)^2
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Level:	\( N \)	\(=\)	\( 3700 = 2^{2} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3700.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{8}^{2} q^{2} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{3} - q^{4} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{6} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{7} - \zeta_{8}^{2} q^{8} - q^{9} + (\zeta_{8}^{3} + \zeta_{8}) q^{12} + \cdots - \zeta_{8}^{2} q^{98} +O(q^{100}) \) q + z^2 * q^2 + (-z^3 - z) * q^3 - q^4 + (-z^3 + z) * q^6 + (-z^3 - z) * q^7 - z^2 * q^8 - q^9 + (z^3 + z) * q^12 + (-z^3 + z) * q^14 + q^16 - z^2 * q^18 + (-z^3 + z) * q^19 - 2 * q^21 + (z^3 - z) * q^24 + (z^3 + z) * q^28 + (-z^3 + z) * q^31 + z^2 * q^32 + q^36 - z^2 * q^37 + (z^3 + z) * q^38 - 2 * q^41 - 2z^2 q^42 + (-z^3 - z) * q^47 + (-z^3 - z) * q^48 - q^49 + (z^3 - z) * q^56 - 2z^2 q^57 + (z^3 - z) * q^59 + (z^3 + z) * q^62 + (z^3 + z) * q^63 - q^64 + (z^3 + z) * q^67 + z^2 * q^72 + q^74 + (z^3 - z) * q^76 + (z^3 - z) * q^79 - q^81 - 2z^2 q^82 + (-z^3 - z) * q^83 + 2 * q^84 - 2z^2 q^93 + (-z^3 + z) * q^94 + (-z^3 + z) * q^96 + 2z^2 q^97 - z^2 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{9} + 4 q^{16} - 8 q^{21} + 4 q^{36} - 8 q^{41} - 4 q^{49} - 4 q^{64} + 4 q^{74} - 4 q^{81} + 8 q^{84}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^9 + 4 * q^16 - 8 * q^21 + 4 * q^36 - 8 * q^41 - 4 * q^49 - 4 * q^64 + 4 * q^74 - 4 * q^81 + 8 * q^84

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	3700.1.b.f

Level	$3700$
Weight	$1$
Character orbit	3700.b
Analytic conductor	$1.847$
Analytic rank	$0$
Dimension	$4$
Projective image	$D_{4}$
CM discriminant	-740
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(1.84654054674\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\zeta_{8})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{4} + 1 \) x^4 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 740)
Projective image:	\(D_{4}\)
Projective field:	Galois closure of \(\Q(\sqrt{26 +2 \sqrt{185}})\)
Artin image:	$C_4\times D_8$
Artin field:	Galois closure of \(\mathbb{Q}[x]/(x^{32} + \cdots)\)

$p$	$F_p(T)$
$2$	\( (T^{2} + 1)^{2} \) (T^2 + 1)^2
$3$	\( (T^{2} + 2)^{2} \) (T^2 + 2)^2
$5$	\( T^{4} \) T^4
$7$	\( (T^{2} + 2)^{2} \) (T^2 + 2)^2
$11$	\( T^{4} \) T^4
$13$	\( T^{4} \) T^4
$17$	\( T^{4} \) T^4
$19$	\( (T^{2} - 2)^{2} \) (T^2 - 2)^2
$23$	\( T^{4} \) T^4
$29$	\( T^{4} \) T^4
$31$	\( (T^{2} - 2)^{2} \) (T^2 - 2)^2
$37$	\( (T^{2} + 1)^{2} \) (T^2 + 1)^2
$41$	\( (T + 2)^{4} \) (T + 2)^4
$43$	\( T^{4} \) T^4
$47$	\( (T^{2} + 2)^{2} \) (T^2 + 2)^2
$53$	\( T^{4} \) T^4
$59$	\( (T^{2} - 2)^{2} \) (T^2 - 2)^2
$61$	\( T^{4} \) T^4
$67$	\( (T^{2} + 2)^{2} \) (T^2 + 2)^2
$71$	\( T^{4} \) T^4
$73$	\( T^{4} \) T^4
$79$	\( (T^{2} - 2)^{2} \) (T^2 - 2)^2
$83$	\( (T^{2} + 2)^{2} \) (T^2 + 2)^2
$89$	\( T^{4} \) T^4
$97$	\( (T^{2} + 4)^{2} \) (T^2 + 4)^2
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\(n\)	\(1001\)	\(1777\)	\(1851\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1\)

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