LMFDB - Newform orbit 2200.1.fr.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2200,1,Mod(107,2200)] 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("2200.107"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2200, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 10, 5, 6])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 2200 = 2^{3} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2200.fr (of order $20$, degree $8$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,-20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	no
Analytic conductor:	$1.09794302779$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{20})$
Coefficient field:	$\Q(\zeta_{40})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{10}$
Projective field:	Galois closure of 10.0.30181730444800000.3

$q$-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_{40}^{9} q^{2} + ( - \zeta_{40}^{11} + \zeta_{40}^{3}) q^{3} + \zeta_{40}^{18} q^{4} + ( - \zeta_{40}^{12} - 1) q^{6} + \zeta_{40}^{7} q^{8} + ( - \zeta_{40}^{14} + \cdots - \zeta_{40}^{2}) q^{9} + \cdots + ( - \zeta_{40}^{18} + \cdots - \zeta_{40}^{6}) q^{99} +O(q^{100}) $ q - z^9 * q^2 + (-z^11 + z^3) * q^3 + z^18 * q^4 + (-z^12 - 1) * q^6 + z^7 * q^8 + (-z^14 + z^6 - z^2) * q^9 - z^12 * q^11 + (z^9 - z) * q^12 - z^16 * q^16 + (z^13 - z^9) * q^17 + (-z^15 + z^11 - z^3) * q^18 + (-z^14 - z^10) * q^19 - z * q^22 + (-z^18 + z^10) * q^24 + (-z^17 + z^13 + z^9 - z^5) * q^27 - z^5 * q^32 + (-z^15 - z^3) * q^33 + (z^18 + z^2) * q^34 + (z^12 - z^4 + 1) * q^36 + (z^19 - z^3) * q^38 + (-z^16 + z^8) * q^41 + (-z^19 - z^11) * q^43 + z^10 * q^44 + (-z^19 - z^7) * q^48 - z^6 * q^49 + (z^16 - z^12 + z^4 - 1) * q^51 + (-z^18 + z^14 - z^6 + z^2) * q^54 + (-z^17 - z^13 - z^5 - z) * q^57 + (-z^10 + z^6) * q^59 + z^14 * q^64 + (z^12 - z^4) * q^66 + (z^17 + z^13) * q^67 + (-z^11 + z^7) * q^68 + (z^13 - z^9 + z) * q^72 + (z^19 - z^7) * q^73 + (z^12 + z^8) * q^76 + (z^16 + z^12 - z^8 + z^4 + 1) * q^81 + (-z^17 - z^5) * q^82 + (z^15 + z^7) * q^83 + (-z^8 - 1) * q^86 - z^19 * q^88 + (z^14 + z^6) * q^89 + (z^16 - z^8) * q^96 + (-z^13 + z^5) * q^97 + z^15 * q^98 + (-z^18 + z^14 - z^6) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 20 q^{6} - 4 q^{11} + 4 q^{16} + 16 q^{36} - 20 q^{51} + 24 q^{81} - 12 q^{86}+O(q^{100}) $ 16 * q - 20 * q^6 - 4 * q^11 + 4 * q^16 + 16 * q^36 - 20 * q^51 + 24 * q^81 - 12 * q^86

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2200\mathbb{Z}\right)^\times$.

$n$	$177$	$551$	$1101$	$1201$
$\chi(n)$	$-\zeta_{40}^{10}$	$-1$	$-1$	$-\zeta_{40}^{8}$

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} $

107.1

0.987688	−	0.156434i
−0.987688	+	0.156434i
0.987688	+	0.156434i
−0.987688	−	0.156434i
0.156434	−	0.987688i
−0.156434	+	0.987688i
−0.453990	+	0.891007i
0.453990	−	0.891007i
−0.891007	−	0.453990i
0.891007	+	0.453990i
−0.891007	+	0.453990i
0.891007	−	0.453990i
0.156434	+	0.987688i
−0.156434	−	0.987688i
−0.453990	−	0.891007i
0.453990	+	0.891007i

−0.156434

0.987688i

1.04744

0.533698i

−0.951057

−

0.309017i

−0.690983

0.951057i

0.453990

−

0.891007i

0.224514

0.309017i

107.2

0.156434

−

0.987688i

−1.04744

−

0.533698i

−0.951057

−

0.309017i

−0.690983

0.951057i

−0.453990

0.891007i

0.224514

0.309017i

843.1

−0.156434

−

0.987688i

1.04744

−

0.533698i

−0.951057

0.309017i

−0.690983

−

0.951057i

0.453990

0.891007i

0.224514

−

0.309017i

843.2

0.156434

0.987688i

−1.04744

0.533698i

−0.951057

0.309017i

−0.690983

−

0.951057i

−0.453990

−

0.891007i

0.224514

−

0.309017i

1107.1

−0.987688

0.156434i

0.533698

1.04744i

0.951057

−

0.309017i

−0.690983

−

0.951057i

−0.891007

0.453990i

−0.224514

0.309017i

1107.2

0.987688

−

0.156434i

−0.533698

−

1.04744i

0.951057

−

0.309017i

−0.690983

−

0.951057i

0.891007

−

0.453990i

−0.224514

0.309017i

1443.1

−0.891007

0.453990i

1.87869

0.297556i

0.587785

−

0.809017i

−1.80902

0.587785i

−0.156434

0.987688i

2.48990

0.809017i

1443.2

0.891007

−

0.453990i

−1.87869

−

0.297556i

0.587785

−

0.809017i

−1.80902

0.587785i

0.156434

−

0.987688i

2.48990

0.809017i

1707.1

−0.453990

−

0.891007i

0.297556

−

1.87869i

−0.587785

0.809017i

−1.80902

0.587785i

0.987688

0.156434i

−2.48990

−

0.809017i

1707.2

0.453990

0.891007i

−0.297556

1.87869i

−0.587785

0.809017i

−1.80902

0.587785i

−0.987688

−

0.156434i

−2.48990

−

0.809017i

1843.1

−0.453990

0.891007i

0.297556

1.87869i

−0.587785

−

0.809017i

−1.80902

−

0.587785i

0.987688

−

0.156434i

−2.48990

0.809017i

1843.2

0.453990

−

0.891007i

−0.297556

−

1.87869i

−0.587785

−

0.809017i

−1.80902

−

0.587785i

−0.987688

0.156434i

−2.48990

0.809017i

2043.1

−0.987688

−

0.156434i

0.533698

−

1.04744i

0.951057

0.309017i

−0.690983

0.951057i

−0.891007

−

0.453990i

−0.224514

−

0.309017i

2043.2

0.987688

0.156434i

−0.533698

1.04744i

0.951057

0.309017i

−0.690983

0.951057i

0.891007

0.453990i

−0.224514

−

0.309017i

2107.1

−0.891007

−

0.453990i

1.87869

−

0.297556i

0.587785

0.809017i

−1.80902

−

0.587785i

−0.156434

−

0.987688i

2.48990

−

0.809017i

2107.2

0.891007

0.453990i

−1.87869

0.297556i

0.587785

0.809017i

−1.80902

−

0.587785i

0.156434

0.987688i

2.48990

−

0.809017i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
5.b	even	2	1	inner
5.c	odd	4	2	inner
11.d	odd	10	1	inner
40.e	odd	2	1	inner
40.k	even	4	2	inner
55.h	odd	10	1	inner
55.l	even	20	2	inner
88.k	even	10	1	inner
440.bm	even	10	1	inner
440.br	odd	20	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2200.1.fr.a	✓	16
5.b	even	2	1	inner	2200.1.fr.a	✓	16
5.c	odd	4	2	inner	2200.1.fr.a	✓	16
8.d	odd	2	1	CM	2200.1.fr.a	✓	16
11.d	odd	10	1	inner	2200.1.fr.a	✓	16
40.e	odd	2	1	inner	2200.1.fr.a	✓	16
40.k	even	4	2	inner	2200.1.fr.a	✓	16
55.h	odd	10	1	inner	2200.1.fr.a	✓	16
55.l	even	20	2	inner	2200.1.fr.a	✓	16
88.k	even	10	1	inner	2200.1.fr.a	✓	16
440.bm	even	10	1	inner	2200.1.fr.a	✓	16
440.br	odd	20	2	inner	2200.1.fr.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2200.1.fr.a	✓	16	1.a	even	1	1	trivial
2200.1.fr.a	✓	16	5.b	even	2	1	inner
2200.1.fr.a	✓	16	5.c	odd	4	2	inner
2200.1.fr.a	✓	16	8.d	odd	2	1	CM
2200.1.fr.a	✓	16	11.d	odd	10	1	inner
2200.1.fr.a	✓	16	40.e	odd	2	1	inner
2200.1.fr.a	✓	16	40.k	even	4	2	inner
2200.1.fr.a	✓	16	55.h	odd	10	1	inner
2200.1.fr.a	✓	16	55.l	even	20	2	inner
2200.1.fr.a	✓	16	88.k	even	10	1	inner
2200.1.fr.a	✓	16	440.bm	even	10	1	inner
2200.1.fr.a	✓	16	440.br	odd	20	2	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{16} - 20T_{3}^{12} + 150T_{3}^{8} + 125T_{3}^{4} + 625 $ T3^16 - 20*T3^12 + 150*T3^8 + 125*T3^4 + 625 acting on $S_{1}^{\mathrm{new}}(2200, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} - T^{12} + \cdots + 1 $ T^16 - T^12 + T^8 - T^4 + 1
$3$	$ T^{16} - 20 T^{12} + \cdots + 625 $ T^16 - 20T^12 + 150T^8 + 125*T^4 + 625
$5$	$ T^{16} $ T^16
$7$	$ T^{16} $ T^16
$11$	$ (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} $ (T^4 + T^3 + T^2 + T + 1)^4
$13$	$ T^{16} $ T^16
$17$	$ T^{16} + 4 T^{12} + \cdots + 1 $ T^16 + 4T^12 + 46T^8 - 11*T^4 + 1
$19$	$ (T^{8} + 5 T^{6} + 10 T^{4} + 25)^{2} $ (T^8 + 5T^6 + 10T^4 + 25)^2
$23$	$ T^{16} $ T^16
$29$	$ T^{16} $ T^16
$31$	$ T^{16} $ T^16
$37$	$ T^{16} $ T^16
$41$	$ (T^{4} - 5 T + 5)^{4} $ (T^4 - 5*T + 5)^4
$43$	$ (T^{8} + 7 T^{4} + 1)^{2} $ (T^8 + 7*T^4 + 1)^2
$47$	$ T^{16} $ T^16
$53$	$ T^{16} $ T^16
$59$	$ (T^{8} + T^{6} + 6 T^{4} + \cdots + 1)^{2} $ (T^8 + T^6 + 6T^4 - 4T^2 + 1)^2
$61$	$ T^{16} $ T^16
$67$	$ (T^{8} + 15 T^{4} + 25)^{2} $ (T^8 + 15*T^4 + 25)^2
$71$	$ T^{16} $ T^16
$73$	$ T^{16} + 4 T^{12} + \cdots + 1 $ T^16 + 4T^12 + 46T^8 - 11*T^4 + 1
$79$	$ T^{16} $ T^16
$83$	$ T^{16} - 11 T^{12} + \cdots + 1 $ T^16 - 11T^12 + 46T^8 + 4*T^4 + 1
$89$	$ (T^{4} + 3 T^{2} + 1)^{4} $ (T^4 + 3*T^2 + 1)^4
$97$	$ T^{16} + 5 T^{12} + \cdots + 625 $ T^16 + 5T^12 + 150T^8 - 500*T^4 + 625
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Higher genus families
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Belyi maps

Level:	\( N \)	\(=\)	\( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2200.fr (of order \(20\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{40}^{9} q^{2} + ( - \zeta_{40}^{11} + \zeta_{40}^{3}) q^{3} + \zeta_{40}^{18} q^{4} + ( - \zeta_{40}^{12} - 1) q^{6} + \zeta_{40}^{7} q^{8} + ( - \zeta_{40}^{14} + \cdots - \zeta_{40}^{2}) q^{9} + \cdots + ( - \zeta_{40}^{18} + \cdots - \zeta_{40}^{6}) q^{99} +O(q^{100}) \) q - z^9 * q^2 + (-z^11 + z^3) * q^3 + z^18 * q^4 + (-z^12 - 1) * q^6 + z^7 * q^8 + (-z^14 + z^6 - z^2) * q^9 - z^12 * q^11 + (z^9 - z) * q^12 - z^16 * q^16 + (z^13 - z^9) * q^17 + (-z^15 + z^11 - z^3) * q^18 + (-z^14 - z^10) * q^19 - z * q^22 + (-z^18 + z^10) * q^24 + (-z^17 + z^13 + z^9 - z^5) * q^27 - z^5 * q^32 + (-z^15 - z^3) * q^33 + (z^18 + z^2) * q^34 + (z^12 - z^4 + 1) * q^36 + (z^19 - z^3) * q^38 + (-z^16 + z^8) * q^41 + (-z^19 - z^11) * q^43 + z^10 * q^44 + (-z^19 - z^7) * q^48 - z^6 * q^49 + (z^16 - z^12 + z^4 - 1) * q^51 + (-z^18 + z^14 - z^6 + z^2) * q^54 + (-z^17 - z^13 - z^5 - z) * q^57 + (-z^10 + z^6) * q^59 + z^14 * q^64 + (z^12 - z^4) * q^66 + (z^17 + z^13) * q^67 + (-z^11 + z^7) * q^68 + (z^13 - z^9 + z) * q^72 + (z^19 - z^7) * q^73 + (z^12 + z^8) * q^76 + (z^16 + z^12 - z^8 + z^4 + 1) * q^81 + (-z^17 - z^5) * q^82 + (z^15 + z^7) * q^83 + (-z^8 - 1) * q^86 - z^19 * q^88 + (z^14 + z^6) * q^89 + (z^16 - z^8) * q^96 + (-z^13 + z^5) * q^97 + z^15 * q^98 + (-z^18 + z^14 - z^6) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 20 q^{6} - 4 q^{11} + 4 q^{16} + 16 q^{36} - 20 q^{51} + 24 q^{81} - 12 q^{86}+O(q^{100}) \) 16 * q - 20 * q^6 - 4 * q^11 + 4 * q^16 + 16 * q^36 - 20 * q^51 + 24 * q^81 - 12 * q^86

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	2200.1.fr.a

Level	$2200$
Weight	$1$
Character orbit	2200.fr
Analytic conductor	$1.098$
Analytic rank	$0$
Dimension	$16$
Projective image	$D_{10}$
CM discriminant	-8
Inner twists	$16$

Self dual:	no
Analytic conductor:	\(1.09794302779\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(2\) over \(\Q(\zeta_{20})\)
Coefficient field:	\(\Q(\zeta_{40})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{10}\)
Projective field:	Galois closure of 10.0.30181730444800000.3

\(n\)	\(177\)	\(551\)	\(1101\)	\(1201\)
\(\chi(n)\)	\(-\zeta_{40}^{10}\)	\(-1\)	\(-1\)	\(-\zeta_{40}^{8}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 107.2
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
5.b	even	2	1	inner
5.c	odd	4	2	inner
11.d	odd	10	1	inner
40.e	odd	2	1	inner
40.k	even	4	2	inner
55.h	odd	10	1	inner
55.l	even	20	2	inner
88.k	even	10	1	inner
440.bm	even	10	1	inner
440.br	odd	20	2	inner

$p$	$F_p(T)$
$2$	\( T^{16} - T^{12} + \cdots + 1 \) T^16 - T^12 + T^8 - T^4 + 1
$3$	\( T^{16} - 20 T^{12} + \cdots + 625 \) T^16 - 20T^12 + 150T^8 + 125*T^4 + 625
$5$	\( T^{16} \) T^16
$7$	\( T^{16} \) T^16
$11$	\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} \) (T^4 + T^3 + T^2 + T + 1)^4
$13$	\( T^{16} \) T^16
$17$	\( T^{16} + 4 T^{12} + \cdots + 1 \) T^16 + 4T^12 + 46T^8 - 11*T^4 + 1
$19$	\( (T^{8} + 5 T^{6} + 10 T^{4} + 25)^{2} \) (T^8 + 5T^6 + 10T^4 + 25)^2
$23$	\( T^{16} \) T^16
$29$	\( T^{16} \) T^16
$31$	\( T^{16} \) T^16
$37$	\( T^{16} \) T^16
$41$	\( (T^{4} - 5 T + 5)^{4} \) (T^4 - 5*T + 5)^4
$43$	\( (T^{8} + 7 T^{4} + 1)^{2} \) (T^8 + 7*T^4 + 1)^2
$47$	\( T^{16} \) T^16
$53$	\( T^{16} \) T^16
$59$	\( (T^{8} + T^{6} + 6 T^{4} + \cdots + 1)^{2} \) (T^8 + T^6 + 6T^4 - 4T^2 + 1)^2
$61$	\( T^{16} \) T^16
$67$	\( (T^{8} + 15 T^{4} + 25)^{2} \) (T^8 + 15*T^4 + 25)^2
$71$	\( T^{16} \) T^16
$73$	\( T^{16} + 4 T^{12} + \cdots + 1 \) T^16 + 4T^12 + 46T^8 - 11*T^4 + 1
$79$	\( T^{16} \) T^16
$83$	\( T^{16} - 11 T^{12} + \cdots + 1 \) T^16 - 11T^12 + 46T^8 + 4*T^4 + 1
$89$	\( (T^{4} + 3 T^{2} + 1)^{4} \) (T^4 + 3*T^2 + 1)^4
$97$	\( T^{16} + 5 T^{12} + \cdots + 625 \) T^16 + 5T^12 + 150T^8 - 500*T^4 + 625
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