LMFDB - Newform orbit 4000.2.c.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4000,2,Mod(1249,4000)] 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("4000.1249"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4000, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4000 = 2^{5} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4000.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

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

Self dual:	no
Analytic conductor:	$31.9401608085$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.268960000.3
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 5x^{6} + 13x^{4} - 20x^{2} + 16 $ x^8 - 5x^6 + 13x^4 - 20*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta_{3} - \beta_1) q^{3} + \beta_{3} q^{7} + (\beta_{2} + 2) q^{9} - \beta_{4} q^{11} - \beta_{6} q^{13} - \beta_{5} q^{17} - \beta_{7} q^{19} - q^{21} + ( - \beta_{3} + 4 \beta_1) q^{23} + (4 \beta_{3} - 3 \beta_1) q^{27}+ \cdots + ( - \beta_{7} - 2 \beta_{4}) q^{99}+O(q^{100}) $ q + (b3 - b1) * q^3 + b3 * q^7 + (b2 + 2) * q^9 - b4 * q^11 - b6 * q^13 - b5 * q^17 - b7 * q^19 - q^21 + (-b3 + 4b1) q^23 + (4b3 - 3b1) * q^27 + (-5b2 + 2) q^29 + b4 * q^31 - b5 * q^33 + (b6 + b5) * q^37 + b7 * q^39 + (3b2 + 4) q^41 + (7b3 + b1) q^43 + (5b3 - 7b1) * q^47 + (-b2 + 5) * q^49 + (-b7 + b4) * q^51 + 2b6 q^53 + (-b6 + b5) * q^57 + (2b7 + b4) q^59 + (-3b2 + 3) q^61 + (2b3 + b1) q^63 + (-4b3 - 4b1) * q^67 + (-4b2 + 1) q^69 + b7 * q^71 + (b6 + 2b5) q^73 + (-b6 - b5) * q^77 + (-b7 - b4) * q^79 + (6b2 + 2) q^81 + (9b3 - 2b1) * q^83 + (7b3 - 12b1) * q^87 + (3b2 - 5) q^89 + (b7 + b4) * q^91 + b5 * q^93 + (b6 + 2b5) q^97 + (-b7 - 2b4) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{9} - 8 q^{21} + 36 q^{29} + 20 q^{41} + 44 q^{49} + 36 q^{61} + 24 q^{69} - 8 q^{81} - 52 q^{89}+O(q^{100}) $ 8 * q + 12 * q^9 - 8 * q^21 + 36 * q^29 + 20 * q^41 + 44 * q^49 + 36 * q^61 + 24 * q^69 - 8 * q^81 - 52 * q^89

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 5x^{6} + 13x^{4} - 20x^{2} + 16 $ x^8 - 5*x^6 + 13*x^4 - 20*x^2 + 16 :

$\beta_{1}$	$=$	$ ( \nu^{7} - \nu^{5} + \nu^{3} ) / 8 $ (v^7 - v^5 + v^3) / 8
$\beta_{2}$	$=$	$ ( -\nu^{6} + 5\nu^{4} - 9\nu^{2} + 8 ) / 4 $ (-v^6 + 5v^4 - 9v^2 + 8) / 4
$\beta_{3}$	$=$	$ ( \nu^{7} - 3\nu^{5} + 7\nu^{3} - 6\nu ) / 4 $ (v^7 - 3v^5 + 7v^3 - 6*v) / 4
$\beta_{4}$	$=$	$ ( -\nu^{7} + 5\nu^{5} - 13\nu^{3} + 28\nu ) / 4 $ (-v^7 + 5v^5 - 13v^3 + 28*v) / 4
$\beta_{5}$	$=$	$ ( \nu^{6} - 5\nu^{4} + 17\nu^{2} - 20 ) / 2 $ (v^6 - 5v^4 + 17v^2 - 20) / 2
$\beta_{6}$	$=$	$ ( 3\nu^{6} - 7\nu^{4} + 11\nu^{2} - 8 ) / 2 $ (3v^6 - 7v^4 + 11*v^2 - 8) / 2
$\beta_{7}$	$=$	$ -\nu^{7} + 4\nu^{5} - 6\nu^{3} + 5\nu $ -v^7 + 4v^5 - 6v^3 + 5*v

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( \beta_{4} + 2\beta_{3} - 2\beta_1 ) / 4 $ (b4 + 2b3 - 2b1) / 4
$\nu^{2}$	$=$	$ ( \beta_{5} + 2\beta_{2} + 6 ) / 4 $ (b5 + 2*b2 + 6) / 4
$\nu^{3}$	$=$	$ ( \beta_{7} + \beta_{4} + 8\beta_{3} - 6\beta_1 ) / 4 $ (b7 + b4 + 8b3 - 6b1) / 4
$\nu^{4}$	$=$	$ ( \beta_{6} + 2\beta_{5} + 10\beta_{2} + 4 ) / 4 $ (b6 + 2b5 + 10b2 + 4) / 4
$\nu^{5}$	$=$	$ ( 3\beta_{7} + 10\beta_{3} + 4\beta_1 ) / 4 $ (3b7 + 10b3 + 4*b1) / 4
$\nu^{6}$	$=$	$ ( 5\beta_{6} + \beta_{5} + 16\beta_{2} - 2 ) / 4 $ (5b6 + b5 + 16b2 - 2) / 4
$\nu^{7}$	$=$	$ ( 2\beta_{7} - \beta_{4} + 2\beta_{3} + 42\beta_1 ) / 4 $ (2b7 - b4 + 2b3 + 42*b1) / 4

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times$.

$n$	$1377$	$2501$	$2751$
$\chi(n)$	$-1$	$1$	$1$

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} $

1249.1

1.15995	−	0.809017i
−1.15995	−	0.809017i
1.38004	−	0.309017i
−1.38004	−	0.309017i
1.38004	+	0.309017i
−1.38004	+	0.309017i
1.15995	+	0.809017i
−1.15995	+	0.809017i

−

1.61803i

−

0.618034i

0.381966

1249.2

−

1.61803i

−

0.618034i

0.381966

1249.3

−

0.618034i

−

1.61803i

2.61803

1249.4

−

0.618034i

−

1.61803i

2.61803

1249.5

0.618034i

1.61803i

2.61803

1249.6

0.618034i

1.61803i

2.61803

1249.7

1.61803i

0.618034i

0.381966

1249.8

1.61803i

0.618034i

0.381966

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.b	even	2	1	inner
20.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4000.2.c.e		8
4.b	odd	2	1	inner	4000.2.c.e		8
5.b	even	2	1	inner	4000.2.c.e		8
5.c	odd	4	1		4000.2.a.c	✓	4
5.c	odd	4	1		4000.2.a.h	yes	4
20.d	odd	2	1	inner	4000.2.c.e		8
20.e	even	4	1		4000.2.a.c	✓	4
20.e	even	4	1		4000.2.a.h	yes	4
40.i	odd	4	1		8000.2.a.bc		4
40.i	odd	4	1		8000.2.a.bp		4
40.k	even	4	1		8000.2.a.bc		4
40.k	even	4	1		8000.2.a.bp		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4000.2.a.c	✓	4	5.c	odd	4	1
4000.2.a.c	✓	4	20.e	even	4	1
4000.2.a.h	yes	4	5.c	odd	4	1
4000.2.a.h	yes	4	20.e	even	4	1
4000.2.c.e		8	1.a	even	1	1	trivial
4000.2.c.e		8	4.b	odd	2	1	inner
4000.2.c.e		8	5.b	even	2	1	inner
4000.2.c.e		8	20.d	odd	2	1	inner
8000.2.a.bc		4	40.i	odd	4	1
8000.2.a.bc		4	40.k	even	4	1
8000.2.a.bp		4	40.i	odd	4	1
8000.2.a.bp		4	40.k	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(4000, [\chi])$:

$ T_{3}^{4} + 3T_{3}^{2} + 1 $

T3^4 + 3*T3^2 + 1

$ T_{7}^{4} + 3T_{7}^{2} + 1 $

T7^4 + 3*T7^2 + 1

$ T_{11}^{4} - 52T_{11}^{2} + 656 $

T11^4 - 52*T11^2 + 656

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ (T^{4} + 3 T^{2} + 1)^{2} $ (T^4 + 3*T^2 + 1)^2
$5$	$ T^{8} $ T^8
$7$	$ (T^{4} + 3 T^{2} + 1)^{2} $ (T^4 + 3*T^2 + 1)^2
$11$	$ (T^{4} - 52 T^{2} + 656)^{2} $ (T^4 - 52*T^2 + 656)^2
$13$	$ (T^{4} + 52 T^{2} + 656)^{2} $ (T^4 + 52*T^2 + 656)^2
$17$	$ (T^{4} + 68 T^{2} + 656)^{2} $ (T^4 + 68*T^2 + 656)^2
$19$	$ (T^{4} - 68 T^{2} + 656)^{2} $ (T^4 - 68*T^2 + 656)^2
$23$	$ (T^{4} + 27 T^{2} + 121)^{2} $ (T^4 + 27*T^2 + 121)^2
$29$	$ (T^{2} - 9 T - 11)^{4} $ (T^2 - 9*T - 11)^4
$31$	$ (T^{4} - 52 T^{2} + 656)^{2} $ (T^4 - 52*T^2 + 656)^2
$37$	$ (T^{4} + 88 T^{2} + 656)^{2} $ (T^4 + 88*T^2 + 656)^2
$41$	$ (T^{2} - 5 T - 5)^{4} $ (T^2 - 5*T - 5)^4
$43$	$ (T^{4} + 163 T^{2} + 1681)^{2} $ (T^4 + 163*T^2 + 1681)^2
$47$	$ (T^{4} + 103 T^{2} + 121)^{2} $ (T^4 + 103*T^2 + 121)^2
$53$	$ (T^{4} + 208 T^{2} + 10496)^{2} $ (T^4 + 208*T^2 + 10496)^2
$59$	$ (T^{4} - 260 T^{2} + 16400)^{2} $ (T^4 - 260*T^2 + 16400)^2
$61$	$ (T^{2} - 9 T + 9)^{4} $ (T^2 - 9*T + 9)^4
$67$	$ (T^{4} + 112 T^{2} + 256)^{2} $ (T^4 + 112*T^2 + 256)^2
$71$	$ (T^{4} - 68 T^{2} + 656)^{2} $ (T^4 - 68*T^2 + 656)^2
$73$	$ (T^{4} + 260 T^{2} + 16400)^{2} $ (T^4 + 260*T^2 + 16400)^2
$79$	$ (T^{4} - 88 T^{2} + 656)^{2} $ (T^4 - 88*T^2 + 656)^2
$83$	$ (T^{4} + 215 T^{2} + 9025)^{2} $ (T^4 + 215*T^2 + 9025)^2
$89$	$ (T^{2} + 13 T + 31)^{4} $ (T^2 + 13*T + 31)^4
$97$	$ (T^{4} + 260 T^{2} + 16400)^{2} $ (T^4 + 260*T^2 + 16400)^2
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Belyi maps

Level:	\( N \)	\(=\)	\( 4000 = 2^{5} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4000.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{3} - \beta_1) q^{3} + \beta_{3} q^{7} + (\beta_{2} + 2) q^{9} - \beta_{4} q^{11} - \beta_{6} q^{13} - \beta_{5} q^{17} - \beta_{7} q^{19} - q^{21} + ( - \beta_{3} + 4 \beta_1) q^{23} + (4 \beta_{3} - 3 \beta_1) q^{27}+ \cdots + ( - \beta_{7} - 2 \beta_{4}) q^{99}+O(q^{100}) \) q + (b3 - b1) * q^3 + b3 * q^7 + (b2 + 2) * q^9 - b4 * q^11 - b6 * q^13 - b5 * q^17 - b7 * q^19 - q^21 + (-b3 + 4b1) q^23 + (4b3 - 3b1) * q^27 + (-5b2 + 2) q^29 + b4 * q^31 - b5 * q^33 + (b6 + b5) * q^37 + b7 * q^39 + (3b2 + 4) q^41 + (7b3 + b1) q^43 + (5b3 - 7b1) * q^47 + (-b2 + 5) * q^49 + (-b7 + b4) * q^51 + 2b6 q^53 + (-b6 + b5) * q^57 + (2b7 + b4) q^59 + (-3b2 + 3) q^61 + (2b3 + b1) q^63 + (-4b3 - 4b1) * q^67 + (-4b2 + 1) q^69 + b7 * q^71 + (b6 + 2b5) q^73 + (-b6 - b5) * q^77 + (-b7 - b4) * q^79 + (6b2 + 2) q^81 + (9b3 - 2b1) * q^83 + (7b3 - 12b1) * q^87 + (3b2 - 5) q^89 + (b7 + b4) * q^91 + b5 * q^93 + (b6 + 2b5) q^97 + (-b7 - 2b4) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{9} - 8 q^{21} + 36 q^{29} + 20 q^{41} + 44 q^{49} + 36 q^{61} + 24 q^{69} - 8 q^{81} - 52 q^{89}+O(q^{100}) \) 8 * q + 12 * q^9 - 8 * q^21 + 36 * q^29 + 20 * q^41 + 44 * q^49 + 36 * q^61 + 24 * q^69 - 8 * q^81 - 52 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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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	4000.2.c.e

Level	$4000$
Weight	$2$
Character orbit	4000.c
Analytic conductor	$31.940$
Analytic rank	$0$
Dimension	$8$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(31.9401608085\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	8.0.268960000.3
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - 5x^{6} + 13x^{4} - 20x^{2} + 16 \) x^8 - 5x^6 + 13x^4 - 20*x^2 + 16
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 2^{8} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

\(\beta_{1}\)	\(=\)	\( ( \nu^{7} - \nu^{5} + \nu^{3} ) / 8 \) (v^7 - v^5 + v^3) / 8
\(\beta_{2}\)	\(=\)	\( ( -\nu^{6} + 5\nu^{4} - 9\nu^{2} + 8 ) / 4 \) (-v^6 + 5v^4 - 9v^2 + 8) / 4
\(\beta_{3}\)	\(=\)	\( ( \nu^{7} - 3\nu^{5} + 7\nu^{3} - 6\nu ) / 4 \) (v^7 - 3v^5 + 7v^3 - 6*v) / 4
\(\beta_{4}\)	\(=\)	\( ( -\nu^{7} + 5\nu^{5} - 13\nu^{3} + 28\nu ) / 4 \) (-v^7 + 5v^5 - 13v^3 + 28*v) / 4
\(\beta_{5}\)	\(=\)	\( ( \nu^{6} - 5\nu^{4} + 17\nu^{2} - 20 ) / 2 \) (v^6 - 5v^4 + 17v^2 - 20) / 2
\(\beta_{6}\)	\(=\)	\( ( 3\nu^{6} - 7\nu^{4} + 11\nu^{2} - 8 ) / 2 \) (3v^6 - 7v^4 + 11*v^2 - 8) / 2
\(\beta_{7}\)	\(=\)	\( -\nu^{7} + 4\nu^{5} - 6\nu^{3} + 5\nu \) -v^7 + 4v^5 - 6v^3 + 5*v

\(\nu\)	\(=\)	\( ( \beta_{4} + 2\beta_{3} - 2\beta_1 ) / 4 \) (b4 + 2b3 - 2b1) / 4
\(\nu^{2}\)	\(=\)	\( ( \beta_{5} + 2\beta_{2} + 6 ) / 4 \) (b5 + 2*b2 + 6) / 4
\(\nu^{3}\)	\(=\)	\( ( \beta_{7} + \beta_{4} + 8\beta_{3} - 6\beta_1 ) / 4 \) (b7 + b4 + 8b3 - 6b1) / 4
\(\nu^{4}\)	\(=\)	\( ( \beta_{6} + 2\beta_{5} + 10\beta_{2} + 4 ) / 4 \) (b6 + 2b5 + 10b2 + 4) / 4
\(\nu^{5}\)	\(=\)	\( ( 3\beta_{7} + 10\beta_{3} + 4\beta_1 ) / 4 \) (3b7 + 10b3 + 4*b1) / 4
\(\nu^{6}\)	\(=\)	\( ( 5\beta_{6} + \beta_{5} + 16\beta_{2} - 2 ) / 4 \) (5b6 + b5 + 16b2 - 2) / 4
\(\nu^{7}\)	\(=\)	\( ( 2\beta_{7} - \beta_{4} + 2\beta_{3} + 42\beta_1 ) / 4 \) (2b7 - b4 + 2b3 + 42*b1) / 4

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1249.8
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} \) T^8
$3$	\( (T^{4} + 3 T^{2} + 1)^{2} \) (T^4 + 3*T^2 + 1)^2
$5$	\( T^{8} \) T^8
$7$	\( (T^{4} + 3 T^{2} + 1)^{2} \) (T^4 + 3*T^2 + 1)^2
$11$	\( (T^{4} - 52 T^{2} + 656)^{2} \) (T^4 - 52*T^2 + 656)^2
$13$	\( (T^{4} + 52 T^{2} + 656)^{2} \) (T^4 + 52*T^2 + 656)^2
$17$	\( (T^{4} + 68 T^{2} + 656)^{2} \) (T^4 + 68*T^2 + 656)^2
$19$	\( (T^{4} - 68 T^{2} + 656)^{2} \) (T^4 - 68*T^2 + 656)^2
$23$	\( (T^{4} + 27 T^{2} + 121)^{2} \) (T^4 + 27*T^2 + 121)^2
$29$	\( (T^{2} - 9 T - 11)^{4} \) (T^2 - 9*T - 11)^4
$31$	\( (T^{4} - 52 T^{2} + 656)^{2} \) (T^4 - 52*T^2 + 656)^2
$37$	\( (T^{4} + 88 T^{2} + 656)^{2} \) (T^4 + 88*T^2 + 656)^2
$41$	\( (T^{2} - 5 T - 5)^{4} \) (T^2 - 5*T - 5)^4
$43$	\( (T^{4} + 163 T^{2} + 1681)^{2} \) (T^4 + 163*T^2 + 1681)^2
$47$	\( (T^{4} + 103 T^{2} + 121)^{2} \) (T^4 + 103*T^2 + 121)^2
$53$	\( (T^{4} + 208 T^{2} + 10496)^{2} \) (T^4 + 208*T^2 + 10496)^2
$59$	\( (T^{4} - 260 T^{2} + 16400)^{2} \) (T^4 - 260*T^2 + 16400)^2
$61$	\( (T^{2} - 9 T + 9)^{4} \) (T^2 - 9*T + 9)^4
$67$	\( (T^{4} + 112 T^{2} + 256)^{2} \) (T^4 + 112*T^2 + 256)^2
$71$	\( (T^{4} - 68 T^{2} + 656)^{2} \) (T^4 - 68*T^2 + 656)^2
$73$	\( (T^{4} + 260 T^{2} + 16400)^{2} \) (T^4 + 260*T^2 + 16400)^2
$79$	\( (T^{4} - 88 T^{2} + 656)^{2} \) (T^4 - 88*T^2 + 656)^2
$83$	\( (T^{4} + 215 T^{2} + 9025)^{2} \) (T^4 + 215*T^2 + 9025)^2
$89$	\( (T^{2} + 13 T + 31)^{4} \) (T^2 + 13*T + 31)^4
$97$	\( (T^{4} + 260 T^{2} + 16400)^{2} \) (T^4 + 260*T^2 + 16400)^2
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\(n\)	\(1377\)	\(2501\)	\(2751\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)