LMFDB - Newform orbit 260.2.c.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [260,2,Mod(209,260)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(260, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("260.209");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 260 = 2^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	260.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$2.07611045255$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
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 $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_{5} q^{3} + ( - \beta_{4} - \beta_1) q^{5} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{7} + ( - \beta_{2} + \beta_1) q^{9} + ( - 3 \beta_{2} - 1) q^{11} - \beta_{3} q^{13} + ( - \beta_{5} + \beta_{4} - 3 \beta_{3} + \cdots - 1) q^{15}+ \cdots + ( - 5 \beta_{2} + 2 \beta_1 + 3) q^{99}+O(q^{100}) $ q - b5 * q^3 + (-b4 - b1) * q^5 + (-b5 + b4 + b3) * q^7 + (-b2 + b1) * q^9 + (-3b2 - 1) q^11 - b3 * q^13 + (-b5 + b4 - 3b3 - b2 - 1) q^15 + (-2b4 + 4b3) * q^17 + (-b2 + 2b1 - 1) q^19 - 2 * q^21 + (-b5 - 2b3) q^23 + (2b4 - 2b3 + 2b2 - 2b1 + 1) * q^25 + (-2b5 + 2b3) * q^27 + (b2 + b1 + 1) * q^29 + (3b2 - 2b1 + 1) * q^31 + (b5 + 3b4 - 3b3) * q^33 + (-2b3 - 3b2 + b1 + 1) * q^35 + (3b5 + b4 + 5b3) * q^37 + b1 * q^39 + (4b2 + 4) q^41 + (3b5 - 6b4 + 4b3) q^43 + (b5 - 2b4 - b3 + b1 - 2) q^45 + (b5 - b4 + 7b3) q^47 + (4b2 - 2b1 + 3) * q^49 + (-2b2 - 4b1 - 2) * q^51 + (2b5 - 8b3) * q^53 + (3b5 - 2b4 - 6b3 + 3b2 + b1 + 3) * q^55 + (3b5 - b4 + 5b3) * q^57 + (-b2 - 4b1 - 1) q^59 + (5b2 + 3b1 - 3) * q^61 + (-b5 + 3b4 + 3b3) * q^63 + (-b5 + b2) * q^65 + (b5 + 3b4 - b3) q^67 + (-b2 + 3b1 - 3) q^69 + (-b2 + 4b1 + 3) q^71 + (3b5 - 3b4 - 3b3) q^73 + (-3b5 - 4b3 + 2b2 + 2b1 + 2) * q^75 + (-2b5 + 8b4 + 2b3) q^77 + (4b2 - 2b1) * q^79 + (-5b2 + 3b1 - 6) * q^81 + (-5b5 + 7b4 - b3) * q^83 + (4b5 + 2b4 - 2b3 - 2b2 - 2b1 - 4) q^85 + (-2b4 + 4b3) * q^87 + (4b2 - 8b1 - 2) * q^89 + (-b2 + b1 + 1) * q^91 + (-3b5 - b4 - 3b3) * q^93 + (b5 - 2b4 - b2 + 3b1 - 5) * q^95 + (2b5 - 8b4 + 6b3) q^97 + (-5b2 + 2b1 + 3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{9} - 4 q^{15} - 4 q^{19} - 12 q^{21} + 2 q^{25} + 4 q^{29} + 12 q^{35} + 16 q^{41} - 12 q^{45} + 10 q^{49} - 8 q^{51} + 12 q^{55} - 4 q^{59} - 28 q^{61} - 2 q^{65} - 16 q^{69} + 20 q^{71} + 8 q^{75}+ \cdots + 28 q^{99}+O(q^{100}) $ 6 * q + 2 * q^9 - 4 * q^15 - 4 * q^19 - 12 * q^21 + 2 * q^25 + 4 * q^29 + 12 * q^35 + 16 * q^41 - 12 * q^45 + 10 * q^49 - 8 * q^51 + 12 * q^55 - 4 * q^59 - 28 * q^61 - 2 * q^65 - 16 * q^69 + 20 * q^71 + 8 * q^75 - 8 * q^79 - 26 * q^81 - 20 * q^85 - 20 * q^89 + 8 * q^91 - 28 * q^95 + 28 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2*x^5 + 2*x^4 + 2*x^3 + 4*x^2 - 4*x + 2 :

$\beta_{1}$	$=$	$ ( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23 $ (-v^5 + 8v^4 - 4v^3 - v^2 + 2*v + 38) / 23
$\beta_{2}$	$=$	$ ( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23 $ (-5v^5 + 17v^4 - 20v^3 - 5v^2 + 10*v + 29) / 23
$\beta_{3}$	$=$	$ ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 $ (7v^5 - 10v^4 + 5v^3 + 30v^2 + 32*v - 13) / 23
$\beta_{4}$	$=$	$ ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 $ (-11v^5 + 19v^4 - 21v^3 - 11v^2 - 70*v + 27) / 23
$\beta_{5}$	$=$	$ ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23 $ (-14v^5 + 20v^4 - 10v^3 - 37v^2 - 64*v + 26) / 23

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

$\nu$	$=$	$ ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 $ (b5 - b4 + b3 + b2 - b1 + 1) / 2
$\nu^{2}$	$=$	$ \beta_{5} + 2\beta_{3} $ b5 + 2*b3
$\nu^{3}$	$=$	$ 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2 $ 2b5 - b4 + 2b3 - b2 + 2*b1 - 2
$\nu^{4}$	$=$	$ -\beta_{2} + 5\beta _1 - 7 $ -b2 + 5*b1 - 7
$\nu^{5}$	$=$	$ -8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9 $ -8b5 + 3b4 - 9b3 - 3b2 + 8*b1 - 9

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/260\mathbb{Z}\right)^\times$.

$n$	$41$	$131$	$157$
$\chi(n)$	$1$	$1$	$-1$

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} $

209.1

1.45161	+	1.45161i
0.403032	−	0.403032i
−0.854638	+	0.854638i
−0.854638	−	0.854638i
0.403032	+	0.403032i
1.45161	−	1.45161i

−

2.21432i

2.21432

−

0.311108i

−

0.903212i

−1.90321

209.2

−

1.67513i

−1.67513

−

1.48119i

−

1.19394i

0.193937

209.3

−

0.539189i

−0.539189

2.17009i

−

3.70928i

2.70928

209.4

0.539189i

−0.539189

−

2.17009i

3.70928i

2.70928

209.5

1.67513i

−1.67513

1.48119i

1.19394i

0.193937

209.6

2.21432i

2.21432

0.311108i

0.903212i

−1.90321

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	260.2.c.a	✓	6
3.b	odd	2	1		2340.2.h.e		6
4.b	odd	2	1		1040.2.d.d		6
5.b	even	2	1	inner	260.2.c.a	✓	6
5.c	odd	4	1		1300.2.a.j		3
5.c	odd	4	1		1300.2.a.k		3
13.b	even	2	1		3380.2.c.c		6
13.d	odd	4	1		3380.2.d.a		6
13.d	odd	4	1		3380.2.d.b		6
15.d	odd	2	1		2340.2.h.e		6
20.d	odd	2	1		1040.2.d.d		6
20.e	even	4	1		5200.2.a.cd		3
20.e	even	4	1		5200.2.a.cg		3
65.d	even	2	1		3380.2.c.c		6
65.g	odd	4	1		3380.2.d.a		6
65.g	odd	4	1		3380.2.d.b		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
260.2.c.a	✓	6	1.a	even	1	1	trivial
260.2.c.a	✓	6	5.b	even	2	1	inner
1040.2.d.d		6	4.b	odd	2	1
1040.2.d.d		6	20.d	odd	2	1
1300.2.a.j		3	5.c	odd	4	1
1300.2.a.k		3	5.c	odd	4	1
2340.2.h.e		6	3.b	odd	2	1
2340.2.h.e		6	15.d	odd	2	1
3380.2.c.c		6	13.b	even	2	1
3380.2.c.c		6	65.d	even	2	1
3380.2.d.a		6	13.d	odd	4	1
3380.2.d.a		6	65.g	odd	4	1
3380.2.d.b		6	13.d	odd	4	1
3380.2.d.b		6	65.g	odd	4	1
5200.2.a.cd		3	20.e	even	4	1
5200.2.a.cg		3	20.e	even	4	1

Hecke kernels

This newform subspace is the entire newspace $S_{2}^{\mathrm{new}}(260, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} + 8 T^{4} + \cdots + 4 $ T^6 + 8T^4 + 16T^2 + 4
$5$	$ T^{6} - T^{4} + \cdots + 125 $ T^6 - T^4 - 16T^3 - 5T^2 + 125
$7$	$ T^{6} + 16 T^{4} + \cdots + 16 $ T^6 + 16T^4 + 32T^2 + 16
$11$	$ (T^{3} - 30 T - 2)^{2} $ (T^3 - 30*T - 2)^2
$13$	$ (T^{2} + 1)^{3} $ (T^2 + 1)^3
$17$	$ T^{6} + 60 T^{4} + \cdots + 64 $ T^6 + 60T^4 + 560T^2 + 64
$19$	$ (T^{3} + 2 T^{2} - 14 T + 10)^{2} $ (T^3 + 2T^2 - 14T + 10)^2
$23$	$ T^{6} + 20 T^{4} + \cdots + 4 $ T^6 + 20T^4 + 40T^2 + 4
$29$	$ (T^{3} - 2 T^{2} - 8 T - 4)^{2} $ (T^3 - 2T^2 - 8T - 4)^2
$31$	$ (T^{3} - 34 T + 62)^{2} $ (T^3 - 34*T + 62)^2
$37$	$ T^{6} + 176 T^{4} + \cdots + 10000 $ T^6 + 176T^4 + 4800T^2 + 10000
$41$	$ (T^{3} - 8 T^{2} + \cdots + 128)^{2} $ (T^3 - 8T^2 - 32T + 128)^2
$43$	$ T^{6} + 252 T^{4} + \cdots + 538756 $ T^6 + 252T^4 + 20472T^2 + 538756
$47$	$ T^{6} + 144 T^{4} + \cdots + 67600 $ T^6 + 144T^4 + 5984T^2 + 67600
$53$	$ T^{6} + 224 T^{4} + \cdots + 135424 $ T^6 + 224T^4 + 13312T^2 + 135424
$59$	$ (T^{3} + 2 T^{2} - 74 T - 74)^{2} $ (T^3 + 2T^2 - 74T - 74)^2
$61$	$ (T^{3} + 14 T^{2} + \cdots - 1292)^{2} $ (T^3 + 14T^2 - 84T - 1292)^2
$67$	$ T^{6} + 80 T^{4} + \cdots + 5776 $ T^6 + 80T^4 + 1600T^2 + 5776
$71$	$ (T^{3} - 10 T^{2} + \cdots + 334)^{2} $ (T^3 - 10T^2 - 26T + 334)^2
$73$	$ T^{6} + 144 T^{4} + \cdots + 11664 $ T^6 + 144T^4 + 2592T^2 + 11664
$79$	$ (T^{3} + 4 T^{2} - 48 T + 80)^{2} $ (T^3 + 4T^2 - 48T + 80)^2
$83$	$ T^{6} + 392 T^{4} + \cdots + 250000 $ T^6 + 392T^4 + 31344T^2 + 250000
$89$	$ (T^{3} + 10 T^{2} + \cdots - 2056)^{2} $ (T^3 + 10T^2 - 212T - 2056)^2
$97$	$ T^{6} + 428 T^{4} + \cdots + 2050624 $ T^6 + 428T^4 + 55536T^2 + 2050624
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	260.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{5} q^{3} + ( - \beta_{4} - \beta_1) q^{5} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{7} + ( - \beta_{2} + \beta_1) q^{9} + ( - 3 \beta_{2} - 1) q^{11} - \beta_{3} q^{13} + ( - \beta_{5} + \beta_{4} - 3 \beta_{3} + \cdots - 1) q^{15}+ \cdots + ( - 5 \beta_{2} + 2 \beta_1 + 3) q^{99}+O(q^{100}) \) q - b5 * q^3 + (-b4 - b1) * q^5 + (-b5 + b4 + b3) * q^7 + (-b2 + b1) * q^9 + (-3b2 - 1) q^11 - b3 * q^13 + (-b5 + b4 - 3b3 - b2 - 1) q^15 + (-2b4 + 4b3) * q^17 + (-b2 + 2b1 - 1) q^19 - 2 * q^21 + (-b5 - 2b3) q^23 + (2b4 - 2b3 + 2b2 - 2b1 + 1) * q^25 + (-2b5 + 2b3) * q^27 + (b2 + b1 + 1) * q^29 + (3b2 - 2b1 + 1) * q^31 + (b5 + 3b4 - 3b3) * q^33 + (-2b3 - 3b2 + b1 + 1) * q^35 + (3b5 + b4 + 5b3) * q^37 + b1 * q^39 + (4b2 + 4) q^41 + (3b5 - 6b4 + 4b3) q^43 + (b5 - 2b4 - b3 + b1 - 2) q^45 + (b5 - b4 + 7b3) q^47 + (4b2 - 2b1 + 3) * q^49 + (-2b2 - 4b1 - 2) * q^51 + (2b5 - 8b3) * q^53 + (3b5 - 2b4 - 6b3 + 3b2 + b1 + 3) * q^55 + (3b5 - b4 + 5b3) * q^57 + (-b2 - 4b1 - 1) q^59 + (5b2 + 3b1 - 3) * q^61 + (-b5 + 3b4 + 3b3) * q^63 + (-b5 + b2) * q^65 + (b5 + 3b4 - b3) q^67 + (-b2 + 3b1 - 3) q^69 + (-b2 + 4b1 + 3) q^71 + (3b5 - 3b4 - 3b3) q^73 + (-3b5 - 4b3 + 2b2 + 2b1 + 2) * q^75 + (-2b5 + 8b4 + 2b3) q^77 + (4b2 - 2b1) * q^79 + (-5b2 + 3b1 - 6) * q^81 + (-5b5 + 7b4 - b3) * q^83 + (4b5 + 2b4 - 2b3 - 2b2 - 2b1 - 4) q^85 + (-2b4 + 4b3) * q^87 + (4b2 - 8b1 - 2) * q^89 + (-b2 + b1 + 1) * q^91 + (-3b5 - b4 - 3b3) * q^93 + (b5 - 2b4 - b2 + 3b1 - 5) * q^95 + (2b5 - 8b4 + 6b3) q^97 + (-5b2 + 2b1 + 3) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{9} - 4 q^{15} - 4 q^{19} - 12 q^{21} + 2 q^{25} + 4 q^{29} + 12 q^{35} + 16 q^{41} - 12 q^{45} + 10 q^{49} - 8 q^{51} + 12 q^{55} - 4 q^{59} - 28 q^{61} - 2 q^{65} - 16 q^{69} + 20 q^{71} + 8 q^{75}+ \cdots + 28 q^{99}+O(q^{100}) \) 6 * q + 2 * q^9 - 4 * q^15 - 4 * q^19 - 12 * q^21 + 2 * q^25 + 4 * q^29 + 12 * q^35 + 16 * q^41 - 12 * q^45 + 10 * q^49 - 8 * q^51 + 12 * q^55 - 4 * q^59 - 28 * q^61 - 2 * q^65 - 16 * q^69 + 20 * q^71 + 8 * q^75 - 8 * q^79 - 26 * q^81 - 20 * q^85 - 20 * q^89 + 8 * q^91 - 28 * q^95 + 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more