LMFDB - Newform orbit 6336.2.f.k

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6336,2,Mod(3169,6336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6336, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6336.3169");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6336 = 2^{6} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6336.f (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:	$50.5932147207$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.303595776.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 $ x^8 + 5x^6 + 16x^4 + 45*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 2112)
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_{7} q^{7}+O(q^{10}) $ q + b7 * q^7	$ q + \beta_{7} q^{7} - \beta_{2} q^{11} + \beta_{3} q^{13} + ( - \beta_{5} + 1) q^{17} + ( - \beta_{6} - \beta_{2}) q^{19} + (2 \beta_{7} - \beta_{4}) q^{23} + 5 q^{25} + ( - \beta_{3} - \beta_1) q^{29} - \beta_{4} q^{31} + (2 \beta_{3} - 2 \beta_1) q^{37} + (\beta_{5} - 1) q^{41} + (\beta_{6} + 5 \beta_{2}) q^{43} + ( - 2 \beta_{7} + 3 \beta_{4}) q^{47} + ( - 2 \beta_{5} + 7) q^{49} + ( - 2 \beta_{3} + 2 \beta_1) q^{53} - 4 \beta_{2} q^{59} + (\beta_{3} + 2 \beta_1) q^{61} - 4 \beta_{2} q^{67} + ( - 2 \beta_{7} + 3 \beta_{4}) q^{71} + 6 q^{73} - \beta_1 q^{77} + ( - 3 \beta_{7} + 2 \beta_{4}) q^{79} + ( - 2 \beta_{6} + 6 \beta_{2}) q^{83} - 10 q^{89} + ( - 2 \beta_{6} + 6 \beta_{2}) q^{91} + (2 \beta_{5} - 4) q^{97}+O(q^{100}) $ q + b7 * q^7 - b2 * q^11 + b3 * q^13 + (-b5 + 1) * q^17 + (-b6 - b2) * q^19 + (2b7 - b4) q^23 + 5 * q^25 + (-b3 - b1) * q^29 - b4 * q^31 + (2b3 - 2b1) * q^37 + (b5 - 1) * q^41 + (b6 + 5b2) q^43 + (-2b7 + 3b4) * q^47 + (-2b5 + 7) q^49 + (-2b3 + 2b1) * q^53 - 4b2 q^59 + (b3 + 2b1) q^61 - 4b2 q^67 + (-2b7 + 3b4) * q^71 + 6 * q^73 - b1 * q^77 + (-3b7 + 2b4) * q^79 + (-2b6 + 6b2) * q^83 - 10 * q^89 + (-2b6 + 6b2) * q^91 + (2b5 - 4) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 8 q^{17} + 40 q^{25} - 8 q^{41} + 56 q^{49} + 48 q^{73} - 80 q^{89} - 32 q^{97}+O(q^{100}) $ 8 * q + 8 * q^17 + 40 * q^25 - 8 * q^41 + 56 * q^49 + 48 * q^73 - 80 * q^89 - 32 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 $ x^8 + 5*x^6 + 16*x^4 + 45*x^2 + 81 :

$\beta_{1}$	$=$	$ ( -\nu^{6} + 4\nu^{4} + 20\nu^{2} + 27 ) / 18 $ (-v^6 + 4v^4 + 20v^2 + 27) / 18
$\beta_{2}$	$=$	$ ( 5\nu^{7} + 16\nu^{5} + 8\nu^{3} + 81\nu ) / 216 $ (5v^7 + 16v^5 + 8v^3 + 81v) / 216
$\beta_{3}$	$=$	$ ( 5\nu^{6} + 16\nu^{4} + 80\nu^{2} + 153 ) / 36 $ (5v^6 + 16v^4 + 80*v^2 + 153) / 36
$\beta_{4}$	$=$	$ ( -\nu^{7} + 4\nu^{5} + 2\nu^{3} - 9\nu ) / 27 $ (-v^7 + 4v^5 + 2v^3 - 9*v) / 27
$\beta_{5}$	$=$	$ ( 2\nu^{6} + 10\nu^{4} + 14\nu^{2} + 45 ) / 9 $ (2v^6 + 10v^4 + 14*v^2 + 45) / 9
$\beta_{6}$	$=$	$ ( 11\nu^{7} + 64\nu^{5} + 248\nu^{3} + 1071\nu ) / 216 $ (11v^7 + 64v^5 + 248v^3 + 1071v) / 216
$\beta_{7}$	$=$	$ ( 2\nu^{7} + 10\nu^{5} + 32\nu^{3} + 36\nu ) / 27 $ (2v^7 + 10v^5 + 32v^3 + 36v) / 27

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

$\nu$	$=$	$ ( -\beta_{7} + \beta_{6} + \beta_{2} ) / 4 $ (-b7 + b6 + b2) / 4
$\nu^{2}$	$=$	$ ( -\beta_{5} + 2\beta_{3} + \beta _1 - 5 ) / 4 $ (-b5 + 2*b3 + b1 - 5) / 4
$\nu^{3}$	$=$	$ ( 2\beta_{7} - \beta_{4} - 8\beta_{2} ) / 2 $ (2b7 - b4 - 8b2) / 2
$\nu^{4}$	$=$	$ ( 5\beta_{5} - 6\beta_{3} + 5\beta _1 - 7 ) / 4 $ (5b5 - 6b3 + 5*b1 - 7) / 4
$\nu^{5}$	$=$	$ ( -\beta_{7} - \beta_{6} + 16\beta_{4} + 31\beta_{2} ) / 4 $ (-b7 - b6 + 16b4 + 31b2) / 4
$\nu^{6}$	$=$	$ 4\beta_{3} - 8\beta _1 - 5 $ 4b3 - 8b1 - 5
$\nu^{7}$	$=$	$ ( 13\beta_{7} - 13\beta_{6} - 48\beta_{4} + 83\beta_{2} ) / 4 $ (13b7 - 13b6 - 48b4 + 83b2) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6336\mathbb{Z}\right)^\times$.

$n$	$1729$	$3521$	$4159$	$4357$
$\chi(n)$	$1$	$1$	$1$	$-1$

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} $

3169.1

1.26217	−	1.18614i
1.26217	+	1.18614i
0.396143	+	1.68614i
0.396143	−	1.68614i
−0.396143	+	1.68614i
−0.396143	−	1.68614i
−1.26217	−	1.18614i
−1.26217	+	1.18614i

−5.04868

−5.04868

−1.58457

−1.58457

1.58457

1.58457

5.04868

5.04868

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	6336.2.f.k		8
3.b	odd	2	1		2112.2.f.h	✓	8
4.b	odd	2	1	inner	6336.2.f.k		8
8.b	even	2	1	inner	6336.2.f.k		8
8.d	odd	2	1	inner	6336.2.f.k		8
12.b	even	2	1		2112.2.f.h	✓	8
24.f	even	2	1		2112.2.f.h	✓	8
24.h	odd	2	1		2112.2.f.h	✓	8
48.i	odd	4	1		8448.2.a.co		4
48.i	odd	4	1		8448.2.a.cr		4
48.k	even	4	1		8448.2.a.co		4
48.k	even	4	1		8448.2.a.cr		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2112.2.f.h	✓	8	3.b	odd	2	1
2112.2.f.h	✓	8	12.b	even	2	1
2112.2.f.h	✓	8	24.f	even	2	1
2112.2.f.h	✓	8	24.h	odd	2	1
6336.2.f.k		8	1.a	even	1	1	trivial
6336.2.f.k		8	4.b	odd	2	1	inner
6336.2.f.k		8	8.b	even	2	1	inner
6336.2.f.k		8	8.d	odd	2	1	inner
8448.2.a.co		4	48.i	odd	4	1
8448.2.a.co		4	48.k	even	4	1
8448.2.a.cr		4	48.i	odd	4	1
8448.2.a.cr		4	48.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 $S_{2}^{\mathrm{new}}(6336, [\chi])$:

$ T_{5} $

$ T_{7}^{4} - 28T_{7}^{2} + 64 $

$ T_{17}^{2} - 2T_{17} - 32 $

$ T_{19}^{4} + 68T_{19}^{2} + 1024 $

$ T_{41}^{2} + 2T_{41} - 32 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} $ T^8
$5$	$ T^{8} $ T^8
$7$	$ (T^{4} - 28 T^{2} + 64)^{2} $ (T^4 - 28*T^2 + 64)^2
$11$	$ (T^{2} + 1)^{4} $ (T^2 + 1)^4
$13$	$ (T^{2} + 12)^{4} $ (T^2 + 12)^4
$17$	$ (T^{2} - 2 T - 32)^{4} $ (T^2 - 2*T - 32)^4
$19$	$ (T^{4} + 68 T^{2} + 1024)^{2} $ (T^4 + 68*T^2 + 1024)^2
$23$	$ (T^{2} - 44)^{4} $ (T^2 - 44)^4
$29$	$ (T^{4} + 76 T^{2} + 256)^{2} $ (T^4 + 76*T^2 + 256)^2
$31$	$ (T^{2} - 12)^{4} $ (T^2 - 12)^4
$37$	$ (T^{4} + 112 T^{2} + 1024)^{2} $ (T^4 + 112*T^2 + 1024)^2
$41$	$ (T^{2} + 2 T - 32)^{4} $ (T^2 + 2*T - 32)^4
$43$	$ (T^{4} + 116 T^{2} + 64)^{2} $ (T^4 + 116*T^2 + 64)^2
$47$	$ (T^{4} - 184 T^{2} + 16)^{2} $ (T^4 - 184*T^2 + 16)^2
$53$	$ (T^{4} + 112 T^{2} + 1024)^{2} $ (T^4 + 112*T^2 + 1024)^2
$59$	$ (T^{2} + 16)^{4} $ (T^2 + 16)^4
$61$	$ (T^{4} + 184 T^{2} + 16)^{2} $ (T^4 + 184*T^2 + 16)^2
$67$	$ (T^{2} + 16)^{4} $ (T^2 + 16)^4
$71$	$ (T^{4} - 184 T^{2} + 16)^{2} $ (T^4 - 184*T^2 + 16)^2
$73$	$ (T - 6)^{8} $ (T - 6)^8
$79$	$ (T^{4} - 204 T^{2} + 9216)^{2} $ (T^4 - 204*T^2 + 9216)^2
$83$	$ (T^{4} + 336 T^{2} + 9216)^{2} $ (T^4 + 336*T^2 + 9216)^2
$89$	$ (T + 10)^{8} $ (T + 10)^8
$97$	$ (T^{2} + 8 T - 116)^{4} $ (T^2 + 8*T - 116)^4
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:	\( N \)	\(=\)	\( 6336 = 2^{6} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6336.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_{7} q^{7}+O(q^{10}) \) q + b7 * q^7	\( q + \beta_{7} q^{7} - \beta_{2} q^{11} + \beta_{3} q^{13} + ( - \beta_{5} + 1) q^{17} + ( - \beta_{6} - \beta_{2}) q^{19} + (2 \beta_{7} - \beta_{4}) q^{23} + 5 q^{25} + ( - \beta_{3} - \beta_1) q^{29} - \beta_{4} q^{31} + (2 \beta_{3} - 2 \beta_1) q^{37} + (\beta_{5} - 1) q^{41} + (\beta_{6} + 5 \beta_{2}) q^{43} + ( - 2 \beta_{7} + 3 \beta_{4}) q^{47} + ( - 2 \beta_{5} + 7) q^{49} + ( - 2 \beta_{3} + 2 \beta_1) q^{53} - 4 \beta_{2} q^{59} + (\beta_{3} + 2 \beta_1) q^{61} - 4 \beta_{2} q^{67} + ( - 2 \beta_{7} + 3 \beta_{4}) q^{71} + 6 q^{73} - \beta_1 q^{77} + ( - 3 \beta_{7} + 2 \beta_{4}) q^{79} + ( - 2 \beta_{6} + 6 \beta_{2}) q^{83} - 10 q^{89} + ( - 2 \beta_{6} + 6 \beta_{2}) q^{91} + (2 \beta_{5} - 4) q^{97}+O(q^{100}) \) q + b7 * q^7 - b2 * q^11 + b3 * q^13 + (-b5 + 1) * q^17 + (-b6 - b2) * q^19 + (2b7 - b4) q^23 + 5 * q^25 + (-b3 - b1) * q^29 - b4 * q^31 + (2b3 - 2b1) * q^37 + (b5 - 1) * q^41 + (b6 + 5b2) q^43 + (-2b7 + 3b4) * q^47 + (-2b5 + 7) q^49 + (-2b3 + 2b1) * q^53 - 4b2 q^59 + (b3 + 2b1) q^61 - 4b2 q^67 + (-2b7 + 3b4) * q^71 + 6 * q^73 - b1 * q^77 + (-3b7 + 2b4) * q^79 + (-2b6 + 6b2) * q^83 - 10 * q^89 + (-2b6 + 6b2) * q^91 + (2b5 - 4) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 8 q^{17} + 40 q^{25} - 8 q^{41} + 56 q^{49} + 48 q^{73} - 80 q^{89} - 32 q^{97}+O(q^{100}) \) 8 * q + 8 * q^17 + 40 * q^25 - 8 * q^41 + 56 * q^49 + 48 * q^73 - 80 * q^89 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more