Properties

Label 1407.1031
Modulus $1407$
Conductor $1407$
Order $66$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1407, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([33,22,20]))
 
pari: [g,chi] = znchar(Mod(1031,1407))
 

Basic properties

Modulus: \(1407\)
Conductor: \(1407\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1407.cg

\(\chi_{1407}(116,\cdot)\) \(\chi_{1407}(170,\cdot)\) \(\chi_{1407}(284,\cdot)\) \(\chi_{1407}(389,\cdot)\) \(\chi_{1407}(569,\cdot)\) \(\chi_{1407}(620,\cdot)\) \(\chi_{1407}(674,\cdot)\) \(\chi_{1407}(758,\cdot)\) \(\chi_{1407}(926,\cdot)\) \(\chi_{1407}(977,\cdot)\) \(\chi_{1407}(1031,\cdot)\) \(\chi_{1407}(1052,\cdot)\) \(\chi_{1407}(1082,\cdot)\) \(\chi_{1407}(1145,\cdot)\) \(\chi_{1407}(1199,\cdot)\) \(\chi_{1407}(1229,\cdot)\) \(\chi_{1407}(1241,\cdot)\) \(\chi_{1407}(1262,\cdot)\) \(\chi_{1407}(1271,\cdot)\) \(\chi_{1407}(1292,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((470,1207,337)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{10}{33}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 1407 }(1031, a) \) \(-1\)\(1\)\(e\left(\frac{31}{66}\right)\)\(e\left(\frac{31}{33}\right)\)\(e\left(\frac{47}{66}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{47}{66}\right)\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{29}{33}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{23}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1407 }(1031,a) \;\) at \(\;a = \) e.g. 2