Properties

Label 1014.67
Modulus $1014$
Conductor $169$
Order $156$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1014, base_ring=CyclotomicField(156))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,37]))
 
pari: [g,chi] = znchar(Mod(67,1014))
 

Basic properties

Modulus: \(1014\)
Conductor: \(169\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(156\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{169}(67,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1014.w

\(\chi_{1014}(7,\cdot)\) \(\chi_{1014}(37,\cdot)\) \(\chi_{1014}(67,\cdot)\) \(\chi_{1014}(85,\cdot)\) \(\chi_{1014}(97,\cdot)\) \(\chi_{1014}(115,\cdot)\) \(\chi_{1014}(145,\cdot)\) \(\chi_{1014}(163,\cdot)\) \(\chi_{1014}(175,\cdot)\) \(\chi_{1014}(193,\cdot)\) \(\chi_{1014}(223,\cdot)\) \(\chi_{1014}(241,\cdot)\) \(\chi_{1014}(253,\cdot)\) \(\chi_{1014}(271,\cdot)\) \(\chi_{1014}(301,\cdot)\) \(\chi_{1014}(331,\cdot)\) \(\chi_{1014}(349,\cdot)\) \(\chi_{1014}(379,\cdot)\) \(\chi_{1014}(397,\cdot)\) \(\chi_{1014}(409,\cdot)\) \(\chi_{1014}(457,\cdot)\) \(\chi_{1014}(475,\cdot)\) \(\chi_{1014}(487,\cdot)\) \(\chi_{1014}(505,\cdot)\) \(\chi_{1014}(535,\cdot)\) \(\chi_{1014}(553,\cdot)\) \(\chi_{1014}(565,\cdot)\) \(\chi_{1014}(583,\cdot)\) \(\chi_{1014}(613,\cdot)\) \(\chi_{1014}(631,\cdot)\) ...

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: $\Q(\zeta_{156})$
Fixed field: Number field defined by a degree 156 polynomial (not computed)

Values on generators

\((677,847)\) → \((1,e\left(\frac{37}{156}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 1014 }(67, a) \) \(-1\)\(1\)\(e\left(\frac{7}{52}\right)\)\(e\left(\frac{59}{156}\right)\)\(e\left(\frac{67}{156}\right)\)\(e\left(\frac{49}{78}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{7}{26}\right)\)\(e\left(\frac{19}{39}\right)\)\(e\left(\frac{51}{52}\right)\)\(e\left(\frac{20}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1014 }(67,a) \;\) at \(\;a = \) e.g. 2