Properties

Label 1337.363
Modulus $1337$
Conductor $1337$
Order $190$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1337, base_ring=CyclotomicField(190))
 
M = H._module
 
chi = DirichletCharacter(H, M([95,96]))
 
pari: [g,chi] = znchar(Mod(363,1337))
 

Basic properties

Modulus: \(1337\)
Conductor: \(1337\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(190\)
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 1337.z

\(\chi_{1337}(13,\cdot)\) \(\chi_{1337}(20,\cdot)\) \(\chi_{1337}(27,\cdot)\) \(\chi_{1337}(34,\cdot)\) \(\chi_{1337}(48,\cdot)\) \(\chi_{1337}(90,\cdot)\) \(\chi_{1337}(97,\cdot)\) \(\chi_{1337}(104,\cdot)\) \(\chi_{1337}(118,\cdot)\) \(\chi_{1337}(195,\cdot)\) \(\chi_{1337}(209,\cdot)\) \(\chi_{1337}(237,\cdot)\) \(\chi_{1337}(251,\cdot)\) \(\chi_{1337}(258,\cdot)\) \(\chi_{1337}(272,\cdot)\) \(\chi_{1337}(293,\cdot)\) \(\chi_{1337}(321,\cdot)\) \(\chi_{1337}(335,\cdot)\) \(\chi_{1337}(349,\cdot)\) \(\chi_{1337}(363,\cdot)\) \(\chi_{1337}(384,\cdot)\) \(\chi_{1337}(391,\cdot)\) \(\chi_{1337}(398,\cdot)\) \(\chi_{1337}(405,\cdot)\) \(\chi_{1337}(433,\cdot)\) \(\chi_{1337}(447,\cdot)\) \(\chi_{1337}(454,\cdot)\) \(\chi_{1337}(461,\cdot)\) \(\chi_{1337}(468,\cdot)\) \(\chi_{1337}(482,\cdot)\) ...

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: $\Q(\zeta_{95})$
Fixed field: Number field defined by a degree 190 polynomial (not computed)

Values on generators

\((192,974)\) → \((-1,e\left(\frac{48}{95}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 1337 }(363, a) \) \(-1\)\(1\)\(e\left(\frac{22}{95}\right)\)\(e\left(\frac{21}{190}\right)\)\(e\left(\frac{44}{95}\right)\)\(e\left(\frac{29}{38}\right)\)\(e\left(\frac{13}{38}\right)\)\(e\left(\frac{66}{95}\right)\)\(e\left(\frac{21}{95}\right)\)\(e\left(\frac{189}{190}\right)\)\(e\left(\frac{18}{19}\right)\)\(e\left(\frac{109}{190}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1337 }(363,a) \;\) at \(\;a = \) e.g. 2