Properties

Label 4011.860
Modulus $4011$
Conductor $4011$
Order $190$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4011, base_ring=CyclotomicField(190))
 
M = H._module
 
chi = DirichletCharacter(H, M([95,95,146]))
 
pari: [g,chi] = znchar(Mod(860,4011))
 

Basic properties

Modulus: \(4011\)
Conductor: \(4011\)
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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4011.by

\(\chi_{4011}(20,\cdot)\) \(\chi_{4011}(104,\cdot)\) \(\chi_{4011}(209,\cdot)\) \(\chi_{4011}(251,\cdot)\) \(\chi_{4011}(272,\cdot)\) \(\chi_{4011}(293,\cdot)\) \(\chi_{4011}(335,\cdot)\) \(\chi_{4011}(398,\cdot)\) \(\chi_{4011}(461,\cdot)\) \(\chi_{4011}(482,\cdot)\) \(\chi_{4011}(545,\cdot)\) \(\chi_{4011}(650,\cdot)\) \(\chi_{4011}(671,\cdot)\) \(\chi_{4011}(776,\cdot)\) \(\chi_{4011}(818,\cdot)\) \(\chi_{4011}(839,\cdot)\) \(\chi_{4011}(860,\cdot)\) \(\chi_{4011}(881,\cdot)\) \(\chi_{4011}(902,\cdot)\) \(\chi_{4011}(965,\cdot)\) \(\chi_{4011}(1070,\cdot)\) \(\chi_{4011}(1154,\cdot)\) \(\chi_{4011}(1196,\cdot)\) \(\chi_{4011}(1238,\cdot)\) \(\chi_{4011}(1280,\cdot)\) \(\chi_{4011}(1364,\cdot)\) \(\chi_{4011}(1385,\cdot)\) \(\chi_{4011}(1427,\cdot)\) \(\chi_{4011}(1532,\cdot)\) \(\chi_{4011}(1574,\cdot)\) ...

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

\((2675,2866,2311)\) → \((-1,-1,e\left(\frac{73}{95}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 4011 }(860, a) \) \(1\)\(1\)\(e\left(\frac{59}{190}\right)\)\(e\left(\frac{59}{95}\right)\)\(e\left(\frac{8}{19}\right)\)\(e\left(\frac{177}{190}\right)\)\(e\left(\frac{139}{190}\right)\)\(e\left(\frac{31}{38}\right)\)\(e\left(\frac{107}{190}\right)\)\(e\left(\frac{23}{95}\right)\)\(e\left(\frac{29}{95}\right)\)\(e\left(\frac{51}{190}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4011 }(860,a) \;\) at \(\;a = \) e.g. 2