Properties

Label 4011.1609
Modulus $4011$
Conductor $1337$
Order $190$
Real no
Primitive no
Minimal yes
Parity odd

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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([0,95,84]))
 
pari: [g,chi] = znchar(Mod(1609,4011))
 

Basic properties

Modulus: \(4011\)
Conductor: \(1337\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(190\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1337}(272,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4011.cc

\(\chi_{4011}(13,\cdot)\) \(\chi_{4011}(34,\cdot)\) \(\chi_{4011}(97,\cdot)\) \(\chi_{4011}(118,\cdot)\) \(\chi_{4011}(349,\cdot)\) \(\chi_{4011}(391,\cdot)\) \(\chi_{4011}(433,\cdot)\) \(\chi_{4011}(454,\cdot)\) \(\chi_{4011}(517,\cdot)\) \(\chi_{4011}(538,\cdot)\) \(\chi_{4011}(706,\cdot)\) \(\chi_{4011}(790,\cdot)\) \(\chi_{4011}(832,\cdot)\) \(\chi_{4011}(958,\cdot)\) \(\chi_{4011}(979,\cdot)\) \(\chi_{4011}(1000,\cdot)\) \(\chi_{4011}(1063,\cdot)\) \(\chi_{4011}(1084,\cdot)\) \(\chi_{4011}(1189,\cdot)\) \(\chi_{4011}(1210,\cdot)\) \(\chi_{4011}(1231,\cdot)\) \(\chi_{4011}(1315,\cdot)\) \(\chi_{4011}(1357,\cdot)\) \(\chi_{4011}(1441,\cdot)\) \(\chi_{4011}(1546,\cdot)\) \(\chi_{4011}(1588,\cdot)\) \(\chi_{4011}(1609,\cdot)\) \(\chi_{4011}(1630,\cdot)\) \(\chi_{4011}(1672,\cdot)\) \(\chi_{4011}(1735,\cdot)\) ...

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

\((2675,2866,2311)\) → \((1,-1,e\left(\frac{42}{95}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 4011 }(1609, a) \) \(-1\)\(1\)\(e\left(\frac{43}{95}\right)\)\(e\left(\frac{86}{95}\right)\)\(e\left(\frac{23}{38}\right)\)\(e\left(\frac{34}{95}\right)\)\(e\left(\frac{11}{190}\right)\)\(e\left(\frac{11}{19}\right)\)\(e\left(\frac{3}{190}\right)\)\(e\left(\frac{77}{95}\right)\)\(e\left(\frac{157}{190}\right)\)\(e\left(\frac{179}{190}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4011 }(1609,a) \;\) at \(\;a = \) e.g. 2