Properties

Label 4011.794
Modulus $4011$
Conductor $4011$
Order $114$
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(114))
 
M = H._module
 
chi = DirichletCharacter(H, M([57,19,12]))
 
pari: [g,chi] = znchar(Mod(794,4011))
 

Basic properties

Modulus: \(4011\)
Conductor: \(4011\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(114\)
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.bs

\(\chi_{4011}(5,\cdot)\) \(\chi_{4011}(227,\cdot)\) \(\chi_{4011}(341,\cdot)\) \(\chi_{4011}(542,\cdot)\) \(\chi_{4011}(605,\cdot)\) \(\chi_{4011}(698,\cdot)\) \(\chi_{4011}(794,\cdot)\) \(\chi_{4011}(941,\cdot)\) \(\chi_{4011}(1076,\cdot)\) \(\chi_{4011}(1109,\cdot)\) \(\chi_{4011}(1151,\cdot)\) \(\chi_{4011}(1487,\cdot)\) \(\chi_{4011}(1517,\cdot)\) \(\chi_{4011}(1580,\cdot)\) \(\chi_{4011}(1664,\cdot)\) \(\chi_{4011}(1844,\cdot)\) \(\chi_{4011}(1916,\cdot)\) \(\chi_{4011}(1979,\cdot)\) \(\chi_{4011}(2063,\cdot)\) \(\chi_{4011}(2126,\cdot)\) \(\chi_{4011}(2222,\cdot)\) \(\chi_{4011}(2399,\cdot)\) \(\chi_{4011}(2663,\cdot)\) \(\chi_{4011}(2726,\cdot)\) \(\chi_{4011}(2810,\cdot)\) \(\chi_{4011}(3062,\cdot)\) \(\chi_{4011}(3092,\cdot)\) \(\chi_{4011}(3125,\cdot)\) \(\chi_{4011}(3209,\cdot)\) \(\chi_{4011}(3272,\cdot)\) ...

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: $\Q(\zeta_{57})$
Fixed field: Number field defined by a degree 114 polynomial (not computed)

Values on generators

\((2675,2866,2311)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{2}{19}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 4011 }(794, a) \) \(1\)\(1\)\(e\left(\frac{53}{114}\right)\)\(e\left(\frac{53}{57}\right)\)\(e\left(\frac{34}{57}\right)\)\(e\left(\frac{15}{38}\right)\)\(e\left(\frac{7}{114}\right)\)\(e\left(\frac{13}{114}\right)\)\(e\left(\frac{11}{38}\right)\)\(e\left(\frac{49}{57}\right)\)\(e\left(\frac{56}{57}\right)\)\(e\left(\frac{107}{114}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4011 }(794,a) \;\) at \(\;a = \) e.g. 2