Properties

Label 4011.914
Modulus $4011$
Conductor $4011$
Order $114$
Real no
Primitive yes
Minimal yes
Parity odd

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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([57,76,42]))
 
pari: [g,chi] = znchar(Mod(914,4011))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4011.bq

\(\chi_{4011}(32,\cdot)\) \(\chi_{4011}(107,\cdot)\) \(\chi_{4011}(221,\cdot)\) \(\chi_{4011}(368,\cdot)\) \(\chi_{4011}(536,\cdot)\) \(\chi_{4011}(578,\cdot)\) \(\chi_{4011}(800,\cdot)\) \(\chi_{4011}(914,\cdot)\) \(\chi_{4011}(1115,\cdot)\) \(\chi_{4011}(1178,\cdot)\) \(\chi_{4011}(1271,\cdot)\) \(\chi_{4011}(1367,\cdot)\) \(\chi_{4011}(1514,\cdot)\) \(\chi_{4011}(1649,\cdot)\) \(\chi_{4011}(1682,\cdot)\) \(\chi_{4011}(1724,\cdot)\) \(\chi_{4011}(2060,\cdot)\) \(\chi_{4011}(2090,\cdot)\) \(\chi_{4011}(2153,\cdot)\) \(\chi_{4011}(2237,\cdot)\) \(\chi_{4011}(2417,\cdot)\) \(\chi_{4011}(2489,\cdot)\) \(\chi_{4011}(2552,\cdot)\) \(\chi_{4011}(2636,\cdot)\) \(\chi_{4011}(2699,\cdot)\) \(\chi_{4011}(2795,\cdot)\) \(\chi_{4011}(2972,\cdot)\) \(\chi_{4011}(3236,\cdot)\) \(\chi_{4011}(3299,\cdot)\) \(\chi_{4011}(3383,\cdot)\) ...

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

\((2675,2866,2311)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{7}{19}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 4011 }(914, a) \) \(-1\)\(1\)\(e\left(\frac{5}{114}\right)\)\(e\left(\frac{5}{57}\right)\)\(e\left(\frac{29}{114}\right)\)\(e\left(\frac{5}{38}\right)\)\(e\left(\frac{17}{57}\right)\)\(e\left(\frac{55}{114}\right)\)\(e\left(\frac{5}{19}\right)\)\(e\left(\frac{10}{57}\right)\)\(e\left(\frac{31}{114}\right)\)\(e\left(\frac{40}{57}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4011 }(914,a) \;\) at \(\;a = \) e.g. 2