Properties

Label 4011.647
Modulus $4011$
Conductor $4011$
Order $570$
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(570))
 
M = H._module
 
chi = DirichletCharacter(H, M([285,95,177]))
 
pari: [g,chi] = znchar(Mod(647,4011))
 

Basic properties

Modulus: \(4011\)
Conductor: \(4011\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(570\)
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.cl

\(\chi_{4011}(47,\cdot)\) \(\chi_{4011}(89,\cdot)\) \(\chi_{4011}(101,\cdot)\) \(\chi_{4011}(110,\cdot)\) \(\chi_{4011}(131,\cdot)\) \(\chi_{4011}(143,\cdot)\) \(\chi_{4011}(164,\cdot)\) \(\chi_{4011}(173,\cdot)\) \(\chi_{4011}(248,\cdot)\) \(\chi_{4011}(278,\cdot)\) \(\chi_{4011}(290,\cdot)\) \(\chi_{4011}(332,\cdot)\) \(\chi_{4011}(362,\cdot)\) \(\chi_{4011}(374,\cdot)\) \(\chi_{4011}(404,\cdot)\) \(\chi_{4011}(458,\cdot)\) \(\chi_{4011}(488,\cdot)\) \(\chi_{4011}(509,\cdot)\) \(\chi_{4011}(530,\cdot)\) \(\chi_{4011}(563,\cdot)\) \(\chi_{4011}(626,\cdot)\) \(\chi_{4011}(635,\cdot)\) \(\chi_{4011}(647,\cdot)\) \(\chi_{4011}(656,\cdot)\) \(\chi_{4011}(668,\cdot)\) \(\chi_{4011}(689,\cdot)\) \(\chi_{4011}(710,\cdot)\) \(\chi_{4011}(719,\cdot)\) \(\chi_{4011}(740,\cdot)\) \(\chi_{4011}(752,\cdot)\) ...

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: $\Q(\zeta_{285})$
Fixed field: Number field defined by a degree 570 polynomial (not computed)

Values on generators

\((2675,2866,2311)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{59}{190}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 4011 }(647, a) \) \(-1\)\(1\)\(e\left(\frac{283}{570}\right)\)\(e\left(\frac{283}{285}\right)\)\(e\left(\frac{49}{57}\right)\)\(e\left(\frac{93}{190}\right)\)\(e\left(\frac{203}{570}\right)\)\(e\left(\frac{32}{57}\right)\)\(e\left(\frac{53}{190}\right)\)\(e\left(\frac{281}{285}\right)\)\(e\left(\frac{28}{285}\right)\)\(e\left(\frac{41}{285}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4011 }(647,a) \;\) at \(\;a = \) e.g. 2