Properties

Label 4011.557
Modulus $4011$
Conductor $4011$
Order $570$
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(570))
 
M = H._module
 
chi = DirichletCharacter(H, M([285,380,243]))
 
pari: [g,chi] = znchar(Mod(557,4011))
 

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

Galois orbit 4011.cj

\(\chi_{4011}(44,\cdot)\) \(\chi_{4011}(53,\cdot)\) \(\chi_{4011}(74,\cdot)\) \(\chi_{4011}(95,\cdot)\) \(\chi_{4011}(116,\cdot)\) \(\chi_{4011}(137,\cdot)\) \(\chi_{4011}(179,\cdot)\) \(\chi_{4011}(212,\cdot)\) \(\chi_{4011}(233,\cdot)\) \(\chi_{4011}(254,\cdot)\) \(\chi_{4011}(284,\cdot)\) \(\chi_{4011}(296,\cdot)\) \(\chi_{4011}(305,\cdot)\) \(\chi_{4011}(317,\cdot)\) \(\chi_{4011}(359,\cdot)\) \(\chi_{4011}(380,\cdot)\) \(\chi_{4011}(401,\cdot)\) \(\chi_{4011}(410,\cdot)\) \(\chi_{4011}(443,\cdot)\) \(\chi_{4011}(473,\cdot)\) \(\chi_{4011}(494,\cdot)\) \(\chi_{4011}(506,\cdot)\) \(\chi_{4011}(527,\cdot)\) \(\chi_{4011}(557,\cdot)\) \(\chi_{4011}(569,\cdot)\) \(\chi_{4011}(620,\cdot)\) \(\chi_{4011}(662,\cdot)\) \(\chi_{4011}(674,\cdot)\) \(\chi_{4011}(683,\cdot)\) \(\chi_{4011}(704,\cdot)\) ...

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

\((2675,2866,2311)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{81}{190}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 4011 }(557, a) \) \(1\)\(1\)\(e\left(\frac{337}{570}\right)\)\(e\left(\frac{52}{285}\right)\)\(e\left(\frac{17}{114}\right)\)\(e\left(\frac{147}{190}\right)\)\(e\left(\frac{211}{285}\right)\)\(e\left(\frac{23}{57}\right)\)\(e\left(\frac{71}{95}\right)\)\(e\left(\frac{104}{285}\right)\)\(e\left(\frac{539}{570}\right)\)\(e\left(\frac{433}{570}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4011 }(557,a) \;\) at \(\;a = \) e.g. 2