Properties

Label 4011.17
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([285,95,294]))
 
pari: [g,chi] = znchar(Mod(17,4011))
 

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.ck

\(\chi_{4011}(17,\cdot)\) \(\chi_{4011}(26,\cdot)\) \(\chi_{4011}(59,\cdot)\) \(\chi_{4011}(68,\cdot)\) \(\chi_{4011}(80,\cdot)\) \(\chi_{4011}(194,\cdot)\) \(\chi_{4011}(206,\cdot)\) \(\chi_{4011}(215,\cdot)\) \(\chi_{4011}(236,\cdot)\) \(\chi_{4011}(269,\cdot)\) \(\chi_{4011}(299,\cdot)\) \(\chi_{4011}(311,\cdot)\) \(\chi_{4011}(320,\cdot)\) \(\chi_{4011}(353,\cdot)\) \(\chi_{4011}(395,\cdot)\) \(\chi_{4011}(416,\cdot)\) \(\chi_{4011}(425,\cdot)\) \(\chi_{4011}(446,\cdot)\) \(\chi_{4011}(467,\cdot)\) \(\chi_{4011}(479,\cdot)\) \(\chi_{4011}(500,\cdot)\) \(\chi_{4011}(551,\cdot)\) \(\chi_{4011}(593,\cdot)\) \(\chi_{4011}(677,\cdot)\) \(\chi_{4011}(731,\cdot)\) \(\chi_{4011}(773,\cdot)\) \(\chi_{4011}(782,\cdot)\) \(\chi_{4011}(815,\cdot)\) \(\chi_{4011}(824,\cdot)\) \(\chi_{4011}(836,\cdot)\) ...

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

\((2675,2866,2311)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{49}{95}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 4011 }(17, a) \) \(1\)\(1\)\(e\left(\frac{301}{570}\right)\)\(e\left(\frac{16}{285}\right)\)\(e\left(\frac{7}{57}\right)\)\(e\left(\frac{111}{190}\right)\)\(e\left(\frac{371}{570}\right)\)\(e\left(\frac{1}{114}\right)\)\(e\left(\frac{51}{190}\right)\)\(e\left(\frac{32}{285}\right)\)\(e\left(\frac{61}{285}\right)\)\(e\left(\frac{199}{570}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4011 }(17,a) \;\) at \(\;a = \) e.g. 2