Properties

Label 4011.79
Modulus $4011$
Conductor $1337$
Order $285$
Real no
Primitive no
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([0,190,186]))
 
pari: [g,chi] = znchar(Mod(79,4011))
 

Basic properties

Modulus: \(4011\)
Conductor: \(1337\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(285\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1337}(79,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4011.ce

\(\chi_{4011}(4,\cdot)\) \(\chi_{4011}(16,\cdot)\) \(\chi_{4011}(46,\cdot)\) \(\chi_{4011}(67,\cdot)\) \(\chi_{4011}(79,\cdot)\) \(\chi_{4011}(100,\cdot)\) \(\chi_{4011}(130,\cdot)\) \(\chi_{4011}(163,\cdot)\) \(\chi_{4011}(172,\cdot)\) \(\chi_{4011}(193,\cdot)\) \(\chi_{4011}(214,\cdot)\) \(\chi_{4011}(256,\cdot)\) \(\chi_{4011}(268,\cdot)\) \(\chi_{4011}(277,\cdot)\) \(\chi_{4011}(289,\cdot)\) \(\chi_{4011}(319,\cdot)\) \(\chi_{4011}(340,\cdot)\) \(\chi_{4011}(361,\cdot)\) \(\chi_{4011}(394,\cdot)\) \(\chi_{4011}(436,\cdot)\) \(\chi_{4011}(457,\cdot)\) \(\chi_{4011}(478,\cdot)\) \(\chi_{4011}(499,\cdot)\) \(\chi_{4011}(520,\cdot)\) \(\chi_{4011}(529,\cdot)\) \(\chi_{4011}(583,\cdot)\) \(\chi_{4011}(613,\cdot)\) \(\chi_{4011}(676,\cdot)\) \(\chi_{4011}(688,\cdot)\) \(\chi_{4011}(772,\cdot)\) ...

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 285 polynomial (not computed)

Values on generators

\((2675,2866,2311)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{31}{95}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 4011 }(79, a) \) \(1\)\(1\)\(e\left(\frac{7}{285}\right)\)\(e\left(\frac{14}{285}\right)\)\(e\left(\frac{56}{57}\right)\)\(e\left(\frac{7}{95}\right)\)\(e\left(\frac{2}{285}\right)\)\(e\left(\frac{4}{57}\right)\)\(e\left(\frac{52}{95}\right)\)\(e\left(\frac{28}{285}\right)\)\(e\left(\frac{89}{285}\right)\)\(e\left(\frac{283}{285}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4011 }(79,a) \;\) at \(\;a = \) e.g. 2