Properties

Label 5850.503
Modulus $5850$
Conductor $975$
Order $60$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5850, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,21,40]))
 
pari: [g,chi] = znchar(Mod(503,5850))
 

Basic properties

Modulus: \(5850\)
Conductor: \(975\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{975}(503,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5850.gh

\(\chi_{5850}(503,\cdot)\) \(\chi_{5850}(737,\cdot)\) \(\chi_{5850}(1277,\cdot)\) \(\chi_{5850}(1673,\cdot)\) \(\chi_{5850}(2213,\cdot)\) \(\chi_{5850}(2447,\cdot)\) \(\chi_{5850}(3077,\cdot)\) \(\chi_{5850}(3383,\cdot)\) \(\chi_{5850}(3617,\cdot)\) \(\chi_{5850}(4013,\cdot)\) \(\chi_{5850}(4247,\cdot)\) \(\chi_{5850}(4553,\cdot)\) \(\chi_{5850}(4787,\cdot)\) \(\chi_{5850}(5183,\cdot)\) \(\chi_{5850}(5417,\cdot)\) \(\chi_{5850}(5723,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((3251,3277,2251)\) → \((-1,e\left(\frac{7}{20}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 5850 }(503, a) \) \(1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{7}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5850 }(503,a) \;\) at \(\;a = \) e.g. 2