Properties

Label 9295.822
Modulus $9295$
Conductor $715$
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(9295, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([15,18,20]))
 
pari: [g,chi] = znchar(Mod(822,9295))
 

Basic properties

Modulus: \(9295\)
Conductor: \(715\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{715}(107,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9295.dq

\(\chi_{9295}(822,\cdot)\) \(\chi_{9295}(1498,\cdot)\) \(\chi_{9295}(1667,\cdot)\) \(\chi_{9295}(1712,\cdot)\) \(\chi_{9295}(3357,\cdot)\) \(\chi_{9295}(4033,\cdot)\) \(\chi_{9295}(4078,\cdot)\) \(\chi_{9295}(4923,\cdot)\) \(\chi_{9295}(5892,\cdot)\) \(\chi_{9295}(5937,\cdot)\) \(\chi_{9295}(6613,\cdot)\) \(\chi_{9295}(6782,\cdot)\) \(\chi_{9295}(8258,\cdot)\) \(\chi_{9295}(8472,\cdot)\) \(\chi_{9295}(9103,\cdot)\) \(\chi_{9295}(9148,\cdot)\)

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

\((7437,4226,6931)\) → \((i,e\left(\frac{3}{10}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(14\)\(16\)
\( \chi_{ 9295 }(822, a) \) \(1\)\(1\)\(e\left(\frac{53}{60}\right)\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{29}{30}\right)\)\(i\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{8}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9295 }(822,a) \;\) at \(\;a = \) e.g. 2