Properties

Label 87362.15
Modulus $87362$
Conductor $43681$
Order $18810$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(87362, base_ring=CyclotomicField(18810))
 
M = H._module
 
chi = DirichletCharacter(H, M([8892,1595]))
 
pari: [g,chi] = znchar(Mod(15,87362))
 

Basic properties

Modulus: \(87362\)
Conductor: \(43681\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18810\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{43681}(15,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 87362.dr

\(\chi_{87362}(15,\cdot)\) \(\chi_{87362}(53,\cdot)\) \(\chi_{87362}(59,\cdot)\) \(\chi_{87362}(71,\cdot)\) \(\chi_{87362}(91,\cdot)\) \(\chi_{87362}(97,\cdot)\) \(\chi_{87362}(135,\cdot)\) \(\chi_{87362}(147,\cdot)\) \(\chi_{87362}(181,\cdot)\) \(\chi_{87362}(185,\cdot)\) \(\chi_{87362}(203,\cdot)\) \(\chi_{87362}(223,\cdot)\) \(\chi_{87362}(257,\cdot)\) \(\chi_{87362}(279,\cdot)\) \(\chi_{87362}(295,\cdot)\) \(\chi_{87362}(317,\cdot)\) \(\chi_{87362}(345,\cdot)\) \(\chi_{87362}(355,\cdot)\) \(\chi_{87362}(357,\cdot)\) \(\chi_{87362}(383,\cdot)\) \(\chi_{87362}(401,\cdot)\) \(\chi_{87362}(421,\cdot)\) \(\chi_{87362}(433,\cdot)\) \(\chi_{87362}(471,\cdot)\) \(\chi_{87362}(489,\cdot)\) \(\chi_{87362}(509,\cdot)\) \(\chi_{87362}(515,\cdot)\) \(\chi_{87362}(553,\cdot)\) \(\chi_{87362}(599,\cdot)\) \(\chi_{87362}(603,\cdot)\) ...

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: $\Q(\zeta_{9405})$
Fixed field: Number field defined by a degree 18810 polynomial (not computed)

Values on generators

\((21661,22023)\) → \((e\left(\frac{26}{55}\right),e\left(\frac{29}{342}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(13\)\(15\)\(17\)\(21\)\(23\)\(25\)
\( \chi_{ 87362 }(15, a) \) \(-1\)\(1\)\(e\left(\frac{661}{1710}\right)\)\(e\left(\frac{6154}{9405}\right)\)\(e\left(\frac{89}{3135}\right)\)\(e\left(\frac{661}{855}\right)\)\(e\left(\frac{12097}{18810}\right)\)\(e\left(\frac{769}{18810}\right)\)\(e\left(\frac{3079}{9405}\right)\)\(e\left(\frac{1561}{3762}\right)\)\(e\left(\frac{886}{1881}\right)\)\(e\left(\frac{2903}{9405}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 87362 }(15,a) \;\) at \(\;a = \) e.g. 2