Properties

Label 87362.35
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([13851,2200]))
 
pari: [g,chi] = znchar(Mod(35,87362))
 

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 \(\chi_{43681}(35,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 87362.dp

\(\chi_{87362}(17,\cdot)\) \(\chi_{87362}(35,\cdot)\) \(\chi_{87362}(61,\cdot)\) \(\chi_{87362}(63,\cdot)\) \(\chi_{87362}(73,\cdot)\) \(\chi_{87362}(85,\cdot)\) \(\chi_{87362}(101,\cdot)\) \(\chi_{87362}(123,\cdot)\) \(\chi_{87362}(139,\cdot)\) \(\chi_{87362}(149,\cdot)\) \(\chi_{87362}(195,\cdot)\) \(\chi_{87362}(237,\cdot)\) \(\chi_{87362}(271,\cdot)\) \(\chi_{87362}(283,\cdot)\) \(\chi_{87362}(321,\cdot)\) \(\chi_{87362}(327,\cdot)\) \(\chi_{87362}(347,\cdot)\) \(\chi_{87362}(359,\cdot)\) \(\chi_{87362}(365,\cdot)\) \(\chi_{87362}(435,\cdot)\) \(\chi_{87362}(453,\cdot)\) \(\chi_{87362}(479,\cdot)\) \(\chi_{87362}(491,\cdot)\) \(\chi_{87362}(503,\cdot)\) \(\chi_{87362}(519,\cdot)\) \(\chi_{87362}(541,\cdot)\) \(\chi_{87362}(557,\cdot)\) \(\chi_{87362}(567,\cdot)\) \(\chi_{87362}(579,\cdot)\) \(\chi_{87362}(613,\cdot)\) ...

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

\((21661,22023)\) → \((e\left(\frac{81}{110}\right),e\left(\frac{20}{171}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(13\)\(15\)\(17\)\(21\)\(23\)\(25\)
\( \chi_{ 87362 }(35, a) \) \(-1\)\(1\)\(e\left(\frac{49}{855}\right)\)\(e\left(\frac{5882}{9405}\right)\)\(e\left(\frac{4379}{6270}\right)\)\(e\left(\frac{98}{855}\right)\)\(e\left(\frac{16031}{18810}\right)\)\(e\left(\frac{6421}{9405}\right)\)\(e\left(\frac{9679}{18810}\right)\)\(e\left(\frac{2843}{3762}\right)\)\(e\left(\frac{1169}{1881}\right)\)\(e\left(\frac{2359}{9405}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 87362 }(35,a) \;\) at \(\;a = \) e.g. 2