Properties

Label 1805.4
Modulus $1805$
Conductor $1805$
Order $342$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1805, base_ring=CyclotomicField(342))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([171,2]))
 
pari: [g,chi] = znchar(Mod(4,1805))
 

Basic properties

Modulus: \(1805\)
Conductor: \(1805\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(342\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1805.bf

\(\chi_{1805}(4,\cdot)\) \(\chi_{1805}(9,\cdot)\) \(\chi_{1805}(24,\cdot)\) \(\chi_{1805}(44,\cdot)\) \(\chi_{1805}(74,\cdot)\) \(\chi_{1805}(104,\cdot)\) \(\chi_{1805}(119,\cdot)\) \(\chi_{1805}(139,\cdot)\) \(\chi_{1805}(149,\cdot)\) \(\chi_{1805}(169,\cdot)\) \(\chi_{1805}(194,\cdot)\) \(\chi_{1805}(199,\cdot)\) \(\chi_{1805}(214,\cdot)\) \(\chi_{1805}(244,\cdot)\) \(\chi_{1805}(264,\cdot)\) \(\chi_{1805}(289,\cdot)\) \(\chi_{1805}(294,\cdot)\) \(\chi_{1805}(309,\cdot)\) \(\chi_{1805}(329,\cdot)\) \(\chi_{1805}(339,\cdot)\) \(\chi_{1805}(359,\cdot)\) \(\chi_{1805}(384,\cdot)\) \(\chi_{1805}(404,\cdot)\) \(\chi_{1805}(424,\cdot)\) \(\chi_{1805}(434,\cdot)\) \(\chi_{1805}(454,\cdot)\) \(\chi_{1805}(479,\cdot)\) \(\chi_{1805}(484,\cdot)\) \(\chi_{1805}(499,\cdot)\) \(\chi_{1805}(519,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{171})$
Fixed field: Number field defined by a degree 342 polynomial (not computed)

Values on generators

\((362,1446)\) → \((-1,e\left(\frac{1}{171}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 1805 }(4, a) \) \(1\)\(1\)\(e\left(\frac{173}{342}\right)\)\(e\left(\frac{107}{342}\right)\)\(e\left(\frac{2}{171}\right)\)\(e\left(\frac{140}{171}\right)\)\(e\left(\frac{43}{114}\right)\)\(e\left(\frac{59}{114}\right)\)\(e\left(\frac{107}{171}\right)\)\(e\left(\frac{34}{57}\right)\)\(e\left(\frac{37}{114}\right)\)\(e\left(\frac{145}{342}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1805 }(4,a) \;\) at \(\;a = \) e.g. 2