Properties

Label 1805.1777
Modulus $1805$
Conductor $95$
Order $36$
Real no
Primitive no
Minimal no
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(36))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([9,34]))
 
pari: [g,chi] = znchar(Mod(1777,1805))
 

Basic properties

Modulus: \(1805\)
Conductor: \(95\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{95}(67,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1805.s

\(\chi_{1805}(127,\cdot)\) \(\chi_{1805}(262,\cdot)\) \(\chi_{1805}(307,\cdot)\) \(\chi_{1805}(333,\cdot)\) \(\chi_{1805}(477,\cdot)\) \(\chi_{1805}(488,\cdot)\) \(\chi_{1805}(623,\cdot)\) \(\chi_{1805}(668,\cdot)\) \(\chi_{1805}(838,\cdot)\) \(\chi_{1805}(1382,\cdot)\) \(\chi_{1805}(1743,\cdot)\) \(\chi_{1805}(1777,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: \(\Q(\zeta_{95})^+\)

Values on generators

\((362,1446)\) → \((i,e\left(\frac{17}{18}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 1805 }(1777, a) \) \(1\)\(1\)\(e\left(\frac{7}{36}\right)\)\(e\left(\frac{1}{36}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{17}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1805 }(1777,a) \;\) at \(\;a = \) e.g. 2