Properties

Label 1805.49
Modulus $1805$
Conductor $1805$
Order $114$
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(114))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([57,100]))
 
pari: [g,chi] = znchar(Mod(49,1805))
 

Basic properties

Modulus: \(1805\)
Conductor: \(1805\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(114\)
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.ba

\(\chi_{1805}(49,\cdot)\) \(\chi_{1805}(64,\cdot)\) \(\chi_{1805}(144,\cdot)\) \(\chi_{1805}(159,\cdot)\) \(\chi_{1805}(239,\cdot)\) \(\chi_{1805}(254,\cdot)\) \(\chi_{1805}(334,\cdot)\) \(\chi_{1805}(349,\cdot)\) \(\chi_{1805}(444,\cdot)\) \(\chi_{1805}(524,\cdot)\) \(\chi_{1805}(539,\cdot)\) \(\chi_{1805}(619,\cdot)\) \(\chi_{1805}(634,\cdot)\) \(\chi_{1805}(714,\cdot)\) \(\chi_{1805}(729,\cdot)\) \(\chi_{1805}(809,\cdot)\) \(\chi_{1805}(824,\cdot)\) \(\chi_{1805}(904,\cdot)\) \(\chi_{1805}(919,\cdot)\) \(\chi_{1805}(999,\cdot)\) \(\chi_{1805}(1094,\cdot)\) \(\chi_{1805}(1109,\cdot)\) \(\chi_{1805}(1189,\cdot)\) \(\chi_{1805}(1204,\cdot)\) \(\chi_{1805}(1284,\cdot)\) \(\chi_{1805}(1299,\cdot)\) \(\chi_{1805}(1379,\cdot)\) \(\chi_{1805}(1394,\cdot)\) \(\chi_{1805}(1474,\cdot)\) \(\chi_{1805}(1489,\cdot)\) ...

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: $\Q(\zeta_{57})$
Fixed field: Number field defined by a degree 114 polynomial (not computed)

Values on generators

\((362,1446)\) → \((-1,e\left(\frac{50}{57}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 1805 }(49, a) \) \(1\)\(1\)\(e\left(\frac{43}{114}\right)\)\(e\left(\frac{49}{114}\right)\)\(e\left(\frac{43}{57}\right)\)\(e\left(\frac{46}{57}\right)\)\(e\left(\frac{3}{38}\right)\)\(e\left(\frac{5}{38}\right)\)\(e\left(\frac{49}{57}\right)\)\(e\left(\frac{9}{19}\right)\)\(e\left(\frac{7}{38}\right)\)\(e\left(\frac{11}{114}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1805 }(49,a) \;\) at \(\;a = \) e.g. 2