Properties

Label 1805.11
Modulus $1805$
Conductor $361$
Order $57$
Real no
Primitive no
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([0,34]))
 
pari: [g,chi] = znchar(Mod(11,1805))
 

Basic properties

Modulus: \(1805\)
Conductor: \(361\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(57\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{361}(11,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1805.w

\(\chi_{1805}(11,\cdot)\) \(\chi_{1805}(26,\cdot)\) \(\chi_{1805}(106,\cdot)\) \(\chi_{1805}(121,\cdot)\) \(\chi_{1805}(201,\cdot)\) \(\chi_{1805}(216,\cdot)\) \(\chi_{1805}(296,\cdot)\) \(\chi_{1805}(311,\cdot)\) \(\chi_{1805}(391,\cdot)\) \(\chi_{1805}(406,\cdot)\) \(\chi_{1805}(486,\cdot)\) \(\chi_{1805}(501,\cdot)\) \(\chi_{1805}(581,\cdot)\) \(\chi_{1805}(596,\cdot)\) \(\chi_{1805}(676,\cdot)\) \(\chi_{1805}(691,\cdot)\) \(\chi_{1805}(771,\cdot)\) \(\chi_{1805}(786,\cdot)\) \(\chi_{1805}(866,\cdot)\) \(\chi_{1805}(881,\cdot)\) \(\chi_{1805}(961,\cdot)\) \(\chi_{1805}(976,\cdot)\) \(\chi_{1805}(1056,\cdot)\) \(\chi_{1805}(1071,\cdot)\) \(\chi_{1805}(1166,\cdot)\) \(\chi_{1805}(1246,\cdot)\) \(\chi_{1805}(1261,\cdot)\) \(\chi_{1805}(1341,\cdot)\) \(\chi_{1805}(1356,\cdot)\) \(\chi_{1805}(1436,\cdot)\) ...

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 57 polynomial

Values on generators

\((362,1446)\) → \((1,e\left(\frac{17}{57}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 1805 }(11, a) \) \(1\)\(1\)\(e\left(\frac{17}{57}\right)\)\(e\left(\frac{26}{57}\right)\)\(e\left(\frac{34}{57}\right)\)\(e\left(\frac{43}{57}\right)\)\(e\left(\frac{14}{19}\right)\)\(e\left(\frac{17}{19}\right)\)\(e\left(\frac{52}{57}\right)\)\(e\left(\frac{8}{19}\right)\)\(e\left(\frac{1}{19}\right)\)\(e\left(\frac{7}{57}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1805 }(11,a) \;\) at \(\;a = \) e.g. 2