Properties

Label 1805.6
Modulus $1805$
Conductor $361$
Order $171$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

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([0,140]))
 
pari: [g,chi] = znchar(Mod(6,1805))
 

Basic properties

Modulus: \(1805\)
Conductor: \(361\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(171\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{361}(6,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1805.bc

\(\chi_{1805}(6,\cdot)\) \(\chi_{1805}(16,\cdot)\) \(\chi_{1805}(36,\cdot)\) \(\chi_{1805}(61,\cdot)\) \(\chi_{1805}(66,\cdot)\) \(\chi_{1805}(81,\cdot)\) \(\chi_{1805}(101,\cdot)\) \(\chi_{1805}(111,\cdot)\) \(\chi_{1805}(131,\cdot)\) \(\chi_{1805}(156,\cdot)\) \(\chi_{1805}(161,\cdot)\) \(\chi_{1805}(176,\cdot)\) \(\chi_{1805}(196,\cdot)\) \(\chi_{1805}(206,\cdot)\) \(\chi_{1805}(226,\cdot)\) \(\chi_{1805}(251,\cdot)\) \(\chi_{1805}(256,\cdot)\) \(\chi_{1805}(271,\cdot)\) \(\chi_{1805}(291,\cdot)\) \(\chi_{1805}(301,\cdot)\) \(\chi_{1805}(321,\cdot)\) \(\chi_{1805}(346,\cdot)\) \(\chi_{1805}(351,\cdot)\) \(\chi_{1805}(366,\cdot)\) \(\chi_{1805}(386,\cdot)\) \(\chi_{1805}(396,\cdot)\) \(\chi_{1805}(416,\cdot)\) \(\chi_{1805}(441,\cdot)\) \(\chi_{1805}(446,\cdot)\) \(\chi_{1805}(461,\cdot)\) ...

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 171 polynomial (not computed)

Values on generators

\((362,1446)\) → \((1,e\left(\frac{70}{171}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 1805 }(6, a) \) \(1\)\(1\)\(e\left(\frac{70}{171}\right)\)\(e\left(\frac{154}{171}\right)\)\(e\left(\frac{140}{171}\right)\)\(e\left(\frac{53}{171}\right)\)\(e\left(\frac{23}{57}\right)\)\(e\left(\frac{13}{57}\right)\)\(e\left(\frac{137}{171}\right)\)\(e\left(\frac{43}{57}\right)\)\(e\left(\frac{41}{57}\right)\)\(e\left(\frac{116}{171}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1805 }(6,a) \;\) at \(\;a = \) e.g. 2