Properties

Label 1441.652
Modulus $1441$
Conductor $1441$
Order $130$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1441, base_ring=CyclotomicField(130))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([104,7]))
 
pari: [g,chi] = znchar(Mod(652,1441))
 

Basic properties

Modulus: \(1441\)
Conductor: \(1441\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(130\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1441.bp

\(\chi_{1441}(14,\cdot)\) \(\chi_{1441}(31,\cdot)\) \(\chi_{1441}(93,\cdot)\) \(\chi_{1441}(97,\cdot)\) \(\chi_{1441}(103,\cdot)\) \(\chi_{1441}(104,\cdot)\) \(\chi_{1441}(119,\cdot)\) \(\chi_{1441}(126,\cdot)\) \(\chi_{1441}(214,\cdot)\) \(\chi_{1441}(268,\cdot)\) \(\chi_{1441}(279,\cdot)\) \(\chi_{1441}(291,\cdot)\) \(\chi_{1441}(312,\cdot)\) \(\chi_{1441}(328,\cdot)\) \(\chi_{1441}(344,\cdot)\) \(\chi_{1441}(350,\cdot)\) \(\chi_{1441}(357,\cdot)\) \(\chi_{1441}(378,\cdot)\) \(\chi_{1441}(449,\cdot)\) \(\chi_{1441}(465,\cdot)\) \(\chi_{1441}(515,\cdot)\) \(\chi_{1441}(520,\cdot)\) \(\chi_{1441}(532,\cdot)\) \(\chi_{1441}(630,\cdot)\) \(\chi_{1441}(642,\cdot)\) \(\chi_{1441}(652,\cdot)\) \(\chi_{1441}(751,\cdot)\) \(\chi_{1441}(775,\cdot)\) \(\chi_{1441}(812,\cdot)\) \(\chi_{1441}(873,\cdot)\) ...

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: $\Q(\zeta_{65})$
Fixed field: Number field defined by a degree 130 polynomial (not computed)

Values on generators

\((1311,133)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{7}{130}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\(-1\)\(1\)\(e\left(\frac{111}{130}\right)\)\(e\left(\frac{18}{65}\right)\)\(e\left(\frac{46}{65}\right)\)\(e\left(\frac{44}{65}\right)\)\(e\left(\frac{17}{130}\right)\)\(e\left(\frac{10}{13}\right)\)\(e\left(\frac{73}{130}\right)\)\(e\left(\frac{36}{65}\right)\)\(e\left(\frac{69}{130}\right)\)\(e\left(\frac{64}{65}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1441 }(652,a) \;\) at \(\;a = \) e.g. 2