Properties

Label 1441.28
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([117,98]))
 
pari: [g,chi] = znchar(Mod(28,1441))
 

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.cc

\(\chi_{1441}(28,\cdot)\) \(\chi_{1441}(35,\cdot)\) \(\chi_{1441}(41,\cdot)\) \(\chi_{1441}(129,\cdot)\) \(\chi_{1441}(134,\cdot)\) \(\chi_{1441}(156,\cdot)\) \(\chi_{1441}(195,\cdot)\) \(\chi_{1441}(266,\cdot)\) \(\chi_{1441}(271,\cdot)\) \(\chi_{1441}(321,\cdot)\) \(\chi_{1441}(337,\cdot)\) \(\chi_{1441}(404,\cdot)\) \(\chi_{1441}(413,\cdot)\) \(\chi_{1441}(436,\cdot)\) \(\chi_{1441}(442,\cdot)\) \(\chi_{1441}(458,\cdot)\) \(\chi_{1441}(501,\cdot)\) \(\chi_{1441}(545,\cdot)\) \(\chi_{1441}(667,\cdot)\) \(\chi_{1441}(688,\cdot)\) \(\chi_{1441}(689,\cdot)\) \(\chi_{1441}(699,\cdot)\) \(\chi_{1441}(755,\cdot)\) \(\chi_{1441}(772,\cdot)\) \(\chi_{1441}(822,\cdot)\) \(\chi_{1441}(832,\cdot)\) \(\chi_{1441}(860,\cdot)\) \(\chi_{1441}(887,\cdot)\) \(\chi_{1441}(888,\cdot)\) \(\chi_{1441}(930,\cdot)\) ...

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

\((1311,133)\) → \((e\left(\frac{9}{10}\right),e\left(\frac{49}{65}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\(-1\)\(1\)\(e\left(\frac{17}{26}\right)\)\(e\left(\frac{31}{65}\right)\)\(e\left(\frac{4}{13}\right)\)\(e\left(\frac{18}{65}\right)\)\(e\left(\frac{17}{130}\right)\)\(e\left(\frac{87}{130}\right)\)\(e\left(\frac{25}{26}\right)\)\(e\left(\frac{62}{65}\right)\)\(e\left(\frac{121}{130}\right)\)\(e\left(\frac{51}{65}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1441 }(28,a) \;\) at \(\;a = \) e.g. 2