Properties

Label 1441.30
Modulus $1441$
Conductor $1441$
Order $130$
Real no
Primitive yes
Minimal yes
Parity even

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([39,119]))
 
pari: [g,chi] = znchar(Mod(30,1441))
 

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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1441.bx

\(\chi_{1441}(29,\cdot)\) \(\chi_{1441}(30,\cdot)\) \(\chi_{1441}(72,\cdot)\) \(\chi_{1441}(85,\cdot)\) \(\chi_{1441}(95,\cdot)\) \(\chi_{1441}(127,\cdot)\) \(\chi_{1441}(162,\cdot)\) \(\chi_{1441}(228,\cdot)\) \(\chi_{1441}(237,\cdot)\) \(\chi_{1441}(250,\cdot)\) \(\chi_{1441}(259,\cdot)\) \(\chi_{1441}(358,\cdot)\) \(\chi_{1441}(365,\cdot)\) \(\chi_{1441}(459,\cdot)\) \(\chi_{1441}(481,\cdot)\) \(\chi_{1441}(513,\cdot)\) \(\chi_{1441}(530,\cdot)\) \(\chi_{1441}(534,\cdot)\) \(\chi_{1441}(541,\cdot)\) \(\chi_{1441}(574,\cdot)\) \(\chi_{1441}(578,\cdot)\) \(\chi_{1441}(580,\cdot)\) \(\chi_{1441}(600,\cdot)\) \(\chi_{1441}(640,\cdot)\) \(\chi_{1441}(646,\cdot)\) \(\chi_{1441}(677,\cdot)\) \(\chi_{1441}(722,\cdot)\) \(\chi_{1441}(788,\cdot)\) \(\chi_{1441}(794,\cdot)\) \(\chi_{1441}(876,\cdot)\) ...

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

\((1311,133)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{119}{130}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\(1\)\(1\)\(e\left(\frac{14}{65}\right)\)\(e\left(\frac{4}{13}\right)\)\(e\left(\frac{28}{65}\right)\)\(e\left(\frac{4}{13}\right)\)\(e\left(\frac{34}{65}\right)\)\(e\left(\frac{127}{130}\right)\)\(e\left(\frac{42}{65}\right)\)\(e\left(\frac{8}{13}\right)\)\(e\left(\frac{34}{65}\right)\)\(e\left(\frac{48}{65}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1441 }(30,a) \;\) at \(\;a = \) e.g. 2