Properties

Label 1441.10
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([65,47]))
 
pari: [g,chi] = znchar(Mod(10,1441))
 

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.ca

\(\chi_{1441}(10,\cdot)\) \(\chi_{1441}(54,\cdot)\) \(\chi_{1441}(76,\cdot)\) \(\chi_{1441}(87,\cdot)\) \(\chi_{1441}(98,\cdot)\) \(\chi_{1441}(120,\cdot)\) \(\chi_{1441}(153,\cdot)\) \(\chi_{1441}(197,\cdot)\) \(\chi_{1441}(219,\cdot)\) \(\chi_{1441}(241,\cdot)\) \(\chi_{1441}(285,\cdot)\) \(\chi_{1441}(318,\cdot)\) \(\chi_{1441}(329,\cdot)\) \(\chi_{1441}(373,\cdot)\) \(\chi_{1441}(384,\cdot)\) \(\chi_{1441}(395,\cdot)\) \(\chi_{1441}(450,\cdot)\) \(\chi_{1441}(483,\cdot)\) \(\chi_{1441}(538,\cdot)\) \(\chi_{1441}(648,\cdot)\) \(\chi_{1441}(681,\cdot)\) \(\chi_{1441}(692,\cdot)\) \(\chi_{1441}(758,\cdot)\) \(\chi_{1441}(868,\cdot)\) \(\chi_{1441}(879,\cdot)\) \(\chi_{1441}(890,\cdot)\) \(\chi_{1441}(901,\cdot)\) \(\chi_{1441}(912,\cdot)\) \(\chi_{1441}(923,\cdot)\) \(\chi_{1441}(934,\cdot)\) ...

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

\((1311,133)\) → \((-1,e\left(\frac{47}{130}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\(1\)\(1\)\(e\left(\frac{56}{65}\right)\)\(e\left(\frac{2}{65}\right)\)\(e\left(\frac{47}{65}\right)\)\(e\left(\frac{41}{65}\right)\)\(e\left(\frac{58}{65}\right)\)\(e\left(\frac{27}{130}\right)\)\(e\left(\frac{38}{65}\right)\)\(e\left(\frac{4}{65}\right)\)\(e\left(\frac{32}{65}\right)\)\(e\left(\frac{49}{65}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1441 }(10,a) \;\) at \(\;a = \) e.g. 2