Properties

Label 1441.13
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([13,18]))
 
pari: [g,chi] = znchar(Mod(13,1441))
 

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.bn

\(\chi_{1441}(7,\cdot)\) \(\chi_{1441}(13,\cdot)\) \(\chi_{1441}(123,\cdot)\) \(\chi_{1441}(140,\cdot)\) \(\chi_{1441}(172,\cdot)\) \(\chi_{1441}(206,\cdot)\) \(\chi_{1441}(248,\cdot)\) \(\chi_{1441}(260,\cdot)\) \(\chi_{1441}(277,\cdot)\) \(\chi_{1441}(305,\cdot)\) \(\chi_{1441}(326,\cdot)\) \(\chi_{1441}(327,\cdot)\) \(\chi_{1441}(336,\cdot)\) \(\chi_{1441}(343,\cdot)\) \(\chi_{1441}(376,\cdot)\) \(\chi_{1441}(387,\cdot)\) \(\chi_{1441}(535,\cdot)\) \(\chi_{1441}(536,\cdot)\) \(\chi_{1441}(552,\cdot)\) \(\chi_{1441}(558,\cdot)\) \(\chi_{1441}(624,\cdot)\) \(\chi_{1441}(633,\cdot)\) \(\chi_{1441}(701,\cdot)\) \(\chi_{1441}(710,\cdot)\) \(\chi_{1441}(732,\cdot)\) \(\chi_{1441}(756,\cdot)\) \(\chi_{1441}(776,\cdot)\) \(\chi_{1441}(877,\cdot)\) \(\chi_{1441}(937,\cdot)\) \(\chi_{1441}(952,\cdot)\) ...

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

\((1311,133)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{9}{65}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\(-1\)\(1\)\(e\left(\frac{31}{130}\right)\)\(e\left(\frac{10}{13}\right)\)\(e\left(\frac{31}{65}\right)\)\(e\left(\frac{10}{13}\right)\)\(e\left(\frac{1}{130}\right)\)\(e\left(\frac{129}{130}\right)\)\(e\left(\frac{93}{130}\right)\)\(e\left(\frac{7}{13}\right)\)\(e\left(\frac{1}{130}\right)\)\(e\left(\frac{16}{65}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1441 }(13,a) \;\) at \(\;a = \) e.g. 2