Properties

Label 1441.1007
Modulus $1441$
Conductor $1441$
Order $130$
Real no
Primitive yes
Minimal yes
Parity even

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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([117,61]))
 
pari: [g,chi] = znchar(Mod(1007,1441))
 

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.cb

\(\chi_{1441}(83,\cdot)\) \(\chi_{1441}(96,\cdot)\) \(\chi_{1441}(139,\cdot)\) \(\chi_{1441}(160,\cdot)\) \(\chi_{1441}(226,\cdot)\) \(\chi_{1441}(249,\cdot)\) \(\chi_{1441}(272,\cdot)\) \(\chi_{1441}(288,\cdot)\) \(\chi_{1441}(316,\cdot)\) \(\chi_{1441}(338,\cdot)\) \(\chi_{1441}(415,\cdot)\) \(\chi_{1441}(480,\cdot)\) \(\chi_{1441}(491,\cdot)\) \(\chi_{1441}(508,\cdot)\) \(\chi_{1441}(634,\cdot)\) \(\chi_{1441}(644,\cdot)\) \(\chi_{1441}(678,\cdot)\) \(\chi_{1441}(695,\cdot)\) \(\chi_{1441}(712,\cdot)\) \(\chi_{1441}(766,\cdot)\) \(\chi_{1441}(800,\cdot)\) \(\chi_{1441}(816,\cdot)\) \(\chi_{1441}(853,\cdot)\) \(\chi_{1441}(871,\cdot)\) \(\chi_{1441}(919,\cdot)\) \(\chi_{1441}(948,\cdot)\) \(\chi_{1441}(1007,\cdot)\) \(\chi_{1441}(1014,\cdot)\) \(\chi_{1441}(1036,\cdot)\) \(\chi_{1441}(1130,\cdot)\) ...

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

\((1311,133)\) → \((e\left(\frac{9}{10}\right),e\left(\frac{61}{130}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 1441 }(1007, a) \) \(1\)\(1\)\(e\left(\frac{24}{65}\right)\)\(e\left(\frac{64}{65}\right)\)\(e\left(\frac{48}{65}\right)\)\(e\left(\frac{12}{65}\right)\)\(e\left(\frac{23}{65}\right)\)\(e\left(\frac{9}{26}\right)\)\(e\left(\frac{7}{65}\right)\)\(e\left(\frac{63}{65}\right)\)\(e\left(\frac{36}{65}\right)\)\(e\left(\frac{47}{65}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1441 }(1007,a) \;\) at \(\;a = \) e.g. 2