Properties

Label 1441.23
Modulus $1441$
Conductor $131$
Order $130$
Real no
Primitive no
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([0,23]))
 
pari: [g,chi] = znchar(Mod(23,1441))
 

Basic properties

Modulus: \(1441\)
Conductor: \(131\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(130\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{131}(23,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1441.bs

\(\chi_{1441}(23,\cdot)\) \(\chi_{1441}(56,\cdot)\) \(\chi_{1441}(67,\cdot)\) \(\chi_{1441}(111,\cdot)\) \(\chi_{1441}(122,\cdot)\) \(\chi_{1441}(133,\cdot)\) \(\chi_{1441}(188,\cdot)\) \(\chi_{1441}(221,\cdot)\) \(\chi_{1441}(276,\cdot)\) \(\chi_{1441}(386,\cdot)\) \(\chi_{1441}(419,\cdot)\) \(\chi_{1441}(430,\cdot)\) \(\chi_{1441}(496,\cdot)\) \(\chi_{1441}(606,\cdot)\) \(\chi_{1441}(617,\cdot)\) \(\chi_{1441}(628,\cdot)\) \(\chi_{1441}(639,\cdot)\) \(\chi_{1441}(650,\cdot)\) \(\chi_{1441}(661,\cdot)\) \(\chi_{1441}(672,\cdot)\) \(\chi_{1441}(705,\cdot)\) \(\chi_{1441}(727,\cdot)\) \(\chi_{1441}(738,\cdot)\) \(\chi_{1441}(771,\cdot)\) \(\chi_{1441}(782,\cdot)\) \(\chi_{1441}(815,\cdot)\) \(\chi_{1441}(826,\cdot)\) \(\chi_{1441}(881,\cdot)\) \(\chi_{1441}(892,\cdot)\) \(\chi_{1441}(914,\cdot)\) ...

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

\((1311,133)\) → \((1,e\left(\frac{23}{130}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\(-1\)\(1\)\(e\left(\frac{23}{130}\right)\)\(e\left(\frac{48}{65}\right)\)\(e\left(\frac{23}{65}\right)\)\(e\left(\frac{9}{65}\right)\)\(e\left(\frac{119}{130}\right)\)\(e\left(\frac{64}{65}\right)\)\(e\left(\frac{69}{130}\right)\)\(e\left(\frac{31}{65}\right)\)\(e\left(\frac{41}{130}\right)\)\(e\left(\frac{6}{65}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1441 }(23,a) \;\) at \(\;a = \) e.g. 2