Properties

Label 1441.37
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([26,41]))
 
pari: [g,chi] = znchar(Mod(37,1441))
 

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.bo

\(\chi_{1441}(37,\cdot)\) \(\chi_{1441}(82,\cdot)\) \(\chi_{1441}(157,\cdot)\) \(\chi_{1441}(203,\cdot)\) \(\chi_{1441}(218,\cdot)\) \(\chi_{1441}(224,\cdot)\) \(\chi_{1441}(229,\cdot)\) \(\chi_{1441}(235,\cdot)\) \(\chi_{1441}(257,\cdot)\) \(\chi_{1441}(258,\cdot)\) \(\chi_{1441}(345,\cdot)\) \(\chi_{1441}(368,\cdot)\) \(\chi_{1441}(372,\cdot)\) \(\chi_{1441}(377,\cdot)\) \(\chi_{1441}(390,\cdot)\) \(\chi_{1441}(416,\cdot)\) \(\chi_{1441}(422,\cdot)\) \(\chi_{1441}(488,\cdot)\) \(\chi_{1441}(489,\cdot)\) \(\chi_{1441}(504,\cdot)\) \(\chi_{1441}(564,\cdot)\) \(\chi_{1441}(665,\cdot)\) \(\chi_{1441}(685,\cdot)\) \(\chi_{1441}(709,\cdot)\) \(\chi_{1441}(731,\cdot)\) \(\chi_{1441}(740,\cdot)\) \(\chi_{1441}(808,\cdot)\) \(\chi_{1441}(817,\cdot)\) \(\chi_{1441}(883,\cdot)\) \(\chi_{1441}(889,\cdot)\) ...

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

\((1311,133)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{41}{130}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\(-1\)\(1\)\(e\left(\frac{67}{130}\right)\)\(e\left(\frac{4}{13}\right)\)\(e\left(\frac{2}{65}\right)\)\(e\left(\frac{4}{13}\right)\)\(e\left(\frac{107}{130}\right)\)\(e\left(\frac{44}{65}\right)\)\(e\left(\frac{71}{130}\right)\)\(e\left(\frac{8}{13}\right)\)\(e\left(\frac{107}{130}\right)\)\(e\left(\frac{22}{65}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1441 }(37,a) \;\) at \(\;a = \) e.g. 2