Properties

Label 1441.1010
Modulus $1441$
Conductor $1441$
Order $130$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1441, base_ring=CyclotomicField(130))
 
M = H._module
 
chi = DirichletCharacter(H, M([78,101]))
 
pari: [g,chi] = znchar(Mod(1010,1441))
 

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.bw

\(\chi_{1441}(115,\cdot)\) \(\chi_{1441}(213,\cdot)\) \(\chi_{1441}(234,\cdot)\) \(\chi_{1441}(251,\cdot)\) \(\chi_{1441}(302,\cdot)\) \(\chi_{1441}(334,\cdot)\) \(\chi_{1441}(389,\cdot)\) \(\chi_{1441}(399,\cdot)\) \(\chi_{1441}(410,\cdot)\) \(\chi_{1441}(423,\cdot)\) \(\chi_{1441}(443,\cdot)\) \(\chi_{1441}(460,\cdot)\) \(\chi_{1441}(478,\cdot)\) \(\chi_{1441}(499,\cdot)\) \(\chi_{1441}(509,\cdot)\) \(\chi_{1441}(521,\cdot)\) \(\chi_{1441}(526,\cdot)\) \(\chi_{1441}(555,\cdot)\) \(\chi_{1441}(614,\cdot)\) \(\chi_{1441}(620,\cdot)\) \(\chi_{1441}(621,\cdot)\) \(\chi_{1441}(643,\cdot)\) \(\chi_{1441}(663,\cdot)\) \(\chi_{1441}(773,\cdot)\) \(\chi_{1441}(779,\cdot)\) \(\chi_{1441}(796,\cdot)\) \(\chi_{1441}(823,\cdot)\) \(\chi_{1441}(840,\cdot)\) \(\chi_{1441}(852,\cdot)\) \(\chi_{1441}(862,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ 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

\((1311,133)\) → \((e\left(\frac{3}{5}\right),e\left(\frac{101}{130}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 1441 }(1010, a) \) \(-1\)\(1\)\(e\left(\frac{49}{130}\right)\)\(e\left(\frac{48}{65}\right)\)\(e\left(\frac{49}{65}\right)\)\(e\left(\frac{9}{65}\right)\)\(e\left(\frac{3}{26}\right)\)\(e\left(\frac{51}{65}\right)\)\(e\left(\frac{17}{130}\right)\)\(e\left(\frac{31}{65}\right)\)\(e\left(\frac{67}{130}\right)\)\(e\left(\frac{32}{65}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1441 }(1010,a) \;\) at \(\;a = \) e.g. 2