Properties

Label 1441.476
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([104,11]))
 
pari: [g,chi] = znchar(Mod(476,1441))
 

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.cd

\(\chi_{1441}(26,\cdot)\) \(\chi_{1441}(124,\cdot)\) \(\chi_{1441}(137,\cdot)\) \(\chi_{1441}(141,\cdot)\) \(\chi_{1441}(148,\cdot)\) \(\chi_{1441}(168,\cdot)\) \(\chi_{1441}(181,\cdot)\) \(\chi_{1441}(185,\cdot)\) \(\chi_{1441}(207,\cdot)\) \(\chi_{1441}(246,\cdot)\) \(\chi_{1441}(247,\cdot)\) \(\chi_{1441}(284,\cdot)\) \(\chi_{1441}(355,\cdot)\) \(\chi_{1441}(366,\cdot)\) \(\chi_{1441}(388,\cdot)\) \(\chi_{1441}(401,\cdot)\) \(\chi_{1441}(433,\cdot)\) \(\chi_{1441}(476,\cdot)\) \(\chi_{1441}(511,\cdot)\) \(\chi_{1441}(553,\cdot)\) \(\chi_{1441}(554,\cdot)\) \(\chi_{1441}(581,\cdot)\) \(\chi_{1441}(609,\cdot)\) \(\chi_{1441}(619,\cdot)\) \(\chi_{1441}(669,\cdot)\) \(\chi_{1441}(686,\cdot)\) \(\chi_{1441}(742,\cdot)\) \(\chi_{1441}(752,\cdot)\) \(\chi_{1441}(753,\cdot)\) \(\chi_{1441}(774,\cdot)\) ...

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

\((1311,133)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{11}{130}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 1441 }(476, a) \) \(-1\)\(1\)\(e\left(\frac{23}{26}\right)\)\(e\left(\frac{32}{65}\right)\)\(e\left(\frac{10}{13}\right)\)\(e\left(\frac{6}{65}\right)\)\(e\left(\frac{49}{130}\right)\)\(e\left(\frac{47}{65}\right)\)\(e\left(\frac{17}{26}\right)\)\(e\left(\frac{64}{65}\right)\)\(e\left(\frac{127}{130}\right)\)\(e\left(\frac{17}{65}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1441 }(476,a) \;\) at \(\;a = \) e.g. 2