Properties

Label 1287.115
Modulus $1287$
Conductor $1287$
Order $60$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1287, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([40,24,35]))
 
pari: [g,chi] = znchar(Mod(115,1287))
 

Basic properties

Modulus: \(1287\)
Conductor: \(1287\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
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 1287.ep

\(\chi_{1287}(58,\cdot)\) \(\chi_{1287}(115,\cdot)\) \(\chi_{1287}(202,\cdot)\) \(\chi_{1287}(223,\cdot)\) \(\chi_{1287}(466,\cdot)\) \(\chi_{1287}(526,\cdot)\) \(\chi_{1287}(553,\cdot)\) \(\chi_{1287}(643,\cdot)\) \(\chi_{1287}(691,\cdot)\) \(\chi_{1287}(808,\cdot)\) \(\chi_{1287}(817,\cdot)\) \(\chi_{1287}(994,\cdot)\) \(\chi_{1287}(1021,\cdot)\) \(\chi_{1287}(1138,\cdot)\) \(\chi_{1287}(1159,\cdot)\) \(\chi_{1287}(1285,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((1145,937,496)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{2}{5}\right),e\left(\frac{7}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 1287 }(115, a) \) \(-1\)\(1\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{11}{60}\right)\)\(e\left(\frac{53}{60}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{7}{60}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1287 }(115,a) \;\) at \(\;a = \) e.g. 2