Properties

Label 1287.37
Modulus $1287$
Conductor $143$
Order $60$
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(1287, base_ring=CyclotomicField(60))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,12,35]))
 
pari: [g,chi] = znchar(Mod(37,1287))
 

Basic properties

Modulus: \(1287\)
Conductor: \(143\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{143}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1287.eg

\(\chi_{1287}(37,\cdot)\) \(\chi_{1287}(136,\cdot)\) \(\chi_{1287}(163,\cdot)\) \(\chi_{1287}(262,\cdot)\) \(\chi_{1287}(280,\cdot)\) \(\chi_{1287}(379,\cdot)\) \(\chi_{1287}(388,\cdot)\) \(\chi_{1287}(487,\cdot)\) \(\chi_{1287}(631,\cdot)\) \(\chi_{1287}(730,\cdot)\) \(\chi_{1287}(856,\cdot)\) \(\chi_{1287}(955,\cdot)\) \(\chi_{1287}(973,\cdot)\) \(\chi_{1287}(982,\cdot)\) \(\chi_{1287}(1072,\cdot)\) \(\chi_{1287}(1081,\cdot)\)

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

Values on generators

\((1145,937,496)\) → \((1,e\left(\frac{1}{5}\right),e\left(\frac{7}{12}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(14\)\(16\)\(17\)\(19\)
\(-1\)\(1\)\(e\left(\frac{47}{60}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{31}{60}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1287 }(37,a) \;\) at \(\;a = \) e.g. 2