Properties

Label 1287.8
Modulus $1287$
Conductor $429$
Order $20$
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(20))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([10,6,5]))
 
pari: [g,chi] = znchar(Mod(8,1287))
 

Basic properties

Modulus: \(1287\)
Conductor: \(429\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{429}(8,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1287.cx

\(\chi_{1287}(8,\cdot)\) \(\chi_{1287}(161,\cdot)\) \(\chi_{1287}(359,\cdot)\) \(\chi_{1287}(512,\cdot)\) \(\chi_{1287}(629,\cdot)\) \(\chi_{1287}(710,\cdot)\) \(\chi_{1287}(827,\cdot)\) \(\chi_{1287}(1097,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: 20.0.16804715511066655991385742163025885437733.1

Values on generators

\((1145,937,496)\) → \((-1,e\left(\frac{3}{10}\right),i)\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(14\)\(16\)\(17\)\(19\)
\(-1\)\(1\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{3}{20}\right)\)\(1\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{3}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1287 }(8,a) \;\) at \(\;a = \) e.g. 2