Properties

Label 1287.530
Modulus $1287$
Conductor $429$
Order $30$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1287, base_ring=CyclotomicField(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([15,3,25]))
 
pari: [g,chi] = znchar(Mod(530,1287))
 

Basic properties

Modulus: \(1287\)
Conductor: \(429\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{429}(101,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1287.dc

\(\chi_{1287}(17,\cdot)\) \(\chi_{1287}(62,\cdot)\) \(\chi_{1287}(134,\cdot)\) \(\chi_{1287}(413,\cdot)\) \(\chi_{1287}(530,\cdot)\) \(\chi_{1287}(602,\cdot)\) \(\chi_{1287}(953,\cdot)\) \(\chi_{1287}(998,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: 30.30.1327410130438875352958596311757100603538624959445253457304779421.1

Values on generators

\((1145,937,496)\) → \((-1,e\left(\frac{1}{10}\right),e\left(\frac{5}{6}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 1287 }(530, a) \) \(1\)\(1\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{7}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1287 }(530,a) \;\) at \(\;a = \) e.g. 2