Properties

Label 1400.387
Modulus $1400$
Conductor $1400$
Order $60$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1400, base_ring=CyclotomicField(60))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([30,30,27,20]))
 
pari: [g,chi] = znchar(Mod(387,1400))
 

Basic properties

Modulus: \(1400\)
Conductor: \(1400\)
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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1400.do

\(\chi_{1400}(67,\cdot)\) \(\chi_{1400}(123,\cdot)\) \(\chi_{1400}(163,\cdot)\) \(\chi_{1400}(347,\cdot)\) \(\chi_{1400}(387,\cdot)\) \(\chi_{1400}(403,\cdot)\) \(\chi_{1400}(627,\cdot)\) \(\chi_{1400}(667,\cdot)\) \(\chi_{1400}(683,\cdot)\) \(\chi_{1400}(723,\cdot)\) \(\chi_{1400}(947,\cdot)\) \(\chi_{1400}(963,\cdot)\) \(\chi_{1400}(1003,\cdot)\) \(\chi_{1400}(1187,\cdot)\) \(\chi_{1400}(1227,\cdot)\) \(\chi_{1400}(1283,\cdot)\)

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

\((351,701,1177,801)\) → \((-1,-1,e\left(\frac{9}{20}\right),e\left(\frac{1}{3}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{11}{60}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{7}{60}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{13}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1400 }(387,a) \;\) at \(\;a = \) e.g. 2