Properties

Label 1476.5
Modulus $1476$
Conductor $369$
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(1476, base_ring=CyclotomicField(60))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,50,33]))
 
pari: [g,chi] = znchar(Mod(5,1476))
 

Basic properties

Modulus: \(1476\)
Conductor: \(369\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{369}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1476.cg

\(\chi_{1476}(5,\cdot)\) \(\chi_{1476}(77,\cdot)\) \(\chi_{1476}(185,\cdot)\) \(\chi_{1476}(389,\cdot)\) \(\chi_{1476}(497,\cdot)\) \(\chi_{1476}(569,\cdot)\) \(\chi_{1476}(617,\cdot)\) \(\chi_{1476}(677,\cdot)\) \(\chi_{1476}(689,\cdot)\) \(\chi_{1476}(869,\cdot)\) \(\chi_{1476}(941,\cdot)\) \(\chi_{1476}(1109,\cdot)\) \(\chi_{1476}(1181,\cdot)\) \(\chi_{1476}(1361,\cdot)\) \(\chi_{1476}(1373,\cdot)\) \(\chi_{1476}(1433,\cdot)\)

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

\((739,821,1441)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{11}{20}\right))\)

Values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 1476 }(5, a) \) \(-1\)\(1\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{47}{60}\right)\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{43}{60}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{41}{60}\right)\)\(e\left(\frac{1}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1476 }(5,a) \;\) at \(\;a = \) e.g. 2