Properties

Label 1502.61
Modulus $1502$
Conductor $751$
Order $75$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1502, base_ring=CyclotomicField(150))
 
M = H._module
 
chi = DirichletCharacter(H, M([112]))
 
pari: [g,chi] = znchar(Mod(61,1502))
 

Basic properties

Modulus: \(1502\)
Conductor: \(751\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(75\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{751}(61,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1502.k

\(\chi_{1502}(61,\cdot)\) \(\chi_{1502}(107,\cdot)\) \(\chi_{1502}(119,\cdot)\) \(\chi_{1502}(121,\cdot)\) \(\chi_{1502}(131,\cdot)\) \(\chi_{1502}(163,\cdot)\) \(\chi_{1502}(197,\cdot)\) \(\chi_{1502}(229,\cdot)\) \(\chi_{1502}(273,\cdot)\) \(\chi_{1502}(299,\cdot)\) \(\chi_{1502}(307,\cdot)\) \(\chi_{1502}(399,\cdot)\) \(\chi_{1502}(405,\cdot)\) \(\chi_{1502}(451,\cdot)\) \(\chi_{1502}(471,\cdot)\) \(\chi_{1502}(631,\cdot)\) \(\chi_{1502}(637,\cdot)\) \(\chi_{1502}(639,\cdot)\) \(\chi_{1502}(643,\cdot)\) \(\chi_{1502}(717,\cdot)\) \(\chi_{1502}(783,\cdot)\) \(\chi_{1502}(803,\cdot)\) \(\chi_{1502}(837,\cdot)\) \(\chi_{1502}(881,\cdot)\) \(\chi_{1502}(931,\cdot)\) \(\chi_{1502}(935,\cdot)\) \(\chi_{1502}(941,\cdot)\) \(\chi_{1502}(945,\cdot)\) \(\chi_{1502}(951,\cdot)\) \(\chi_{1502}(1035,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{75})$
Fixed field: Number field defined by a degree 75 polynomial

Values on generators

\(3\) → \(e\left(\frac{56}{75}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 1502 }(61, a) \) \(1\)\(1\)\(e\left(\frac{56}{75}\right)\)\(e\left(\frac{41}{75}\right)\)\(e\left(\frac{2}{25}\right)\)\(e\left(\frac{37}{75}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{23}{75}\right)\)\(e\left(\frac{22}{75}\right)\)\(e\left(\frac{49}{75}\right)\)\(e\left(\frac{8}{75}\right)\)\(e\left(\frac{62}{75}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1502 }(61,a) \;\) at \(\;a = \) e.g. 2