Properties

Label 1350.773
Modulus $1350$
Conductor $225$
Order $60$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1350, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([50,33]))
 
pari: [g,chi] = znchar(Mod(773,1350))
 

Basic properties

Modulus: \(1350\)
Conductor: \(225\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{225}(23,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1350.bd

\(\chi_{1350}(17,\cdot)\) \(\chi_{1350}(197,\cdot)\) \(\chi_{1350}(233,\cdot)\) \(\chi_{1350}(287,\cdot)\) \(\chi_{1350}(413,\cdot)\) \(\chi_{1350}(467,\cdot)\) \(\chi_{1350}(503,\cdot)\) \(\chi_{1350}(683,\cdot)\) \(\chi_{1350}(737,\cdot)\) \(\chi_{1350}(773,\cdot)\) \(\chi_{1350}(827,\cdot)\) \(\chi_{1350}(953,\cdot)\) \(\chi_{1350}(1097,\cdot)\) \(\chi_{1350}(1223,\cdot)\) \(\chi_{1350}(1277,\cdot)\) \(\chi_{1350}(1313,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((1001,1027)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{11}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 1350 }(773, a) \) \(1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{7}{60}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{11}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1350 }(773,a) \;\) at \(\;a = \) e.g. 2