Properties

Label 4225.654
Modulus $4225$
Conductor $325$
Order $30$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4225, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([3,5]))
 
pari: [g,chi] = znchar(Mod(654,4225))
 

Basic properties

Modulus: \(4225\)
Conductor: \(325\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{325}(4,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4225.bl

\(\chi_{4225}(654,\cdot)\) \(\chi_{4225}(1544,\cdot)\) \(\chi_{4225}(2344,\cdot)\) \(\chi_{4225}(2389,\cdot)\) \(\chi_{4225}(3189,\cdot)\) \(\chi_{4225}(3234,\cdot)\) \(\chi_{4225}(4034,\cdot)\) \(\chi_{4225}(4079,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: 30.30.3133675547880273211567620924800081638750270940363407135009765625.1

Values on generators

\((677,3551)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(14\)
\( \chi_{ 4225 }(654, a) \) \(1\)\(1\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{3}{5}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4225 }(654,a) \;\) at \(\;a = \) e.g. 2