Properties

Label 4225.28
Modulus $4225$
Conductor $4225$
Order $780$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4225, base_ring=CyclotomicField(780))
 
M = H._module
 
chi = DirichletCharacter(H, M([273,545]))
 
pari: [g,chi] = znchar(Mod(28,4225))
 

Basic properties

Modulus: \(4225\)
Conductor: \(4225\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(780\)
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 4225.db

\(\chi_{4225}(28,\cdot)\) \(\chi_{4225}(37,\cdot)\) \(\chi_{4225}(58,\cdot)\) \(\chi_{4225}(72,\cdot)\) \(\chi_{4225}(102,\cdot)\) \(\chi_{4225}(123,\cdot)\) \(\chi_{4225}(137,\cdot)\) \(\chi_{4225}(158,\cdot)\) \(\chi_{4225}(167,\cdot)\) \(\chi_{4225}(202,\cdot)\) \(\chi_{4225}(223,\cdot)\) \(\chi_{4225}(253,\cdot)\) \(\chi_{4225}(267,\cdot)\) \(\chi_{4225}(288,\cdot)\) \(\chi_{4225}(297,\cdot)\) \(\chi_{4225}(353,\cdot)\) \(\chi_{4225}(362,\cdot)\) \(\chi_{4225}(383,\cdot)\) \(\chi_{4225}(397,\cdot)\) \(\chi_{4225}(448,\cdot)\) \(\chi_{4225}(462,\cdot)\) \(\chi_{4225}(483,\cdot)\) \(\chi_{4225}(492,\cdot)\) \(\chi_{4225}(513,\cdot)\) \(\chi_{4225}(527,\cdot)\) \(\chi_{4225}(548,\cdot)\) \(\chi_{4225}(578,\cdot)\) \(\chi_{4225}(592,\cdot)\) \(\chi_{4225}(613,\cdot)\) \(\chi_{4225}(622,\cdot)\) ...

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: $\Q(\zeta_{780})$
Fixed field: Number field defined by a degree 780 polynomial (not computed)

Values on generators

\((677,3551)\) → \((e\left(\frac{7}{20}\right),e\left(\frac{109}{156}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(14\)
\( \chi_{ 4225 }(28, a) \) \(1\)\(1\)\(e\left(\frac{19}{390}\right)\)\(e\left(\frac{71}{780}\right)\)\(e\left(\frac{19}{195}\right)\)\(e\left(\frac{109}{780}\right)\)\(e\left(\frac{20}{39}\right)\)\(e\left(\frac{19}{130}\right)\)\(e\left(\frac{71}{390}\right)\)\(e\left(\frac{443}{780}\right)\)\(e\left(\frac{49}{260}\right)\)\(e\left(\frac{73}{130}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4225 }(28,a) \;\) at \(\;a = \) e.g. 2