Properties

Label 4592.25
Modulus $4592$
Conductor $2296$
Order $30$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4592, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,15,20,3]))
 
pari: [g,chi] = znchar(Mod(25,4592))
 

Basic properties

Modulus: \(4592\)
Conductor: \(2296\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2296}(1173,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4592.fz

\(\chi_{4592}(25,\cdot)\) \(\chi_{4592}(681,\cdot)\) \(\chi_{4592}(3161,\cdot)\) \(\chi_{4592}(3385,\cdot)\) \(\chi_{4592}(3721,\cdot)\) \(\chi_{4592}(3817,\cdot)\) \(\chi_{4592}(4041,\cdot)\) \(\chi_{4592}(4377,\cdot)\)

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: Number field defined by a degree 30 polynomial

Values on generators

\((575,3445,3937,785)\) → \((1,-1,e\left(\frac{2}{3}\right),e\left(\frac{1}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 4592 }(25, a) \) \(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{1}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4592 }(25,a) \;\) at \(\;a = \) e.g. 2