Properties

Label 4030.19
Modulus $4030$
Conductor $2015$
Order $60$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4030, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,25,8]))
 
pari: [g,chi] = znchar(Mod(19,4030))
 

Basic properties

Modulus: \(4030\)
Conductor: \(2015\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2015}(19,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4030.hr

\(\chi_{4030}(19,\cdot)\) \(\chi_{4030}(59,\cdot)\) \(\chi_{4030}(609,\cdot)\) \(\chi_{4030}(1099,\cdot)\) \(\chi_{4030}(1289,\cdot)\) \(\chi_{4030}(1619,\cdot)\) \(\chi_{4030}(1879,\cdot)\) \(\chi_{4030}(1909,\cdot)\) \(\chi_{4030}(2229,\cdot)\) \(\chi_{4030}(2459,\cdot)\) \(\chi_{4030}(3079,\cdot)\) \(\chi_{4030}(3109,\cdot)\) \(\chi_{4030}(3269,\cdot)\) \(\chi_{4030}(3729,\cdot)\) \(\chi_{4030}(3789,\cdot)\) \(\chi_{4030}(4019,\cdot)\)

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

\((807,1861,2731)\) → \((-1,e\left(\frac{5}{12}\right),e\left(\frac{2}{15}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 4030 }(19, a) \) \(-1\)\(1\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{7}{60}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{13}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4030 }(19,a) \;\) at \(\;a = \) e.g. 2