Properties

Label 3150.1997
Modulus $3150$
Conductor $525$
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(3150, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,51,20]))
 
pari: [g,chi] = znchar(Mod(1997,3150))
 

Basic properties

Modulus: \(3150\)
Conductor: \(525\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{525}(422,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3150.ed

\(\chi_{3150}(53,\cdot)\) \(\chi_{3150}(233,\cdot)\) \(\chi_{3150}(683,\cdot)\) \(\chi_{3150}(737,\cdot)\) \(\chi_{3150}(863,\cdot)\) \(\chi_{3150}(1187,\cdot)\) \(\chi_{3150}(1313,\cdot)\) \(\chi_{3150}(1367,\cdot)\) \(\chi_{3150}(1817,\cdot)\) \(\chi_{3150}(1997,\cdot)\) \(\chi_{3150}(2123,\cdot)\) \(\chi_{3150}(2447,\cdot)\) \(\chi_{3150}(2573,\cdot)\) \(\chi_{3150}(2627,\cdot)\) \(\chi_{3150}(2753,\cdot)\) \(\chi_{3150}(3077,\cdot)\)

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

\((2801,127,451)\) → \((-1,e\left(\frac{17}{20}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 3150 }(1997, a) \) \(1\)\(1\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{53}{60}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{19}{60}\right)\)\(e\left(\frac{9}{10}\right)\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3150 }(1997,a) \;\) at \(\;a = \) e.g. 2