Properties

Label 2475.38
Modulus $2475$
Conductor $2475$
Order $60$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,57,24]))
 
pari: [g,chi] = znchar(Mod(38,2475))
 

Basic properties

Modulus: \(2475\)
Conductor: \(2475\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
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 2475.gd

\(\chi_{2475}(38,\cdot)\) \(\chi_{2475}(137,\cdot)\) \(\chi_{2475}(542,\cdot)\) \(\chi_{2475}(608,\cdot)\) \(\chi_{2475}(653,\cdot)\) \(\chi_{2475}(977,\cdot)\) \(\chi_{2475}(1247,\cdot)\) \(\chi_{2475}(1373,\cdot)\) \(\chi_{2475}(1433,\cdot)\) \(\chi_{2475}(1478,\cdot)\) \(\chi_{2475}(1688,\cdot)\) \(\chi_{2475}(1787,\cdot)\) \(\chi_{2475}(1802,\cdot)\) \(\chi_{2475}(2072,\cdot)\) \(\chi_{2475}(2192,\cdot)\) \(\chi_{2475}(2198,\cdot)\)

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

\((551,2377,2026)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{19}{20}\right),e\left(\frac{2}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(13\)\(14\)\(16\)\(17\)\(19\)\(23\)
\( \chi_{ 2475 }(38, a) \) \(1\)\(1\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{47}{60}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{17}{60}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2475 }(38,a) \;\) at \(\;a = \) e.g. 2