Properties

Label 4025.1908
Modulus $4025$
Conductor $4025$
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(4025, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([9,40,30]))
 
pari: [g,chi] = znchar(Mod(1908,4025))
 

Basic properties

Modulus: \(4025\)
Conductor: \(4025\)
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 4025.cg

\(\chi_{4025}(137,\cdot)\) \(\chi_{4025}(298,\cdot)\) \(\chi_{4025}(597,\cdot)\) \(\chi_{4025}(758,\cdot)\) \(\chi_{4025}(942,\cdot)\) \(\chi_{4025}(1103,\cdot)\) \(\chi_{4025}(1402,\cdot)\) \(\chi_{4025}(1563,\cdot)\) \(\chi_{4025}(1747,\cdot)\) \(\chi_{4025}(1908,\cdot)\) \(\chi_{4025}(2552,\cdot)\) \(\chi_{4025}(2713,\cdot)\) \(\chi_{4025}(3012,\cdot)\) \(\chi_{4025}(3173,\cdot)\) \(\chi_{4025}(3817,\cdot)\) \(\chi_{4025}(3978,\cdot)\)

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

\((2577,1151,3501)\) → \((e\left(\frac{3}{20}\right),e\left(\frac{2}{3}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 4025 }(1908, a) \) \(1\)\(1\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{43}{60}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{41}{60}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{14}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4025 }(1908,a) \;\) at \(\;a = \) e.g. 2