Properties

Label 38025.71
Modulus $38025$
Conductor $12675$
Order $780$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(38025, base_ring=CyclotomicField(780))
 
M = H._module
 
chi = DirichletCharacter(H, M([390,468,685]))
 
pari: [g,chi] = znchar(Mod(71,38025))
 

Basic properties

Modulus: \(38025\)
Conductor: \(12675\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(780\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{12675}(71,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 38025.ok

\(\chi_{38025}(71,\cdot)\) \(\chi_{38025}(206,\cdot)\) \(\chi_{38025}(431,\cdot)\) \(\chi_{38025}(566,\cdot)\) \(\chi_{38025}(656,\cdot)\) \(\chi_{38025}(791,\cdot)\) \(\chi_{38025}(1016,\cdot)\) \(\chi_{38025}(1241,\cdot)\) \(\chi_{38025}(1736,\cdot)\) \(\chi_{38025}(1961,\cdot)\) \(\chi_{38025}(2186,\cdot)\) \(\chi_{38025}(2321,\cdot)\) \(\chi_{38025}(2411,\cdot)\) \(\chi_{38025}(2546,\cdot)\) \(\chi_{38025}(2771,\cdot)\) \(\chi_{38025}(2906,\cdot)\) \(\chi_{38025}(2996,\cdot)\) \(\chi_{38025}(3356,\cdot)\) \(\chi_{38025}(3491,\cdot)\) \(\chi_{38025}(3581,\cdot)\) \(\chi_{38025}(3716,\cdot)\) \(\chi_{38025}(3941,\cdot)\) \(\chi_{38025}(4166,\cdot)\) \(\chi_{38025}(4661,\cdot)\) \(\chi_{38025}(4886,\cdot)\) \(\chi_{38025}(5111,\cdot)\) \(\chi_{38025}(5246,\cdot)\) \(\chi_{38025}(5336,\cdot)\) \(\chi_{38025}(5471,\cdot)\) \(\chi_{38025}(5696,\cdot)\) ...

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: $\Q(\zeta_{780})$
Fixed field: Number field defined by a degree 780 polynomial (not computed)

Values on generators

\((29576,9127,37351)\) → \((-1,e\left(\frac{3}{5}\right),e\left(\frac{137}{156}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(14\)\(16\)\(17\)\(19\)\(22\)
\( \chi_{ 38025 }(71, a) \) \(1\)\(1\)\(e\left(\frac{763}{780}\right)\)\(e\left(\frac{373}{390}\right)\)\(e\left(\frac{151}{156}\right)\)\(e\left(\frac{243}{260}\right)\)\(e\left(\frac{433}{780}\right)\)\(e\left(\frac{123}{130}\right)\)\(e\left(\frac{178}{195}\right)\)\(e\left(\frac{101}{195}\right)\)\(e\left(\frac{53}{60}\right)\)\(e\left(\frac{8}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 38025 }(71,a) \;\) at \(\;a = \) e.g. 2