Properties

Label 38025.62
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([390,351,110]))
 
pari: [g,chi] = znchar(Mod(62,38025))
 

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 \(\chi_{12675}(62,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 38025.oj

\(\chi_{38025}(17,\cdot)\) \(\chi_{38025}(62,\cdot)\) \(\chi_{38025}(413,\cdot)\) \(\chi_{38025}(602,\cdot)\) \(\chi_{38025}(647,\cdot)\) \(\chi_{38025}(953,\cdot)\) \(\chi_{38025}(998,\cdot)\) \(\chi_{38025}(1187,\cdot)\) \(\chi_{38025}(1538,\cdot)\) \(\chi_{38025}(1583,\cdot)\) \(\chi_{38025}(1772,\cdot)\) \(\chi_{38025}(1817,\cdot)\) \(\chi_{38025}(2123,\cdot)\) \(\chi_{38025}(2402,\cdot)\) \(\chi_{38025}(2708,\cdot)\) \(\chi_{38025}(2753,\cdot)\) \(\chi_{38025}(2942,\cdot)\) \(\chi_{38025}(2987,\cdot)\) \(\chi_{38025}(3338,\cdot)\) \(\chi_{38025}(3878,\cdot)\) \(\chi_{38025}(3923,\cdot)\) \(\chi_{38025}(4112,\cdot)\) \(\chi_{38025}(4463,\cdot)\) \(\chi_{38025}(4508,\cdot)\) \(\chi_{38025}(4697,\cdot)\) \(\chi_{38025}(4742,\cdot)\) \(\chi_{38025}(5327,\cdot)\) \(\chi_{38025}(5633,\cdot)\) \(\chi_{38025}(5678,\cdot)\) \(\chi_{38025}(5867,\cdot)\) ...

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

\((29576,9127,37351)\) → \((-1,e\left(\frac{9}{20}\right),e\left(\frac{11}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(14\)\(16\)\(17\)\(19\)\(22\)
\( \chi_{ 38025 }(62, a) \) \(1\)\(1\)\(e\left(\frac{71}{780}\right)\)\(e\left(\frac{71}{390}\right)\)\(e\left(\frac{53}{156}\right)\)\(e\left(\frac{71}{260}\right)\)\(e\left(\frac{44}{195}\right)\)\(e\left(\frac{28}{65}\right)\)\(e\left(\frac{71}{195}\right)\)\(e\left(\frac{733}{780}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{19}{60}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 38025 }(62,a) \;\) at \(\;a = \) e.g. 2