Properties

Label 3525.14
Modulus $3525$
Conductor $3525$
Order $230$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3525, base_ring=CyclotomicField(230))
 
M = H._module
 
chi = DirichletCharacter(H, M([115,69,20]))
 
pari: [g,chi] = znchar(Mod(14,3525))
 

Basic properties

Modulus: \(3525\)
Conductor: \(3525\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(230\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3525.bq

\(\chi_{3525}(14,\cdot)\) \(\chi_{3525}(59,\cdot)\) \(\chi_{3525}(89,\cdot)\) \(\chi_{3525}(119,\cdot)\) \(\chi_{3525}(194,\cdot)\) \(\chi_{3525}(209,\cdot)\) \(\chi_{3525}(239,\cdot)\) \(\chi_{3525}(269,\cdot)\) \(\chi_{3525}(284,\cdot)\) \(\chi_{3525}(314,\cdot)\) \(\chi_{3525}(404,\cdot)\) \(\chi_{3525}(479,\cdot)\) \(\chi_{3525}(494,\cdot)\) \(\chi_{3525}(554,\cdot)\) \(\chi_{3525}(614,\cdot)\) \(\chi_{3525}(629,\cdot)\) \(\chi_{3525}(719,\cdot)\) \(\chi_{3525}(764,\cdot)\) \(\chi_{3525}(779,\cdot)\) \(\chi_{3525}(794,\cdot)\) \(\chi_{3525}(854,\cdot)\) \(\chi_{3525}(914,\cdot)\) \(\chi_{3525}(929,\cdot)\) \(\chi_{3525}(944,\cdot)\) \(\chi_{3525}(989,\cdot)\) \(\chi_{3525}(1004,\cdot)\) \(\chi_{3525}(1019,\cdot)\) \(\chi_{3525}(1109,\cdot)\) \(\chi_{3525}(1184,\cdot)\) \(\chi_{3525}(1229,\cdot)\) ...

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: $\Q(\zeta_{115})$
Fixed field: Number field defined by a degree 230 polynomial (not computed)

Values on generators

\((2351,1552,2026)\) → \((-1,e\left(\frac{3}{10}\right),e\left(\frac{2}{23}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 3525 }(14, a) \) \(-1\)\(1\)\(e\left(\frac{42}{115}\right)\)\(e\left(\frac{84}{115}\right)\)\(e\left(\frac{13}{46}\right)\)\(e\left(\frac{11}{115}\right)\)\(e\left(\frac{209}{230}\right)\)\(e\left(\frac{151}{230}\right)\)\(e\left(\frac{149}{230}\right)\)\(e\left(\frac{53}{115}\right)\)\(e\left(\frac{91}{115}\right)\)\(e\left(\frac{36}{115}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3525 }(14,a) \;\) at \(\;a = \) e.g. 2