Properties

Label 3525.4
Modulus $3525$
Conductor $1175$
Order $230$
Real no
Primitive no
Minimal yes
Parity even

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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([0,23,180]))
 
pari: [g,chi] = znchar(Mod(4,3525))
 

Basic properties

Modulus: \(3525\)
Conductor: \(1175\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(230\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1175}(4,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3525.bo

\(\chi_{3525}(4,\cdot)\) \(\chi_{3525}(34,\cdot)\) \(\chi_{3525}(64,\cdot)\) \(\chi_{3525}(79,\cdot)\) \(\chi_{3525}(169,\cdot)\) \(\chi_{3525}(244,\cdot)\) \(\chi_{3525}(259,\cdot)\) \(\chi_{3525}(289,\cdot)\) \(\chi_{3525}(319,\cdot)\) \(\chi_{3525}(379,\cdot)\) \(\chi_{3525}(394,\cdot)\) \(\chi_{3525}(439,\cdot)\) \(\chi_{3525}(484,\cdot)\) \(\chi_{3525}(529,\cdot)\) \(\chi_{3525}(544,\cdot)\) \(\chi_{3525}(559,\cdot)\) \(\chi_{3525}(589,\cdot)\) \(\chi_{3525}(619,\cdot)\) \(\chi_{3525}(664,\cdot)\) \(\chi_{3525}(679,\cdot)\) \(\chi_{3525}(694,\cdot)\) \(\chi_{3525}(709,\cdot)\) \(\chi_{3525}(739,\cdot)\) \(\chi_{3525}(754,\cdot)\) \(\chi_{3525}(769,\cdot)\) \(\chi_{3525}(784,\cdot)\) \(\chi_{3525}(964,\cdot)\) \(\chi_{3525}(994,\cdot)\) \(\chi_{3525}(1084,\cdot)\) \(\chi_{3525}(1144,\cdot)\) ...

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

\((2351,1552,2026)\) → \((1,e\left(\frac{1}{10}\right),e\left(\frac{18}{23}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 3525 }(4, a) \) \(1\)\(1\)\(e\left(\frac{43}{230}\right)\)\(e\left(\frac{43}{115}\right)\)\(e\left(\frac{25}{46}\right)\)\(e\left(\frac{129}{230}\right)\)\(e\left(\frac{9}{115}\right)\)\(e\left(\frac{117}{230}\right)\)\(e\left(\frac{84}{115}\right)\)\(e\left(\frac{86}{115}\right)\)\(e\left(\frac{189}{230}\right)\)\(e\left(\frac{2}{115}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3525 }(4,a) \;\) at \(\;a = \) e.g. 2