Properties

Label 3525.37
Modulus $3525$
Conductor $1175$
Order $460$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3525, base_ring=CyclotomicField(460))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,207,420]))
 
pari: [g,chi] = znchar(Mod(37,3525))
 

Basic properties

Modulus: \(3525\)
Conductor: \(1175\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(460\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1175}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3525.bu

\(\chi_{3525}(28,\cdot)\) \(\chi_{3525}(37,\cdot)\) \(\chi_{3525}(97,\cdot)\) \(\chi_{3525}(103,\cdot)\) \(\chi_{3525}(112,\cdot)\) \(\chi_{3525}(148,\cdot)\) \(\chi_{3525}(178,\cdot)\) \(\chi_{3525}(202,\cdot)\) \(\chi_{3525}(238,\cdot)\) \(\chi_{3525}(247,\cdot)\) \(\chi_{3525}(253,\cdot)\) \(\chi_{3525}(262,\cdot)\) \(\chi_{3525}(277,\cdot)\) \(\chi_{3525}(298,\cdot)\) \(\chi_{3525}(337,\cdot)\) \(\chi_{3525}(388,\cdot)\) \(\chi_{3525}(397,\cdot)\) \(\chi_{3525}(403,\cdot)\) \(\chi_{3525}(412,\cdot)\) \(\chi_{3525}(427,\cdot)\) \(\chi_{3525}(448,\cdot)\) \(\chi_{3525}(472,\cdot)\) \(\chi_{3525}(478,\cdot)\) \(\chi_{3525}(487,\cdot)\) \(\chi_{3525}(502,\cdot)\) \(\chi_{3525}(523,\cdot)\) \(\chi_{3525}(538,\cdot)\) \(\chi_{3525}(553,\cdot)\) \(\chi_{3525}(592,\cdot)\) \(\chi_{3525}(598,\cdot)\) ...

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: $\Q(\zeta_{460})$
Fixed field: Number field defined by a degree 460 polynomial (not computed)

Values on generators

\((2351,1552,2026)\) → \((1,e\left(\frac{9}{20}\right),e\left(\frac{21}{23}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 3525 }(37, a) \) \(-1\)\(1\)\(e\left(\frac{407}{460}\right)\)\(e\left(\frac{177}{230}\right)\)\(e\left(\frac{43}{92}\right)\)\(e\left(\frac{301}{460}\right)\)\(e\left(\frac{68}{115}\right)\)\(e\left(\frac{273}{460}\right)\)\(e\left(\frac{81}{230}\right)\)\(e\left(\frac{62}{115}\right)\)\(e\left(\frac{211}{460}\right)\)\(e\left(\frac{43}{230}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3525 }(37,a) \;\) at \(\;a = \) e.g. 2