Properties

Label 3525.38
Modulus $3525$
Conductor $3525$
Order $460$
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(460))
 
M = H._module
 
chi = DirichletCharacter(H, M([230,437,170]))
 
pari: [g,chi] = znchar(Mod(38,3525))
 

Basic properties

Modulus: \(3525\)
Conductor: \(3525\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(460\)
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.bv

\(\chi_{3525}(23,\cdot)\) \(\chi_{3525}(38,\cdot)\) \(\chi_{3525}(62,\cdot)\) \(\chi_{3525}(77,\cdot)\) \(\chi_{3525}(92,\cdot)\) \(\chi_{3525}(113,\cdot)\) \(\chi_{3525}(137,\cdot)\) \(\chi_{3525}(152,\cdot)\) \(\chi_{3525}(167,\cdot)\) \(\chi_{3525}(203,\cdot)\) \(\chi_{3525}(227,\cdot)\) \(\chi_{3525}(233,\cdot)\) \(\chi_{3525}(248,\cdot)\) \(\chi_{3525}(278,\cdot)\) \(\chi_{3525}(287,\cdot)\) \(\chi_{3525}(302,\cdot)\) \(\chi_{3525}(308,\cdot)\) \(\chi_{3525}(317,\cdot)\) \(\chi_{3525}(323,\cdot)\) \(\chi_{3525}(362,\cdot)\) \(\chi_{3525}(398,\cdot)\) \(\chi_{3525}(428,\cdot)\) \(\chi_{3525}(452,\cdot)\) \(\chi_{3525}(458,\cdot)\) \(\chi_{3525}(467,\cdot)\) \(\chi_{3525}(503,\cdot)\) \(\chi_{3525}(527,\cdot)\) \(\chi_{3525}(548,\cdot)\) \(\chi_{3525}(587,\cdot)\) \(\chi_{3525}(602,\cdot)\) ...

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

\((2351,1552,2026)\) → \((-1,e\left(\frac{19}{20}\right),e\left(\frac{17}{46}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 3525 }(38, a) \) \(-1\)\(1\)\(e\left(\frac{47}{460}\right)\)\(e\left(\frac{47}{230}\right)\)\(e\left(\frac{53}{92}\right)\)\(e\left(\frac{141}{460}\right)\)\(e\left(\frac{33}{115}\right)\)\(e\left(\frac{53}{460}\right)\)\(e\left(\frac{78}{115}\right)\)\(e\left(\frac{47}{115}\right)\)\(e\left(\frac{351}{460}\right)\)\(e\left(\frac{84}{115}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3525 }(38,a) \;\) at \(\;a = \) e.g. 2