Properties

Label 3525.7
Modulus $3525$
Conductor $235$
Order $92$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3525, base_ring=CyclotomicField(92))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,23,64]))
 
pari: [g,chi] = znchar(Mod(7,3525))
 

Basic properties

Modulus: \(3525\)
Conductor: \(235\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(92\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{235}(7,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3525.bh

\(\chi_{3525}(7,\cdot)\) \(\chi_{3525}(118,\cdot)\) \(\chi_{3525}(157,\cdot)\) \(\chi_{3525}(307,\cdot)\) \(\chi_{3525}(343,\cdot)\) \(\chi_{3525}(382,\cdot)\) \(\chi_{3525}(418,\cdot)\) \(\chi_{3525}(457,\cdot)\) \(\chi_{3525}(568,\cdot)\) \(\chi_{3525}(643,\cdot)\) \(\chi_{3525}(682,\cdot)\) \(\chi_{3525}(907,\cdot)\) \(\chi_{3525}(943,\cdot)\) \(\chi_{3525}(982,\cdot)\) \(\chi_{3525}(1093,\cdot)\) \(\chi_{3525}(1132,\cdot)\) \(\chi_{3525}(1207,\cdot)\) \(\chi_{3525}(1243,\cdot)\) \(\chi_{3525}(1318,\cdot)\) \(\chi_{3525}(1507,\cdot)\) \(\chi_{3525}(1657,\cdot)\) \(\chi_{3525}(1807,\cdot)\) \(\chi_{3525}(1882,\cdot)\) \(\chi_{3525}(2143,\cdot)\) \(\chi_{3525}(2218,\cdot)\) \(\chi_{3525}(2293,\cdot)\) \(\chi_{3525}(2368,\cdot)\) \(\chi_{3525}(2518,\cdot)\) \(\chi_{3525}(2593,\cdot)\) \(\chi_{3525}(2668,\cdot)\) ...

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: $\Q(\zeta_{92})$
Fixed field: Number field defined by a degree 92 polynomial

Values on generators

\((2351,1552,2026)\) → \((1,i,e\left(\frac{16}{23}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 3525 }(7, a) \) \(-1\)\(1\)\(e\left(\frac{71}{92}\right)\)\(e\left(\frac{25}{46}\right)\)\(e\left(\frac{47}{92}\right)\)\(e\left(\frac{29}{92}\right)\)\(e\left(\frac{20}{23}\right)\)\(e\left(\frac{37}{92}\right)\)\(e\left(\frac{13}{46}\right)\)\(e\left(\frac{2}{23}\right)\)\(e\left(\frac{35}{92}\right)\)\(e\left(\frac{37}{46}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3525 }(7,a) \;\) at \(\;a = \) e.g. 2