Properties

Label 3525.32
Modulus $3525$
Conductor $705$
Order $92$
Real no
Primitive no
Minimal no
Parity even

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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([46,23,88]))
 
pari: [g,chi] = znchar(Mod(32,3525))
 

Basic properties

Modulus: \(3525\)
Conductor: \(705\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(92\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{705}(32,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3525.bi

\(\chi_{3525}(32,\cdot)\) \(\chi_{3525}(68,\cdot)\) \(\chi_{3525}(143,\cdot)\) \(\chi_{3525}(332,\cdot)\) \(\chi_{3525}(482,\cdot)\) \(\chi_{3525}(632,\cdot)\) \(\chi_{3525}(707,\cdot)\) \(\chi_{3525}(968,\cdot)\) \(\chi_{3525}(1043,\cdot)\) \(\chi_{3525}(1118,\cdot)\) \(\chi_{3525}(1193,\cdot)\) \(\chi_{3525}(1343,\cdot)\) \(\chi_{3525}(1418,\cdot)\) \(\chi_{3525}(1493,\cdot)\) \(\chi_{3525}(1532,\cdot)\) \(\chi_{3525}(1568,\cdot)\) \(\chi_{3525}(1607,\cdot)\) \(\chi_{3525}(1682,\cdot)\) \(\chi_{3525}(1757,\cdot)\) \(\chi_{3525}(1793,\cdot)\) \(\chi_{3525}(1907,\cdot)\) \(\chi_{3525}(1943,\cdot)\) \(\chi_{3525}(1982,\cdot)\) \(\chi_{3525}(2057,\cdot)\) \(\chi_{3525}(2093,\cdot)\) \(\chi_{3525}(2132,\cdot)\) \(\chi_{3525}(2168,\cdot)\) \(\chi_{3525}(2243,\cdot)\) \(\chi_{3525}(2357,\cdot)\) \(\chi_{3525}(2468,\cdot)\) ...

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

\((2351,1552,2026)\) → \((-1,i,e\left(\frac{22}{23}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 3525 }(32, a) \) \(1\)\(1\)\(e\left(\frac{89}{92}\right)\)\(e\left(\frac{43}{46}\right)\)\(e\left(\frac{79}{92}\right)\)\(e\left(\frac{83}{92}\right)\)\(e\left(\frac{9}{46}\right)\)\(e\left(\frac{25}{92}\right)\)\(e\left(\frac{19}{23}\right)\)\(e\left(\frac{20}{23}\right)\)\(e\left(\frac{5}{92}\right)\)\(e\left(\frac{25}{46}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3525 }(32,a) \;\) at \(\;a = \) e.g. 2