Properties

Label 3525.22
Modulus $3525$
Conductor $1175$
Order $460$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3525, base_ring=CyclotomicField(460))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,391,250]))
 
pari: [g,chi] = znchar(Mod(22,3525))
 

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 \(\chi_{1175}(22,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3525.bs

\(\chi_{3525}(13,\cdot)\) \(\chi_{3525}(22,\cdot)\) \(\chi_{3525}(52,\cdot)\) \(\chi_{3525}(58,\cdot)\) \(\chi_{3525}(67,\cdot)\) \(\chi_{3525}(73,\cdot)\) \(\chi_{3525}(88,\cdot)\) \(\chi_{3525}(127,\cdot)\) \(\chi_{3525}(133,\cdot)\) \(\chi_{3525}(163,\cdot)\) \(\chi_{3525}(172,\cdot)\) \(\chi_{3525}(208,\cdot)\) \(\chi_{3525}(217,\cdot)\) \(\chi_{3525}(223,\cdot)\) \(\chi_{3525}(292,\cdot)\) \(\chi_{3525}(313,\cdot)\) \(\chi_{3525}(322,\cdot)\) \(\chi_{3525}(352,\cdot)\) \(\chi_{3525}(358,\cdot)\) \(\chi_{3525}(367,\cdot)\) \(\chi_{3525}(373,\cdot)\) \(\chi_{3525}(433,\cdot)\) \(\chi_{3525}(442,\cdot)\) \(\chi_{3525}(463,\cdot)\) \(\chi_{3525}(508,\cdot)\) \(\chi_{3525}(547,\cdot)\) \(\chi_{3525}(562,\cdot)\) \(\chi_{3525}(577,\cdot)\) \(\chi_{3525}(583,\cdot)\) \(\chi_{3525}(622,\cdot)\) ...

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

\((2351,1552,2026)\) → \((1,e\left(\frac{17}{20}\right),e\left(\frac{25}{46}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 3525 }(22, a) \) \(1\)\(1\)\(e\left(\frac{291}{460}\right)\)\(e\left(\frac{61}{230}\right)\)\(e\left(\frac{59}{92}\right)\)\(e\left(\frac{413}{460}\right)\)\(e\left(\frac{93}{230}\right)\)\(e\left(\frac{59}{460}\right)\)\(e\left(\frac{63}{230}\right)\)\(e\left(\frac{61}{115}\right)\)\(e\left(\frac{343}{460}\right)\)\(e\left(\frac{87}{115}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3525 }(22,a) \;\) at \(\;a = \) e.g. 2