Properties

Label 3525.11
Modulus $3525$
Conductor $3525$
Order $230$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3525, base_ring=CyclotomicField(230))
 
M = H._module
 
chi = DirichletCharacter(H, M([115,184,35]))
 
pari: [g,chi] = znchar(Mod(11,3525))
 

Basic properties

Modulus: \(3525\)
Conductor: \(3525\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(230\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3525.bm

\(\chi_{3525}(11,\cdot)\) \(\chi_{3525}(41,\cdot)\) \(\chi_{3525}(86,\cdot)\) \(\chi_{3525}(116,\cdot)\) \(\chi_{3525}(146,\cdot)\) \(\chi_{3525}(161,\cdot)\) \(\chi_{3525}(221,\cdot)\) \(\chi_{3525}(266,\cdot)\) \(\chi_{3525}(311,\cdot)\) \(\chi_{3525}(386,\cdot)\) \(\chi_{3525}(416,\cdot)\) \(\chi_{3525}(446,\cdot)\) \(\chi_{3525}(461,\cdot)\) \(\chi_{3525}(536,\cdot)\) \(\chi_{3525}(641,\cdot)\) \(\chi_{3525}(656,\cdot)\) \(\chi_{3525}(671,\cdot)\) \(\chi_{3525}(716,\cdot)\) \(\chi_{3525}(731,\cdot)\) \(\chi_{3525}(746,\cdot)\) \(\chi_{3525}(791,\cdot)\) \(\chi_{3525}(821,\cdot)\) \(\chi_{3525}(866,\cdot)\) \(\chi_{3525}(881,\cdot)\) \(\chi_{3525}(971,\cdot)\) \(\chi_{3525}(1016,\cdot)\) \(\chi_{3525}(1031,\cdot)\) \(\chi_{3525}(1091,\cdot)\) \(\chi_{3525}(1121,\cdot)\) \(\chi_{3525}(1166,\cdot)\) ...

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: $\Q(\zeta_{115})$
Fixed field: Number field defined by a degree 230 polynomial (not computed)

Values on generators

\((2351,1552,2026)\) → \((-1,e\left(\frac{4}{5}\right),e\left(\frac{7}{46}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 3525 }(11, a) \) \(1\)\(1\)\(e\left(\frac{9}{230}\right)\)\(e\left(\frac{9}{115}\right)\)\(e\left(\frac{20}{23}\right)\)\(e\left(\frac{27}{230}\right)\)\(e\left(\frac{42}{115}\right)\)\(e\left(\frac{201}{230}\right)\)\(e\left(\frac{209}{230}\right)\)\(e\left(\frac{18}{115}\right)\)\(e\left(\frac{77}{230}\right)\)\(e\left(\frac{57}{230}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3525 }(11,a) \;\) at \(\;a = \) e.g. 2