Properties

Label 3525.56
Modulus $3525$
Conductor $3525$
Order $230$
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(230))
 
M = H._module
 
chi = DirichletCharacter(H, M([115,92,200]))
 
pari: [g,chi] = znchar(Mod(56,3525))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3525.bl

\(\chi_{3525}(56,\cdot)\) \(\chi_{3525}(71,\cdot)\) \(\chi_{3525}(131,\cdot)\) \(\chi_{3525}(191,\cdot)\) \(\chi_{3525}(206,\cdot)\) \(\chi_{3525}(296,\cdot)\) \(\chi_{3525}(341,\cdot)\) \(\chi_{3525}(356,\cdot)\) \(\chi_{3525}(371,\cdot)\) \(\chi_{3525}(431,\cdot)\) \(\chi_{3525}(491,\cdot)\) \(\chi_{3525}(506,\cdot)\) \(\chi_{3525}(521,\cdot)\) \(\chi_{3525}(566,\cdot)\) \(\chi_{3525}(581,\cdot)\) \(\chi_{3525}(596,\cdot)\) \(\chi_{3525}(686,\cdot)\) \(\chi_{3525}(761,\cdot)\) \(\chi_{3525}(806,\cdot)\) \(\chi_{3525}(836,\cdot)\) \(\chi_{3525}(896,\cdot)\) \(\chi_{3525}(911,\cdot)\) \(\chi_{3525}(956,\cdot)\) \(\chi_{3525}(1046,\cdot)\) \(\chi_{3525}(1061,\cdot)\) \(\chi_{3525}(1106,\cdot)\) \(\chi_{3525}(1136,\cdot)\) \(\chi_{3525}(1181,\cdot)\) \(\chi_{3525}(1196,\cdot)\) \(\chi_{3525}(1211,\cdot)\) ...

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

\((2351,1552,2026)\) → \((-1,e\left(\frac{2}{5}\right),e\left(\frac{20}{23}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 3525 }(56, a) \) \(-1\)\(1\)\(e\left(\frac{127}{230}\right)\)\(e\left(\frac{12}{115}\right)\)\(e\left(\frac{19}{23}\right)\)\(e\left(\frac{151}{230}\right)\)\(e\left(\frac{227}{230}\right)\)\(e\left(\frac{19}{115}\right)\)\(e\left(\frac{87}{230}\right)\)\(e\left(\frac{24}{115}\right)\)\(e\left(\frac{141}{230}\right)\)\(e\left(\frac{38}{115}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3525 }(56,a) \;\) at \(\;a = \) e.g. 2