Properties

Label 1375.658
Modulus $1375$
Conductor $1375$
Order $100$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1375, base_ring=CyclotomicField(100))
 
M = H._module
 
chi = DirichletCharacter(H, M([83,60]))
 
pari: [g,chi] = znchar(Mod(658,1375))
 

Basic properties

Modulus: \(1375\)
Conductor: \(1375\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(100\)
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 1375.cp

\(\chi_{1375}(42,\cdot)\) \(\chi_{1375}(48,\cdot)\) \(\chi_{1375}(108,\cdot)\) \(\chi_{1375}(113,\cdot)\) \(\chi_{1375}(202,\cdot)\) \(\chi_{1375}(203,\cdot)\) \(\chi_{1375}(212,\cdot)\) \(\chi_{1375}(247,\cdot)\) \(\chi_{1375}(317,\cdot)\) \(\chi_{1375}(323,\cdot)\) \(\chi_{1375}(383,\cdot)\) \(\chi_{1375}(388,\cdot)\) \(\chi_{1375}(477,\cdot)\) \(\chi_{1375}(478,\cdot)\) \(\chi_{1375}(487,\cdot)\) \(\chi_{1375}(522,\cdot)\) \(\chi_{1375}(592,\cdot)\) \(\chi_{1375}(598,\cdot)\) \(\chi_{1375}(658,\cdot)\) \(\chi_{1375}(663,\cdot)\) \(\chi_{1375}(752,\cdot)\) \(\chi_{1375}(753,\cdot)\) \(\chi_{1375}(762,\cdot)\) \(\chi_{1375}(797,\cdot)\) \(\chi_{1375}(867,\cdot)\) \(\chi_{1375}(873,\cdot)\) \(\chi_{1375}(933,\cdot)\) \(\chi_{1375}(938,\cdot)\) \(\chi_{1375}(1027,\cdot)\) \(\chi_{1375}(1028,\cdot)\) ...

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: $\Q(\zeta_{100})$
Fixed field: Number field defined by a degree 100 polynomial

Values on generators

\((1002,376)\) → \((e\left(\frac{83}{100}\right),e\left(\frac{3}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 1375 }(658, a) \) \(-1\)\(1\)\(e\left(\frac{43}{100}\right)\)\(e\left(\frac{61}{100}\right)\)\(e\left(\frac{43}{50}\right)\)\(e\left(\frac{1}{25}\right)\)\(-i\)\(e\left(\frac{29}{100}\right)\)\(e\left(\frac{11}{50}\right)\)\(e\left(\frac{47}{100}\right)\)\(e\left(\frac{97}{100}\right)\)\(e\left(\frac{9}{50}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1375 }(658,a) \;\) at \(\;a = \) e.g. 2