Properties

Label 1375.772
Modulus $1375$
Conductor $1375$
Order $100$
Real no
Primitive yes
Minimal yes
Parity even

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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([77,10]))
 
pari: [g,chi] = znchar(Mod(772,1375))
 

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

Galois orbit 1375.co

\(\chi_{1375}(17,\cdot)\) \(\chi_{1375}(52,\cdot)\) \(\chi_{1375}(62,\cdot)\) \(\chi_{1375}(83,\cdot)\) \(\chi_{1375}(173,\cdot)\) \(\chi_{1375}(178,\cdot)\) \(\chi_{1375}(222,\cdot)\) \(\chi_{1375}(238,\cdot)\) \(\chi_{1375}(292,\cdot)\) \(\chi_{1375}(327,\cdot)\) \(\chi_{1375}(337,\cdot)\) \(\chi_{1375}(358,\cdot)\) \(\chi_{1375}(448,\cdot)\) \(\chi_{1375}(453,\cdot)\) \(\chi_{1375}(497,\cdot)\) \(\chi_{1375}(513,\cdot)\) \(\chi_{1375}(567,\cdot)\) \(\chi_{1375}(602,\cdot)\) \(\chi_{1375}(612,\cdot)\) \(\chi_{1375}(633,\cdot)\) \(\chi_{1375}(723,\cdot)\) \(\chi_{1375}(728,\cdot)\) \(\chi_{1375}(772,\cdot)\) \(\chi_{1375}(788,\cdot)\) \(\chi_{1375}(842,\cdot)\) \(\chi_{1375}(877,\cdot)\) \(\chi_{1375}(887,\cdot)\) \(\chi_{1375}(908,\cdot)\) \(\chi_{1375}(998,\cdot)\) \(\chi_{1375}(1003,\cdot)\) ...

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

\((1002,376)\) → \((e\left(\frac{77}{100}\right),e\left(\frac{1}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 1375 }(772, a) \) \(1\)\(1\)\(e\left(\frac{87}{100}\right)\)\(e\left(\frac{19}{100}\right)\)\(e\left(\frac{37}{50}\right)\)\(e\left(\frac{3}{50}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{61}{100}\right)\)\(e\left(\frac{19}{50}\right)\)\(e\left(\frac{93}{100}\right)\)\(e\left(\frac{13}{100}\right)\)\(e\left(\frac{1}{50}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1375 }(772,a) \;\) at \(\;a = \) e.g. 2