Properties

Label 1375.106
Modulus $1375$
Conductor $1375$
Order $50$
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(50))
 
M = H._module
 
chi = DirichletCharacter(H, M([34,35]))
 
pari: [g,chi] = znchar(Mod(106,1375))
 

Basic properties

Modulus: \(1375\)
Conductor: \(1375\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(50\)
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.cd

\(\chi_{1375}(61,\cdot)\) \(\chi_{1375}(96,\cdot)\) \(\chi_{1375}(106,\cdot)\) \(\chi_{1375}(266,\cdot)\) \(\chi_{1375}(336,\cdot)\) \(\chi_{1375}(371,\cdot)\) \(\chi_{1375}(381,\cdot)\) \(\chi_{1375}(541,\cdot)\) \(\chi_{1375}(611,\cdot)\) \(\chi_{1375}(646,\cdot)\) \(\chi_{1375}(656,\cdot)\) \(\chi_{1375}(816,\cdot)\) \(\chi_{1375}(886,\cdot)\) \(\chi_{1375}(921,\cdot)\) \(\chi_{1375}(931,\cdot)\) \(\chi_{1375}(1091,\cdot)\) \(\chi_{1375}(1161,\cdot)\) \(\chi_{1375}(1196,\cdot)\) \(\chi_{1375}(1206,\cdot)\) \(\chi_{1375}(1366,\cdot)\)

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: \(\Q(\zeta_{25})\)
Fixed field: Number field defined by a degree 50 polynomial

Values on generators

\((1002,376)\) → \((e\left(\frac{17}{25}\right),e\left(\frac{7}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 1375 }(106, a) \) \(-1\)\(1\)\(e\left(\frac{19}{50}\right)\)\(e\left(\frac{9}{25}\right)\)\(e\left(\frac{19}{25}\right)\)\(e\left(\frac{37}{50}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{7}{50}\right)\)\(e\left(\frac{18}{25}\right)\)\(e\left(\frac{3}{25}\right)\)\(e\left(\frac{11}{50}\right)\)\(e\left(\frac{2}{25}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1375 }(106,a) \;\) at \(\;a = \) e.g. 2