Properties

Label 1375.406
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([24,25]))
 
pari: [g,chi] = znchar(Mod(406,1375))
 

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.ch

\(\chi_{1375}(21,\cdot)\) \(\chi_{1375}(131,\cdot)\) \(\chi_{1375}(186,\cdot)\) \(\chi_{1375}(241,\cdot)\) \(\chi_{1375}(296,\cdot)\) \(\chi_{1375}(406,\cdot)\) \(\chi_{1375}(461,\cdot)\) \(\chi_{1375}(516,\cdot)\) \(\chi_{1375}(571,\cdot)\) \(\chi_{1375}(681,\cdot)\) \(\chi_{1375}(736,\cdot)\) \(\chi_{1375}(791,\cdot)\) \(\chi_{1375}(846,\cdot)\) \(\chi_{1375}(956,\cdot)\) \(\chi_{1375}(1011,\cdot)\) \(\chi_{1375}(1066,\cdot)\) \(\chi_{1375}(1121,\cdot)\) \(\chi_{1375}(1231,\cdot)\) \(\chi_{1375}(1286,\cdot)\) \(\chi_{1375}(1341,\cdot)\)

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

\((1002,376)\) → \((e\left(\frac{12}{25}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 1375 }(406, a) \) \(-1\)\(1\)\(e\left(\frac{49}{50}\right)\)\(e\left(\frac{9}{25}\right)\)\(e\left(\frac{24}{25}\right)\)\(e\left(\frac{17}{50}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{47}{50}\right)\)\(e\left(\frac{18}{25}\right)\)\(e\left(\frac{8}{25}\right)\)\(e\left(\frac{11}{50}\right)\)\(e\left(\frac{7}{25}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1375 }(406,a) \;\) at \(\;a = \) e.g. 2