Properties

Label 1275.131
Modulus $1275$
Conductor $1275$
Order $80$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1275, base_ring=CyclotomicField(80))
 
M = H._module
 
chi = DirichletCharacter(H, M([40,32,65]))
 
pari: [g,chi] = znchar(Mod(131,1275))
 

Basic properties

Modulus: \(1275\)
Conductor: \(1275\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(80\)
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 1275.cp

\(\chi_{1275}(11,\cdot)\) \(\chi_{1275}(41,\cdot)\) \(\chi_{1275}(56,\cdot)\) \(\chi_{1275}(71,\cdot)\) \(\chi_{1275}(116,\cdot)\) \(\chi_{1275}(131,\cdot)\) \(\chi_{1275}(146,\cdot)\) \(\chi_{1275}(266,\cdot)\) \(\chi_{1275}(296,\cdot)\) \(\chi_{1275}(311,\cdot)\) \(\chi_{1275}(371,\cdot)\) \(\chi_{1275}(386,\cdot)\) \(\chi_{1275}(431,\cdot)\) \(\chi_{1275}(521,\cdot)\) \(\chi_{1275}(566,\cdot)\) \(\chi_{1275}(581,\cdot)\) \(\chi_{1275}(641,\cdot)\) \(\chi_{1275}(656,\cdot)\) \(\chi_{1275}(686,\cdot)\) \(\chi_{1275}(806,\cdot)\) \(\chi_{1275}(821,\cdot)\) \(\chi_{1275}(836,\cdot)\) \(\chi_{1275}(881,\cdot)\) \(\chi_{1275}(896,\cdot)\) \(\chi_{1275}(911,\cdot)\) \(\chi_{1275}(941,\cdot)\) \(\chi_{1275}(1031,\cdot)\) \(\chi_{1275}(1061,\cdot)\) \(\chi_{1275}(1091,\cdot)\) \(\chi_{1275}(1136,\cdot)\) ...

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: $\Q(\zeta_{80})$
Fixed field: Number field defined by a degree 80 polynomial

Values on generators

\((851,52,751)\) → \((-1,e\left(\frac{2}{5}\right),e\left(\frac{13}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(19\)\(22\)
\( \chi_{ 1275 }(131, a) \) \(1\)\(1\)\(e\left(\frac{11}{40}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{33}{40}\right)\)\(e\left(\frac{47}{80}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{17}{80}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{23}{40}\right)\)\(e\left(\frac{69}{80}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1275 }(131,a) \;\) at \(\;a = \) e.g. 2