Properties

Label 1350.1133
Modulus $1350$
Conductor $75$
Order $20$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1350, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,3]))
 
pari: [g,chi] = znchar(Mod(1133,1350))
 

Basic properties

Modulus: \(1350\)
Conductor: \(75\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{75}(8,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1350.w

\(\chi_{1350}(53,\cdot)\) \(\chi_{1350}(323,\cdot)\) \(\chi_{1350}(377,\cdot)\) \(\chi_{1350}(647,\cdot)\) \(\chi_{1350}(863,\cdot)\) \(\chi_{1350}(917,\cdot)\) \(\chi_{1350}(1133,\cdot)\) \(\chi_{1350}(1187,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: \(\Q(\zeta_{75})^+\)

Values on generators

\((1001,1027)\) → \((-1,e\left(\frac{3}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 1350 }(1133, a) \) \(1\)\(1\)\(-i\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{1}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1350 }(1133,a) \;\) at \(\;a = \) e.g. 2