Properties

Label 1050.67
Modulus $1050$
Conductor $175$
Order $60$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,39,40]))
 
pari: [g,chi] = znchar(Mod(67,1050))
 

Basic properties

Modulus: \(1050\)
Conductor: \(175\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{175}(67,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1050.bu

\(\chi_{1050}(37,\cdot)\) \(\chi_{1050}(67,\cdot)\) \(\chi_{1050}(163,\cdot)\) \(\chi_{1050}(247,\cdot)\) \(\chi_{1050}(277,\cdot)\) \(\chi_{1050}(373,\cdot)\) \(\chi_{1050}(403,\cdot)\) \(\chi_{1050}(487,\cdot)\) \(\chi_{1050}(583,\cdot)\) \(\chi_{1050}(613,\cdot)\) \(\chi_{1050}(667,\cdot)\) \(\chi_{1050}(697,\cdot)\) \(\chi_{1050}(823,\cdot)\) \(\chi_{1050}(877,\cdot)\) \(\chi_{1050}(1003,\cdot)\) \(\chi_{1050}(1033,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((701,127,451)\) → \((1,e\left(\frac{13}{20}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 1050 }(67, a) \) \(-1\)\(1\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{7}{60}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{11}{60}\right)\)\(e\left(\frac{3}{5}\right)\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1050 }(67,a) \;\) at \(\;a = \) e.g. 2