Properties

Label 6930.61
Modulus $6930$
Conductor $693$
Order $30$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6930, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([20,0,25,27]))
 
pari: [g,chi] = znchar(Mod(61,6930))
 

Basic properties

Modulus: \(6930\)
Conductor: \(693\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{693}(61,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6930.hn

\(\chi_{6930}(61,\cdot)\) \(\chi_{6930}(1921,\cdot)\) \(\chi_{6930}(2581,\cdot)\) \(\chi_{6930}(3181,\cdot)\) \(\chi_{6930}(3841,\cdot)\) \(\chi_{6930}(4441,\cdot)\) \(\chi_{6930}(5101,\cdot)\) \(\chi_{6930}(6331,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: 30.30.213748801558268665141942640259330854834575418378058033233791144352997.2

Values on generators

\((1541,1387,2971,2521)\) → \((e\left(\frac{2}{3}\right),1,e\left(\frac{5}{6}\right),e\left(\frac{9}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)\(47\)
\( \chi_{ 6930 }(61, a) \) \(1\)\(1\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{13}{15}\right)\)\(1\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6930 }(61,a) \;\) at \(\;a = \) e.g. 2