Properties

Label 6930.41
Modulus $6930$
Conductor $693$
Order $30$
Real no
Primitive no
Minimal yes
Parity odd

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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([25,0,15,9]))
 
pari: [g,chi] = znchar(Mod(41,6930))
 

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 \(\chi_{693}(41,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6930.hp

\(\chi_{6930}(41,\cdot)\) \(\chi_{6930}(1091,\cdot)\) \(\chi_{6930}(1931,\cdot)\) \(\chi_{6930}(2351,\cdot)\) \(\chi_{6930}(4241,\cdot)\) \(\chi_{6930}(4451,\cdot)\) \(\chi_{6930}(5711,\cdot)\) \(\chi_{6930}(6761,\cdot)\)

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: Number field defined by a degree 30 polynomial

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)\(47\)
\( \chi_{ 6930 }(41, a) \) \(-1\)\(1\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{11}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6930 }(41,a) \;\) at \(\;a = \) e.g. 2