Properties

Label 3630.371
Modulus $3630$
Conductor $363$
Order $110$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3630, base_ring=CyclotomicField(110))
 
M = H._module
 
chi = DirichletCharacter(H, M([55,0,3]))
 
pari: [g,chi] = znchar(Mod(371,3630))
 

Basic properties

Modulus: \(3630\)
Conductor: \(363\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{363}(8,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3630.bn

\(\chi_{3630}(41,\cdot)\) \(\chi_{3630}(101,\cdot)\) \(\chi_{3630}(281,\cdot)\) \(\chi_{3630}(371,\cdot)\) \(\chi_{3630}(431,\cdot)\) \(\chi_{3630}(491,\cdot)\) \(\chi_{3630}(611,\cdot)\) \(\chi_{3630}(701,\cdot)\) \(\chi_{3630}(761,\cdot)\) \(\chi_{3630}(821,\cdot)\) \(\chi_{3630}(1031,\cdot)\) \(\chi_{3630}(1091,\cdot)\) \(\chi_{3630}(1151,\cdot)\) \(\chi_{3630}(1271,\cdot)\) \(\chi_{3630}(1361,\cdot)\) \(\chi_{3630}(1421,\cdot)\) \(\chi_{3630}(1481,\cdot)\) \(\chi_{3630}(1601,\cdot)\) \(\chi_{3630}(1751,\cdot)\) \(\chi_{3630}(1811,\cdot)\) \(\chi_{3630}(1931,\cdot)\) \(\chi_{3630}(2021,\cdot)\) \(\chi_{3630}(2081,\cdot)\) \(\chi_{3630}(2141,\cdot)\) \(\chi_{3630}(2261,\cdot)\) \(\chi_{3630}(2351,\cdot)\) \(\chi_{3630}(2471,\cdot)\) \(\chi_{3630}(2591,\cdot)\) \(\chi_{3630}(2681,\cdot)\) \(\chi_{3630}(2741,\cdot)\) ...

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: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 110 polynomial (not computed)

Values on generators

\((1211,727,3511)\) → \((-1,1,e\left(\frac{3}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 3630 }(371, a) \) \(1\)\(1\)\(e\left(\frac{21}{110}\right)\)\(e\left(\frac{83}{110}\right)\)\(e\left(\frac{46}{55}\right)\)\(e\left(\frac{29}{110}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{53}{55}\right)\)\(e\left(\frac{19}{55}\right)\)\(e\left(\frac{8}{55}\right)\)\(e\left(\frac{7}{55}\right)\)\(e\left(\frac{15}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3630 }(371,a) \;\) at \(\;a = \) e.g. 2