Properties

Label 205.bb
Modulus $205$
Conductor $205$
Order $40$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(205, base_ring=CyclotomicField(40))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,31]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(13,205))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(205\)
Conductor: \(205\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(40\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{40})\)
Fixed field: 40.40.737344664758480175980241689422890987152982395783688920879217945970594882965087890625.1

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(6\) \(7\) \(8\) \(9\) \(11\) \(12\) \(13\)
\(\chi_{205}(13,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{31}{40}\right)\) \(e\left(\frac{39}{40}\right)\) \(e\left(\frac{7}{10}\right)\) \(-i\) \(e\left(\frac{13}{40}\right)\) \(e\left(\frac{27}{40}\right)\) \(e\left(\frac{11}{40}\right)\)
\(\chi_{205}(17,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{33}{40}\right)\) \(e\left(\frac{17}{40}\right)\) \(e\left(\frac{1}{10}\right)\) \(i\) \(e\left(\frac{19}{40}\right)\) \(e\left(\frac{21}{40}\right)\) \(e\left(\frac{13}{40}\right)\)
\(\chi_{205}(22,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{29}{40}\right)\) \(e\left(\frac{21}{40}\right)\) \(e\left(\frac{3}{10}\right)\) \(i\) \(e\left(\frac{7}{40}\right)\) \(e\left(\frac{33}{40}\right)\) \(e\left(\frac{9}{40}\right)\)
\(\chi_{205}(28,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{11}{40}\right)\) \(e\left(\frac{19}{40}\right)\) \(e\left(\frac{7}{10}\right)\) \(-i\) \(e\left(\frac{33}{40}\right)\) \(e\left(\frac{7}{40}\right)\) \(e\left(\frac{31}{40}\right)\)
\(\chi_{205}(47,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{1}{40}\right)\) \(e\left(\frac{9}{40}\right)\) \(e\left(\frac{7}{10}\right)\) \(i\) \(e\left(\frac{3}{40}\right)\) \(e\left(\frac{37}{40}\right)\) \(e\left(\frac{21}{40}\right)\)
\(\chi_{205}(48,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{39}{40}\right)\) \(e\left(\frac{31}{40}\right)\) \(e\left(\frac{3}{10}\right)\) \(-i\) \(e\left(\frac{37}{40}\right)\) \(e\left(\frac{3}{40}\right)\) \(e\left(\frac{19}{40}\right)\)
\(\chi_{205}(53,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{27}{40}\right)\) \(e\left(\frac{3}{40}\right)\) \(e\left(\frac{9}{10}\right)\) \(-i\) \(e\left(\frac{1}{40}\right)\) \(e\left(\frac{39}{40}\right)\) \(e\left(\frac{7}{40}\right)\)
\(\chi_{205}(67,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{17}{40}\right)\) \(e\left(\frac{33}{40}\right)\) \(e\left(\frac{9}{10}\right)\) \(i\) \(e\left(\frac{11}{40}\right)\) \(e\left(\frac{29}{40}\right)\) \(e\left(\frac{37}{40}\right)\)
\(\chi_{205}(93,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{3}{40}\right)\) \(e\left(\frac{27}{40}\right)\) \(e\left(\frac{1}{10}\right)\) \(-i\) \(e\left(\frac{9}{40}\right)\) \(e\left(\frac{31}{40}\right)\) \(e\left(\frac{23}{40}\right)\)
\(\chi_{205}(97,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{37}{40}\right)\) \(e\left(\frac{13}{40}\right)\) \(e\left(\frac{9}{10}\right)\) \(i\) \(e\left(\frac{31}{40}\right)\) \(e\left(\frac{9}{40}\right)\) \(e\left(\frac{17}{40}\right)\)
\(\chi_{205}(117,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{21}{40}\right)\) \(e\left(\frac{29}{40}\right)\) \(e\left(\frac{7}{10}\right)\) \(i\) \(e\left(\frac{23}{40}\right)\) \(e\left(\frac{17}{40}\right)\) \(e\left(\frac{1}{40}\right)\)
\(\chi_{205}(142,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{9}{40}\right)\) \(e\left(\frac{1}{40}\right)\) \(e\left(\frac{3}{10}\right)\) \(i\) \(e\left(\frac{27}{40}\right)\) \(e\left(\frac{13}{40}\right)\) \(e\left(\frac{29}{40}\right)\)
\(\chi_{205}(147,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{13}{40}\right)\) \(e\left(\frac{37}{40}\right)\) \(e\left(\frac{1}{10}\right)\) \(i\) \(e\left(\frac{39}{40}\right)\) \(e\left(\frac{1}{40}\right)\) \(e\left(\frac{33}{40}\right)\)
\(\chi_{205}(153,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{23}{40}\right)\) \(e\left(\frac{7}{40}\right)\) \(e\left(\frac{1}{10}\right)\) \(-i\) \(e\left(\frac{29}{40}\right)\) \(e\left(\frac{11}{40}\right)\) \(e\left(\frac{3}{40}\right)\)
\(\chi_{205}(193,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{7}{40}\right)\) \(e\left(\frac{23}{40}\right)\) \(e\left(\frac{9}{10}\right)\) \(-i\) \(e\left(\frac{21}{40}\right)\) \(e\left(\frac{19}{40}\right)\) \(e\left(\frac{27}{40}\right)\)
\(\chi_{205}(198,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{19}{40}\right)\) \(e\left(\frac{11}{40}\right)\) \(e\left(\frac{3}{10}\right)\) \(-i\) \(e\left(\frac{17}{40}\right)\) \(e\left(\frac{23}{40}\right)\) \(e\left(\frac{39}{40}\right)\)