Properties

Label 41.h
Modulus $41$
Conductor $41$
Order $40$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(41, base_ring=CyclotomicField(40))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(6,41))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(41\)
Conductor: \(41\)
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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{40})\)
Fixed field: \(\Q(\zeta_{41})\)

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(11\)
\(\chi_{41}(6,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{1}{40}\right)\) \(e\left(\frac{39}{40}\right)\) \(e\left(\frac{19}{20}\right)\) \(-i\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{3}{40}\right)\)
\(\chi_{41}(7,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{39}{40}\right)\) \(e\left(\frac{1}{40}\right)\) \(e\left(\frac{1}{20}\right)\) \(i\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{37}{40}\right)\)
\(\chi_{41}(11,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{40}\right)\) \(e\left(\frac{37}{40}\right)\) \(e\left(\frac{17}{20}\right)\) \(i\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{9}{40}\right)\)
\(\chi_{41}(12,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{27}{40}\right)\) \(e\left(\frac{13}{40}\right)\) \(e\left(\frac{13}{20}\right)\) \(i\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{1}{40}\right)\)
\(\chi_{41}(13,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{31}{40}\right)\) \(e\left(\frac{9}{40}\right)\) \(e\left(\frac{9}{20}\right)\) \(i\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{13}{40}\right)\)
\(\chi_{41}(15,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{37}{40}\right)\) \(e\left(\frac{3}{40}\right)\) \(e\left(\frac{3}{20}\right)\) \(-i\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{31}{40}\right)\)
\(\chi_{41}(17,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{33}{40}\right)\) \(e\left(\frac{7}{40}\right)\) \(e\left(\frac{7}{20}\right)\) \(-i\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{19}{40}\right)\)
\(\chi_{41}(19,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{9}{40}\right)\) \(e\left(\frac{31}{40}\right)\) \(e\left(\frac{11}{20}\right)\) \(-i\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{27}{40}\right)\)
\(\chi_{41}(22,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{29}{40}\right)\) \(e\left(\frac{11}{40}\right)\) \(e\left(\frac{11}{20}\right)\) \(-i\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{7}{40}\right)\)
\(\chi_{41}(24,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{13}{40}\right)\) \(e\left(\frac{27}{40}\right)\) \(e\left(\frac{7}{20}\right)\) \(-i\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{39}{40}\right)\)
\(\chi_{41}(26,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{17}{40}\right)\) \(e\left(\frac{23}{40}\right)\) \(e\left(\frac{3}{20}\right)\) \(-i\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{11}{40}\right)\)
\(\chi_{41}(28,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{11}{40}\right)\) \(e\left(\frac{29}{40}\right)\) \(e\left(\frac{9}{20}\right)\) \(i\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{33}{40}\right)\)
\(\chi_{41}(29,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{7}{40}\right)\) \(e\left(\frac{33}{40}\right)\) \(e\left(\frac{13}{20}\right)\) \(i\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{21}{40}\right)\)
\(\chi_{41}(30,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{23}{40}\right)\) \(e\left(\frac{17}{40}\right)\) \(e\left(\frac{17}{20}\right)\) \(i\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{29}{40}\right)\)
\(\chi_{41}(34,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{19}{40}\right)\) \(e\left(\frac{21}{40}\right)\) \(e\left(\frac{1}{20}\right)\) \(i\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{17}{40}\right)\)
\(\chi_{41}(35,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{21}{40}\right)\) \(e\left(\frac{19}{40}\right)\) \(e\left(\frac{19}{20}\right)\) \(-i\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{23}{40}\right)\)