↑

Properties

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

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character orbit
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(41, base_ring=CyclotomicField(40)) M = H._module chi = DirichletCharacter(H, M([1])) chi.galois_orbit()
 
Copy content gp:[g,chi] = znchar(Mod(6, 41)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("41.6"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Basic properties

Modulus: \(41\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(41\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(40\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: yes
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Minimal: yes
Parity: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Related number fields

Field of values: \(\Q(\zeta_{40})\)
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 40 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Characters in Galois orbit

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