↑

Properties

Label 441.z
Modulus $441$
Conductor $441$
Order $21$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(42)) M = H._module chi = DirichletCharacter(H, M([14,10])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(4,441)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(441\)
Conductor: \(441\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(21\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 21 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(4\) \(5\) \(8\) \(10\) \(11\) \(13\) \(16\) \(17\) \(19\)
\(\chi_{441}(4,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{441}(16,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{441}(130,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{441}(142,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{441}(193,\cdot)\) \(1\) \(1\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{441}(205,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{441}(256,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{441}(268,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{441}(319,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{441}(331,\cdot)\) \(1\) \(1\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{441}(382,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{441}(394,\cdot)\) \(1\) \(1\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{2}{3}\right)\)