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Properties

Label 441.bn
Modulus $441$
Conductor $441$
Order $42$
Real no
Primitive yes
Minimal yes
Parity even

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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([35,29])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(5,441)) 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: \(42\)
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 42 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(4\) \(5\) \(8\) \(10\) \(11\) \(13\) \(16\) \(17\) \(19\)
\(\chi_{441}(5,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{441}(38,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{441}(101,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{441}(131,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{441}(164,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{441}(194,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{441}(257,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{441}(290,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{441}(320,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{441}(353,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{441}(383,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{441}(416,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{5}{6}\right)\)