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Properties

Label 8820.hk
Modulus $8820$
Conductor $1764$
Order $42$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8820, base_ring=CyclotomicField(42)) M = H._module chi = DirichletCharacter(H, M([21,7,0,40])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(11,8820)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(8820\)
Conductor: \(1764\)
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: no, induced from 1764.cm
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: 42.42.84986492613867422664202192058457487324646368884457377881199611944629462212222730322047304527187202914910208.1

Characters in Galois orbit

Character \(-1\) \(1\) \(11\) \(13\) \(17\) \(19\) \(23\) \(29\) \(31\) \(37\) \(41\) \(43\)
\(\chi_{8820}(11,\cdot)\) \(1\) \(1\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{13}{42}\right)\) \(-1\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{37}{42}\right)\)
\(\chi_{8820}(1031,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{41}{42}\right)\) \(-1\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{23}{42}\right)\)
\(\chi_{8820}(1271,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{25}{42}\right)\) \(-1\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{13}{42}\right)\)
\(\chi_{8820}(2291,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{23}{42}\right)\) \(-1\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{17}{42}\right)\)
\(\chi_{8820}(2531,\cdot)\) \(1\) \(1\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{37}{42}\right)\) \(-1\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{31}{42}\right)\)
\(\chi_{8820}(3551,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{5}{42}\right)\) \(-1\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{11}{42}\right)\)
\(\chi_{8820}(4811,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{29}{42}\right)\) \(-1\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{5}{42}\right)\)
\(\chi_{8820}(5051,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{19}{42}\right)\) \(-1\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{25}{42}\right)\)
\(\chi_{8820}(6071,\cdot)\) \(1\) \(1\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{11}{42}\right)\) \(-1\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{41}{42}\right)\)
\(\chi_{8820}(6311,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{31}{42}\right)\) \(-1\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{1}{42}\right)\)
\(\chi_{8820}(7571,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{1}{42}\right)\) \(-1\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{19}{42}\right)\)
\(\chi_{8820}(8591,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{17}{42}\right)\) \(-1\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{29}{42}\right)\)