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Properties

Label 4998.by
Modulus $4998$
Conductor $833$
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(4998, base_ring=CyclotomicField(42)) M = H._module chi = DirichletCharacter(H, M([0,22,21])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(781,4998)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(4998\)
Conductor: \(833\)
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 833.z
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 \(-1\) \(1\) \(5\) \(11\) \(13\) \(19\) \(23\) \(25\) \(29\) \(31\) \(37\) \(41\)
\(\chi_{4998}(781,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{5}{14}\right)\)
\(\chi_{4998}(1087,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{3}{14}\right)\)
\(\chi_{4998}(1495,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{3}{14}\right)\)
\(\chi_{4998}(1801,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{13}{14}\right)\)
\(\chi_{4998}(2209,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{1}{14}\right)\)
\(\chi_{4998}(2515,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{9}{14}\right)\)
\(\chi_{4998}(2923,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{13}{14}\right)\)
\(\chi_{4998}(3229,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{5}{14}\right)\)
\(\chi_{4998}(3637,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{11}{14}\right)\)
\(\chi_{4998}(3943,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{1}{14}\right)\)
\(\chi_{4998}(4351,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{9}{14}\right)\)
\(\chi_{4998}(4657,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{11}{14}\right)\)