Properties

Label 4212.eu
Modulus $4212$
Conductor $4212$
Order $108$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4212, base_ring=CyclotomicField(108))
 
M = H._module
 
chi = DirichletCharacter(H, M([54,32,99]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(7,4212))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(4212\)
Conductor: \(4212\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(108\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: $\Q(\zeta_{108})$
Fixed field: Number field defined by a degree 108 polynomial (not computed)

First 31 of 36 characters in Galois orbit

Character \(-1\) \(1\) \(5\) \(7\) \(11\) \(17\) \(19\) \(23\) \(25\) \(29\) \(31\) \(35\)
\(\chi_{4212}(7,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{108}\right)\) \(e\left(\frac{35}{108}\right)\) \(e\left(\frac{83}{108}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{7}{54}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{73}{108}\right)\) \(e\left(\frac{7}{18}\right)\)
\(\chi_{4212}(67,\cdot)\) \(1\) \(1\) \(e\left(\frac{53}{108}\right)\) \(e\left(\frac{49}{108}\right)\) \(e\left(\frac{73}{108}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{53}{54}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{59}{108}\right)\) \(e\left(\frac{17}{18}\right)\)
\(\chi_{4212}(331,\cdot)\) \(1\) \(1\) \(e\left(\frac{61}{108}\right)\) \(e\left(\frac{89}{108}\right)\) \(e\left(\frac{29}{108}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{7}{54}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{19}{108}\right)\) \(e\left(\frac{7}{18}\right)\)
\(\chi_{4212}(427,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{108}\right)\) \(e\left(\frac{7}{108}\right)\) \(e\left(\frac{103}{108}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{23}{54}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{101}{108}\right)\) \(e\left(\frac{5}{18}\right)\)
\(\chi_{4212}(475,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{108}\right)\) \(e\left(\frac{47}{108}\right)\) \(e\left(\frac{59}{108}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{31}{54}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{61}{108}\right)\) \(e\left(\frac{13}{18}\right)\)
\(\chi_{4212}(535,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{108}\right)\) \(e\left(\frac{97}{108}\right)\) \(e\left(\frac{85}{108}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{41}{54}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{11}{108}\right)\) \(e\left(\frac{5}{18}\right)\)
\(\chi_{4212}(799,\cdot)\) \(1\) \(1\) \(e\left(\frac{85}{108}\right)\) \(e\left(\frac{101}{108}\right)\) \(e\left(\frac{5}{108}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{31}{54}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{7}{108}\right)\) \(e\left(\frac{13}{18}\right)\)
\(\chi_{4212}(895,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{108}\right)\) \(e\left(\frac{55}{108}\right)\) \(e\left(\frac{7}{108}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{11}{54}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{53}{108}\right)\) \(e\left(\frac{11}{18}\right)\)
\(\chi_{4212}(943,\cdot)\) \(1\) \(1\) \(e\left(\frac{55}{108}\right)\) \(e\left(\frac{59}{108}\right)\) \(e\left(\frac{35}{108}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{1}{54}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{49}{108}\right)\) \(e\left(\frac{1}{18}\right)\)
\(\chi_{4212}(1003,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{108}\right)\) \(e\left(\frac{37}{108}\right)\) \(e\left(\frac{97}{108}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{29}{54}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{71}{108}\right)\) \(e\left(\frac{11}{18}\right)\)
\(\chi_{4212}(1267,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{108}\right)\) \(e\left(\frac{5}{108}\right)\) \(e\left(\frac{89}{108}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{1}{54}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{103}{108}\right)\) \(e\left(\frac{1}{18}\right)\)
\(\chi_{4212}(1363,\cdot)\) \(1\) \(1\) \(e\left(\frac{107}{108}\right)\) \(e\left(\frac{103}{108}\right)\) \(e\left(\frac{19}{108}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{53}{54}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{5}{108}\right)\) \(e\left(\frac{17}{18}\right)\)
\(\chi_{4212}(1411,\cdot)\) \(1\) \(1\) \(e\left(\frac{79}{108}\right)\) \(e\left(\frac{71}{108}\right)\) \(e\left(\frac{11}{108}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{25}{54}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{37}{108}\right)\) \(e\left(\frac{7}{18}\right)\)
\(\chi_{4212}(1471,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{108}\right)\) \(e\left(\frac{85}{108}\right)\) \(e\left(\frac{1}{108}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{17}{54}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{23}{108}\right)\) \(e\left(\frac{17}{18}\right)\)
\(\chi_{4212}(1735,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{108}\right)\) \(e\left(\frac{17}{108}\right)\) \(e\left(\frac{65}{108}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{25}{54}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{91}{108}\right)\) \(e\left(\frac{7}{18}\right)\)
\(\chi_{4212}(1831,\cdot)\) \(1\) \(1\) \(e\left(\frac{95}{108}\right)\) \(e\left(\frac{43}{108}\right)\) \(e\left(\frac{31}{108}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{41}{54}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{65}{108}\right)\) \(e\left(\frac{5}{18}\right)\)
\(\chi_{4212}(1879,\cdot)\) \(1\) \(1\) \(e\left(\frac{103}{108}\right)\) \(e\left(\frac{83}{108}\right)\) \(e\left(\frac{95}{108}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{49}{54}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{25}{108}\right)\) \(e\left(\frac{13}{18}\right)\)
\(\chi_{4212}(1939,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{108}\right)\) \(e\left(\frac{25}{108}\right)\) \(e\left(\frac{13}{108}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{5}{54}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{83}{108}\right)\) \(e\left(\frac{5}{18}\right)\)
\(\chi_{4212}(2203,\cdot)\) \(1\) \(1\) \(e\left(\frac{49}{108}\right)\) \(e\left(\frac{29}{108}\right)\) \(e\left(\frac{41}{108}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{49}{54}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{79}{108}\right)\) \(e\left(\frac{13}{18}\right)\)
\(\chi_{4212}(2299,\cdot)\) \(1\) \(1\) \(e\left(\frac{83}{108}\right)\) \(e\left(\frac{91}{108}\right)\) \(e\left(\frac{43}{108}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{29}{54}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{17}{108}\right)\) \(e\left(\frac{11}{18}\right)\)
\(\chi_{4212}(2347,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{108}\right)\) \(e\left(\frac{95}{108}\right)\) \(e\left(\frac{71}{108}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{19}{54}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{13}{108}\right)\) \(e\left(\frac{1}{18}\right)\)
\(\chi_{4212}(2407,\cdot)\) \(1\) \(1\) \(e\left(\frac{101}{108}\right)\) \(e\left(\frac{73}{108}\right)\) \(e\left(\frac{25}{108}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{47}{54}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{35}{108}\right)\) \(e\left(\frac{11}{18}\right)\)
\(\chi_{4212}(2671,\cdot)\) \(1\) \(1\) \(e\left(\frac{73}{108}\right)\) \(e\left(\frac{41}{108}\right)\) \(e\left(\frac{17}{108}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{19}{54}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{67}{108}\right)\) \(e\left(\frac{1}{18}\right)\)
\(\chi_{4212}(2767,\cdot)\) \(1\) \(1\) \(e\left(\frac{71}{108}\right)\) \(e\left(\frac{31}{108}\right)\) \(e\left(\frac{55}{108}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{17}{54}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{77}{108}\right)\) \(e\left(\frac{17}{18}\right)\)
\(\chi_{4212}(2815,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{108}\right)\) \(e\left(\frac{107}{108}\right)\) \(e\left(\frac{47}{108}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{43}{54}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{1}{108}\right)\) \(e\left(\frac{7}{18}\right)\)
\(\chi_{4212}(2875,\cdot)\) \(1\) \(1\) \(e\left(\frac{89}{108}\right)\) \(e\left(\frac{13}{108}\right)\) \(e\left(\frac{37}{108}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{35}{54}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{95}{108}\right)\) \(e\left(\frac{17}{18}\right)\)
\(\chi_{4212}(3139,\cdot)\) \(1\) \(1\) \(e\left(\frac{97}{108}\right)\) \(e\left(\frac{53}{108}\right)\) \(e\left(\frac{101}{108}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{43}{54}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{55}{108}\right)\) \(e\left(\frac{7}{18}\right)\)
\(\chi_{4212}(3235,\cdot)\) \(1\) \(1\) \(e\left(\frac{59}{108}\right)\) \(e\left(\frac{79}{108}\right)\) \(e\left(\frac{67}{108}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{5}{54}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{29}{108}\right)\) \(e\left(\frac{5}{18}\right)\)
\(\chi_{4212}(3283,\cdot)\) \(1\) \(1\) \(e\left(\frac{67}{108}\right)\) \(e\left(\frac{11}{108}\right)\) \(e\left(\frac{23}{108}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{13}{54}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{97}{108}\right)\) \(e\left(\frac{13}{18}\right)\)
\(\chi_{4212}(3343,\cdot)\) \(1\) \(1\) \(e\left(\frac{77}{108}\right)\) \(e\left(\frac{61}{108}\right)\) \(e\left(\frac{49}{108}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{23}{54}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{47}{108}\right)\) \(e\left(\frac{5}{18}\right)\)
\(\chi_{4212}(3607,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{108}\right)\) \(e\left(\frac{65}{108}\right)\) \(e\left(\frac{77}{108}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{13}{54}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{43}{108}\right)\) \(e\left(\frac{13}{18}\right)\)