Properties

Label 2205.em
Modulus $2205$
Conductor $2205$
Order $84$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2205, base_ring=CyclotomicField(84))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([14,63,38]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(38,2205))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(2205\)
Conductor: \(2205\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(4\) \(8\) \(11\) \(13\) \(16\) \(17\) \(19\) \(22\) \(23\)
\(\chi_{2205}(38,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{43}{84}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{47}{84}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{79}{84}\right)\) \(e\left(\frac{23}{84}\right)\)
\(\chi_{2205}(257,\cdot)\) \(-1\) \(1\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{5}{84}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{25}{84}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{17}{84}\right)\) \(e\left(\frac{73}{84}\right)\)
\(\chi_{2205}(353,\cdot)\) \(-1\) \(1\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{67}{84}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{83}{84}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{43}{84}\right)\) \(e\left(\frac{71}{84}\right)\)
\(\chi_{2205}(383,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{83}{84}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{79}{84}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{47}{84}\right)\) \(e\left(\frac{19}{84}\right)\)
\(\chi_{2205}(542,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{73}{84}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{29}{84}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{13}{84}\right)\) \(e\left(\frac{41}{84}\right)\)
\(\chi_{2205}(572,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{53}{84}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{13}{84}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{29}{84}\right)\) \(e\left(\frac{1}{84}\right)\)
\(\chi_{2205}(698,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{13}{42}\right)\) \(e\left(\frac{47}{84}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{67}{84}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{59}{84}\right)\) \(e\left(\frac{31}{84}\right)\)
\(\chi_{2205}(857,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{13}{84}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{65}{84}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{61}{84}\right)\) \(e\left(\frac{5}{84}\right)\)
\(\chi_{2205}(887,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{17}{84}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{1}{84}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{41}{84}\right)\) \(e\left(\frac{13}{84}\right)\)
\(\chi_{2205}(983,\cdot)\) \(-1\) \(1\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{5}{42}\right)\) \(e\left(\frac{31}{84}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{71}{84}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{55}{84}\right)\) \(e\left(\frac{83}{84}\right)\)
\(\chi_{2205}(1013,\cdot)\) \(-1\) \(1\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{37}{42}\right)\) \(e\left(\frac{11}{84}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{55}{84}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{71}{84}\right)\) \(e\left(\frac{43}{84}\right)\)
\(\chi_{2205}(1172,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{37}{84}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{17}{84}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{25}{84}\right)\) \(e\left(\frac{53}{84}\right)\)
\(\chi_{2205}(1202,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{65}{84}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{73}{84}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{53}{84}\right)\) \(e\left(\frac{25}{84}\right)\)
\(\chi_{2205}(1298,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{17}{42}\right)\) \(e\left(\frac{55}{84}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{23}{84}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{19}{84}\right)\) \(e\left(\frac{47}{84}\right)\)
\(\chi_{2205}(1328,\cdot)\) \(-1\) \(1\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{19}{42}\right)\) \(e\left(\frac{59}{84}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{43}{84}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{83}{84}\right)\) \(e\left(\frac{55}{84}\right)\)
\(\chi_{2205}(1487,\cdot)\) \(-1\) \(1\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{61}{84}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{53}{84}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{73}{84}\right)\) \(e\left(\frac{17}{84}\right)\)
\(\chi_{2205}(1517,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{29}{84}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{61}{84}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{65}{84}\right)\) \(e\left(\frac{37}{84}\right)\)
\(\chi_{2205}(1613,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{29}{42}\right)\) \(e\left(\frac{79}{84}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{59}{84}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{67}{84}\right)\) \(e\left(\frac{11}{84}\right)\)
\(\chi_{2205}(1643,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{1}{42}\right)\) \(e\left(\frac{23}{84}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{31}{84}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{84}\right)\) \(e\left(\frac{67}{84}\right)\)
\(\chi_{2205}(1802,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{11}{42}\right)\) \(e\left(\frac{1}{84}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{5}{84}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{37}{84}\right)\) \(e\left(\frac{65}{84}\right)\)
\(\chi_{2205}(1928,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{41}{42}\right)\) \(e\left(\frac{19}{84}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{11}{84}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{31}{84}\right)\) \(e\left(\frac{59}{84}\right)\)
\(\chi_{2205}(1958,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{25}{42}\right)\) \(e\left(\frac{71}{84}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{19}{84}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{23}{84}\right)\) \(e\left(\frac{79}{84}\right)\)
\(\chi_{2205}(2117,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{23}{42}\right)\) \(e\left(\frac{25}{84}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{41}{84}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{84}\right)\) \(e\left(\frac{29}{84}\right)\)
\(\chi_{2205}(2147,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{31}{42}\right)\) \(e\left(\frac{41}{84}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{37}{84}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{84}\right)\) \(e\left(\frac{61}{84}\right)\)