↑

Properties

Label 163.j
Modulus 163163
Conductor 163163
Order 162162
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(163, base_ring=CyclotomicField(162))
 
M = H._module
 
chi = DirichletCharacter(H, M([1]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(2,163))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: 163163
Conductor: 163163
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 162162
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(ζ81)\Q(\zeta_{81})
Fixed field: Number field defined by a degree 162 polynomial (not computed)

First 31 of 54 characters in Galois orbit

Character −1-1 11 22 33 44 55 66 77 88 99 1010 1111
χ163(2,⋅)\chi_{163}(2,\cdot) −1-1 11 e(1162)e\left(\frac{1}{162}\right) e(101162)e\left(\frac{101}{162}\right) e(181)e\left(\frac{1}{81}\right) e(554)e\left(\frac{5}{54}\right) e(1727)e\left(\frac{17}{27}\right) e(73162)e\left(\frac{73}{162}\right) e(154)e\left(\frac{1}{54}\right) e(2081)e\left(\frac{20}{81}\right) e(881)e\left(\frac{8}{81}\right) e(47162)e\left(\frac{47}{162}\right)
χ163(3,⋅)\chi_{163}(3,\cdot) −1-1 11 e(101162)e\left(\frac{101}{162}\right) e(157162)e\left(\frac{157}{162}\right) e(2081)e\left(\frac{20}{81}\right) e(1954)e\left(\frac{19}{54}\right) e(1627)e\left(\frac{16}{27}\right) e(83162)e\left(\frac{83}{162}\right) e(4754)e\left(\frac{47}{54}\right) e(7681)e\left(\frac{76}{81}\right) e(7981)e\left(\frac{79}{81}\right) e(49162)e\left(\frac{49}{162}\right)
χ163(7,⋅)\chi_{163}(7,\cdot) −1-1 11 e(73162)e\left(\frac{73}{162}\right) e(83162)e\left(\frac{83}{162}\right) e(7381)e\left(\frac{73}{81}\right) e(4154)e\left(\frac{41}{54}\right) e(2627)e\left(\frac{26}{27}\right) e(145162)e\left(\frac{145}{162}\right) e(1954)e\left(\frac{19}{54}\right) e(281)e\left(\frac{2}{81}\right) e(1781)e\left(\frac{17}{81}\right) e(29162)e\left(\frac{29}{162}\right)
χ163(11,⋅)\chi_{163}(11,\cdot) −1-1 11 e(47162)e\left(\frac{47}{162}\right) e(49162)e\left(\frac{49}{162}\right) e(4781)e\left(\frac{47}{81}\right) e(1954)e\left(\frac{19}{54}\right) e(1627)e\left(\frac{16}{27}\right) e(29162)e\left(\frac{29}{162}\right) e(4754)e\left(\frac{47}{54}\right) e(4981)e\left(\frac{49}{81}\right) e(5281)e\left(\frac{52}{81}\right) e(103162)e\left(\frac{103}{162}\right)
χ163(12,⋅)\chi_{163}(12,\cdot) −1-1 11 e(103162)e\left(\frac{103}{162}\right) e(35162)e\left(\frac{35}{162}\right) e(2281)e\left(\frac{22}{81}\right) e(2954)e\left(\frac{29}{54}\right) e(2327)e\left(\frac{23}{27}\right) e(67162)e\left(\frac{67}{162}\right) e(4954)e\left(\frac{49}{54}\right) e(3581)e\left(\frac{35}{81}\right) e(1481)e\left(\frac{14}{81}\right) e(143162)e\left(\frac{143}{162}\right)
χ163(18,⋅)\chi_{163}(18,\cdot) −1-1 11 e(41162)e\left(\frac{41}{162}\right) e(91162)e\left(\frac{91}{162}\right) e(4181)e\left(\frac{41}{81}\right) e(4354)e\left(\frac{43}{54}\right) e(2227)e\left(\frac{22}{27}\right) e(77162)e\left(\frac{77}{162}\right) e(4154)e\left(\frac{41}{54}\right) e(1081)e\left(\frac{10}{81}\right) e(481)e\left(\frac{4}{81}\right) e(145162)e\left(\frac{145}{162}\right)
χ163(19,⋅)\chi_{163}(19,\cdot) −1-1 11 e(125162)e\left(\frac{125}{162}\right) e(151162)e\left(\frac{151}{162}\right) e(4481)e\left(\frac{44}{81}\right) e(3154)e\left(\frac{31}{54}\right) e(1927)e\left(\frac{19}{27}\right) e(53162)e\left(\frac{53}{162}\right) e(1754)e\left(\frac{17}{54}\right) e(7081)e\left(\frac{70}{81}\right) e(2881)e\left(\frac{28}{81}\right) e(43162)e\left(\frac{43}{162}\right)
χ163(20,⋅)\chi_{163}(20,\cdot) −1-1 11 e(17162)e\left(\frac{17}{162}\right) e(97162)e\left(\frac{97}{162}\right) e(1781)e\left(\frac{17}{81}\right) e(3154)e\left(\frac{31}{54}\right) e(1927)e\left(\frac{19}{27}\right) e(107162)e\left(\frac{107}{162}\right) e(1754)e\left(\frac{17}{54}\right) e(1681)e\left(\frac{16}{81}\right) e(5581)e\left(\frac{55}{81}\right) e(151162)e\left(\frac{151}{162}\right)
χ163(29,⋅)\chi_{163}(29,\cdot) −1-1 11 e(107162)e\left(\frac{107}{162}\right) e(115162)e\left(\frac{115}{162}\right) e(2681)e\left(\frac{26}{81}\right) e(4954)e\left(\frac{49}{54}\right) e(1027)e\left(\frac{10}{27}\right) e(35162)e\left(\frac{35}{162}\right) e(5354)e\left(\frac{53}{54}\right) e(3481)e\left(\frac{34}{81}\right) e(4681)e\left(\frac{46}{81}\right) e(7162)e\left(\frac{7}{162}\right)
χ163(32,⋅)\chi_{163}(32,\cdot) −1-1 11 e(5162)e\left(\frac{5}{162}\right) e(19162)e\left(\frac{19}{162}\right) e(581)e\left(\frac{5}{81}\right) e(2554)e\left(\frac{25}{54}\right) e(427)e\left(\frac{4}{27}\right) e(41162)e\left(\frac{41}{162}\right) e(554)e\left(\frac{5}{54}\right) e(1981)e\left(\frac{19}{81}\right) e(4081)e\left(\frac{40}{81}\right) e(73162)e\left(\frac{73}{162}\right)
χ163(42,⋅)\chi_{163}(42,\cdot) −1-1 11 e(13162)e\left(\frac{13}{162}\right) e(17162)e\left(\frac{17}{162}\right) e(1381)e\left(\frac{13}{81}\right) e(1154)e\left(\frac{11}{54}\right) e(527)e\left(\frac{5}{27}\right) e(139162)e\left(\frac{139}{162}\right) e(1354)e\left(\frac{13}{54}\right) e(1781)e\left(\frac{17}{81}\right) e(2381)e\left(\frac{23}{81}\right) e(125162)e\left(\frac{125}{162}\right)
χ163(44,⋅)\chi_{163}(44,\cdot) −1-1 11 e(49162)e\left(\frac{49}{162}\right) e(89162)e\left(\frac{89}{162}\right) e(4981)e\left(\frac{49}{81}\right) e(2954)e\left(\frac{29}{54}\right) e(2327)e\left(\frac{23}{27}\right) e(13162)e\left(\frac{13}{162}\right) e(4954)e\left(\frac{49}{54}\right) e(881)e\left(\frac{8}{81}\right) e(6881)e\left(\frac{68}{81}\right) e(35162)e\left(\frac{35}{162}\right)
χ163(45,⋅)\chi_{163}(45,\cdot) −1-1 11 e(55162)e\left(\frac{55}{162}\right) e(47162)e\left(\frac{47}{162}\right) e(5581)e\left(\frac{55}{81}\right) e(554)e\left(\frac{5}{54}\right) e(1727)e\left(\frac{17}{27}\right) e(127162)e\left(\frac{127}{162}\right) e(154)e\left(\frac{1}{54}\right) e(4781)e\left(\frac{47}{81}\right) e(3581)e\left(\frac{35}{81}\right) e(155162)e\left(\frac{155}{162}\right)
χ163(50,⋅)\chi_{163}(50,\cdot) −1-1 11 e(31162)e\left(\frac{31}{162}\right) e(53162)e\left(\frac{53}{162}\right) e(3181)e\left(\frac{31}{81}\right) e(4754)e\left(\frac{47}{54}\right) e(1427)e\left(\frac{14}{27}\right) e(157162)e\left(\frac{157}{162}\right) e(3154)e\left(\frac{31}{54}\right) e(5381)e\left(\frac{53}{81}\right) e(581)e\left(\frac{5}{81}\right) e(161162)e\left(\frac{161}{162}\right)
χ163(52,⋅)\chi_{163}(52,\cdot) −1-1 11 e(53162)e\left(\frac{53}{162}\right) e(7162)e\left(\frac{7}{162}\right) e(5381)e\left(\frac{53}{81}\right) e(4954)e\left(\frac{49}{54}\right) e(1027)e\left(\frac{10}{27}\right) e(143162)e\left(\frac{143}{162}\right) e(5354)e\left(\frac{53}{54}\right) e(781)e\left(\frac{7}{81}\right) e(1981)e\left(\frac{19}{81}\right) e(61162)e\left(\frac{61}{162}\right)
χ163(63,⋅)\chi_{163}(63,\cdot) −1-1 11 e(113162)e\left(\frac{113}{162}\right) e(73162)e\left(\frac{73}{162}\right) e(3281)e\left(\frac{32}{81}\right) e(2554)e\left(\frac{25}{54}\right) e(427)e\left(\frac{4}{27}\right) e(149162)e\left(\frac{149}{162}\right) e(554)e\left(\frac{5}{54}\right) e(7381)e\left(\frac{73}{81}\right) e(1381)e\left(\frac{13}{81}\right) e(127162)e\left(\frac{127}{162}\right)
χ163(66,⋅)\chi_{163}(66,\cdot) −1-1 11 e(149162)e\left(\frac{149}{162}\right) e(145162)e\left(\frac{145}{162}\right) e(6881)e\left(\frac{68}{81}\right) e(4354)e\left(\frac{43}{54}\right) e(2227)e\left(\frac{22}{27}\right) e(23162)e\left(\frac{23}{162}\right) e(4154)e\left(\frac{41}{54}\right) e(6481)e\left(\frac{64}{81}\right) e(5881)e\left(\frac{58}{81}\right) e(37162)e\left(\frac{37}{162}\right)
χ163(67,⋅)\chi_{163}(67,\cdot) −1-1 11 e(25162)e\left(\frac{25}{162}\right) e(95162)e\left(\frac{95}{162}\right) e(2581)e\left(\frac{25}{81}\right) e(1754)e\left(\frac{17}{54}\right) e(2027)e\left(\frac{20}{27}\right) e(43162)e\left(\frac{43}{162}\right) e(2554)e\left(\frac{25}{54}\right) e(1481)e\left(\frac{14}{81}\right) e(3881)e\left(\frac{38}{81}\right) e(41162)e\left(\frac{41}{162}\right)
χ163(68,⋅)\chi_{163}(68,\cdot) −1-1 11 e(59162)e\left(\frac{59}{162}\right) e(127162)e\left(\frac{127}{162}\right) e(5981)e\left(\frac{59}{81}\right) e(2554)e\left(\frac{25}{54}\right) e(427)e\left(\frac{4}{27}\right) e(95162)e\left(\frac{95}{162}\right) e(554)e\left(\frac{5}{54}\right) e(4681)e\left(\frac{46}{81}\right) e(6781)e\left(\frac{67}{81}\right) e(19162)e\left(\frac{19}{162}\right)
χ163(70,⋅)\chi_{163}(70,\cdot) −1-1 11 e(89162)e\left(\frac{89}{162}\right) e(79162)e\left(\frac{79}{162}\right) e(881)e\left(\frac{8}{81}\right) e(1354)e\left(\frac{13}{54}\right) e(127)e\left(\frac{1}{27}\right) e(17162)e\left(\frac{17}{162}\right) e(3554)e\left(\frac{35}{54}\right) e(7981)e\left(\frac{79}{81}\right) e(6481)e\left(\frac{64}{81}\right) e(133162)e\left(\frac{133}{162}\right)
χ163(72,⋅)\chi_{163}(72,\cdot) −1-1 11 e(43162)e\left(\frac{43}{162}\right) e(131162)e\left(\frac{131}{162}\right) e(4381)e\left(\frac{43}{81}\right) e(5354)e\left(\frac{53}{54}\right) e(227)e\left(\frac{2}{27}\right) e(61162)e\left(\frac{61}{162}\right) e(4354)e\left(\frac{43}{54}\right) e(5081)e\left(\frac{50}{81}\right) e(2081)e\left(\frac{20}{81}\right) e(77162)e\left(\frac{77}{162}\right)
χ163(73,⋅)\chi_{163}(73,\cdot) −1-1 11 e(137162)e\left(\frac{137}{162}\right) e(67162)e\left(\frac{67}{162}\right) e(5681)e\left(\frac{56}{81}\right) e(3754)e\left(\frac{37}{54}\right) e(727)e\left(\frac{7}{27}\right) e(119162)e\left(\frac{119}{162}\right) e(2954)e\left(\frac{29}{54}\right) e(6781)e\left(\frac{67}{81}\right) e(4381)e\left(\frac{43}{81}\right) e(121162)e\left(\frac{121}{162}\right)
χ163(75,⋅)\chi_{163}(75,\cdot) −1-1 11 e(131162)e\left(\frac{131}{162}\right) e(109162)e\left(\frac{109}{162}\right) e(5081)e\left(\frac{50}{81}\right) e(754)e\left(\frac{7}{54}\right) e(1327)e\left(\frac{13}{27}\right) e(5162)e\left(\frac{5}{162}\right) e(2354)e\left(\frac{23}{54}\right) e(2881)e\left(\frac{28}{81}\right) e(7681)e\left(\frac{76}{81}\right) e(1162)e\left(\frac{1}{162}\right)
χ163(76,⋅)\chi_{163}(76,\cdot) −1-1 11 e(127162)e\left(\frac{127}{162}\right) e(29162)e\left(\frac{29}{162}\right) e(4681)e\left(\frac{46}{81}\right) e(4154)e\left(\frac{41}{54}\right) e(2627)e\left(\frac{26}{27}\right) e(37162)e\left(\frac{37}{162}\right) e(1954)e\left(\frac{19}{54}\right) e(2981)e\left(\frac{29}{81}\right) e(4481)e\left(\frac{44}{81}\right) e(137162)e\left(\frac{137}{162}\right)
χ163(79,⋅)\chi_{163}(79,\cdot) −1-1 11 e(95162)e\left(\frac{95}{162}\right) e(37162)e\left(\frac{37}{162}\right) e(1481)e\left(\frac{14}{81}\right) e(4354)e\left(\frac{43}{54}\right) e(2227)e\left(\frac{22}{27}\right) e(131162)e\left(\frac{131}{162}\right) e(4154)e\left(\frac{41}{54}\right) e(3781)e\left(\frac{37}{81}\right) e(3181)e\left(\frac{31}{81}\right) e(91162)e\left(\frac{91}{162}\right)
χ163(80,⋅)\chi_{163}(80,\cdot) −1-1 11 e(19162)e\left(\frac{19}{162}\right) e(137162)e\left(\frac{137}{162}\right) e(1981)e\left(\frac{19}{81}\right) e(4154)e\left(\frac{41}{54}\right) e(2627)e\left(\frac{26}{27}\right) e(91162)e\left(\frac{91}{162}\right) e(1954)e\left(\frac{19}{54}\right) e(5681)e\left(\frac{56}{81}\right) e(7181)e\left(\frac{71}{81}\right) e(83162)e\left(\frac{83}{162}\right)
χ163(82,⋅)\chi_{163}(82,\cdot) −1-1 11 e(161162)e\left(\frac{161}{162}\right) e(61162)e\left(\frac{61}{162}\right) e(8081)e\left(\frac{80}{81}\right) e(4954)e\left(\frac{49}{54}\right) e(1027)e\left(\frac{10}{27}\right) e(89162)e\left(\frac{89}{162}\right) e(5354)e\left(\frac{53}{54}\right) e(6181)e\left(\frac{61}{81}\right) e(7381)e\left(\frac{73}{81}\right) e(115162)e\left(\frac{115}{162}\right)
χ163(89,⋅)\chi_{163}(89,\cdot) −1-1 11 e(115162)e\left(\frac{115}{162}\right) e(113162)e\left(\frac{113}{162}\right) e(3481)e\left(\frac{34}{81}\right) e(3554)e\left(\frac{35}{54}\right) e(1127)e\left(\frac{11}{27}\right) e(133162)e\left(\frac{133}{162}\right) e(754)e\left(\frac{7}{54}\right) e(3281)e\left(\frac{32}{81}\right) e(2981)e\left(\frac{29}{81}\right) e(59162)e\left(\frac{59}{162}\right)
χ163(92,⋅)\chi_{163}(92,\cdot) −1-1 11 e(11162)e\left(\frac{11}{162}\right) e(139162)e\left(\frac{139}{162}\right) e(1181)e\left(\frac{11}{81}\right) e(154)e\left(\frac{1}{54}\right) e(2527)e\left(\frac{25}{27}\right) e(155162)e\left(\frac{155}{162}\right) e(1154)e\left(\frac{11}{54}\right) e(5881)e\left(\frac{58}{81}\right) e(781)e\left(\frac{7}{81}\right) e(31162)e\left(\frac{31}{162}\right)
χ163(94,⋅)\chi_{163}(94,\cdot) −1-1 11 e(29162)e\left(\frac{29}{162}\right) e(13162)e\left(\frac{13}{162}\right) e(2981)e\left(\frac{29}{81}\right) e(3754)e\left(\frac{37}{54}\right) e(727)e\left(\frac{7}{27}\right) e(11162)e\left(\frac{11}{162}\right) e(2954)e\left(\frac{29}{54}\right) e(1381)e\left(\frac{13}{81}\right) e(7081)e\left(\frac{70}{81}\right) e(67162)e\left(\frac{67}{162}\right)
χ163(101,⋅)\chi_{163}(101,\cdot) −1-1 11 e(151162)e\left(\frac{151}{162}\right) e(23162)e\left(\frac{23}{162}\right) e(7081)e\left(\frac{70}{81}\right) e(5354)e\left(\frac{53}{54}\right) e(227)e\left(\frac{2}{27}\right) e(7162)e\left(\frac{7}{162}\right) e(4354)e\left(\frac{43}{54}\right) e(2381)e\left(\frac{23}{81}\right) e(7481)e\left(\frac{74}{81}\right) e(131162)e\left(\frac{131}{162}\right)