from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1836, base_ring=CyclotomicField(144))
M = H._module
chi = DirichletCharacter(H, M([0,128,27]))
chi.galois_orbit()
[g,chi] = znchar(Mod(61,1836))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1836\) | |
| Conductor: | \(459\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(144\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 459.bc | 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: | $\Q(\zeta_{144})$ |
| Fixed field: | Number field defined by a degree 144 polynomial (not computed) |
First 31 of 48 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1836}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{144}\right)\) | \(e\left(\frac{41}{144}\right)\) | \(e\left(\frac{125}{144}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{85}{144}\right)\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{47}{144}\right)\) | \(e\left(\frac{67}{144}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1836}(97,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{144}\right)\) | \(e\left(\frac{71}{144}\right)\) | \(e\left(\frac{83}{144}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{91}{144}\right)\) | \(e\left(\frac{25}{72}\right)\) | \(e\left(\frac{113}{144}\right)\) | \(e\left(\frac{109}{144}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1836}(133,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{144}\right)\) | \(e\left(\frac{11}{144}\right)\) | \(e\left(\frac{23}{144}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{79}{144}\right)\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{125}{144}\right)\) | \(e\left(\frac{25}{144}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1836}(193,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{144}\right)\) | \(e\left(\frac{13}{144}\right)\) | \(e\left(\frac{1}{144}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{41}{144}\right)\) | \(e\left(\frac{35}{72}\right)\) | \(e\left(\frac{43}{144}\right)\) | \(e\left(\frac{95}{144}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1836}(241,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{144}\right)\) | \(e\left(\frac{83}{144}\right)\) | \(e\left(\frac{95}{144}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{7}{144}\right)\) | \(e\left(\frac{13}{72}\right)\) | \(e\left(\frac{53}{144}\right)\) | \(e\left(\frac{97}{144}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1836}(265,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{119}{144}\right)\) | \(e\left(\frac{73}{144}\right)\) | \(e\left(\frac{61}{144}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{53}{144}\right)\) | \(e\left(\frac{47}{72}\right)\) | \(e\left(\frac{31}{144}\right)\) | \(e\left(\frac{35}{144}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1836}(277,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{144}\right)\) | \(e\left(\frac{95}{144}\right)\) | \(e\left(\frac{107}{144}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{67}{144}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{137}{144}\right)\) | \(e\left(\frac{85}{144}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1836}(301,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{144}\right)\) | \(e\left(\frac{103}{144}\right)\) | \(e\left(\frac{19}{144}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{59}{144}\right)\) | \(e\left(\frac{17}{72}\right)\) | \(e\left(\frac{97}{144}\right)\) | \(e\left(\frac{77}{144}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1836}(313,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{144}\right)\) | \(e\left(\frac{17}{144}\right)\) | \(e\left(\frac{101}{144}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{109}{144}\right)\) | \(e\left(\frac{7}{72}\right)\) | \(e\left(\frac{23}{144}\right)\) | \(e\left(\frac{91}{144}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1836}(337,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{144}\right)\) | \(e\left(\frac{43}{144}\right)\) | \(e\left(\frac{103}{144}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{47}{144}\right)\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{109}{144}\right)\) | \(e\left(\frac{137}{144}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1836}(385,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{144}\right)\) | \(e\left(\frac{5}{144}\right)\) | \(e\left(\frac{89}{144}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{49}{144}\right)\) | \(e\left(\frac{19}{72}\right)\) | \(e\left(\frac{83}{144}\right)\) | \(e\left(\frac{103}{144}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1836}(445,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{144}\right)\) | \(e\left(\frac{115}{144}\right)\) | \(e\left(\frac{31}{144}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{119}{144}\right)\) | \(e\left(\frac{5}{72}\right)\) | \(e\left(\frac{37}{144}\right)\) | \(e\left(\frac{65}{144}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1836}(481,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{144}\right)\) | \(e\left(\frac{127}{144}\right)\) | \(e\left(\frac{43}{144}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{35}{144}\right)\) | \(e\left(\frac{65}{72}\right)\) | \(e\left(\frac{121}{144}\right)\) | \(e\left(\frac{53}{144}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1836}(517,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{143}{144}\right)\) | \(e\left(\frac{49}{144}\right)\) | \(e\left(\frac{37}{144}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{77}{144}\right)\) | \(e\left(\frac{71}{72}\right)\) | \(e\left(\frac{7}{144}\right)\) | \(e\left(\frac{59}{144}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1836}(589,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{144}\right)\) | \(e\left(\frac{37}{144}\right)\) | \(e\left(\frac{25}{144}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{17}{144}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{67}{144}\right)\) | \(e\left(\frac{71}{144}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1836}(601,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{144}\right)\) | \(e\left(\frac{77}{144}\right)\) | \(e\left(\frac{17}{144}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{121}{144}\right)\) | \(e\left(\frac{19}{72}\right)\) | \(e\left(\frac{11}{144}\right)\) | \(e\left(\frac{31}{144}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1836}(673,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{144}\right)\) | \(e\left(\frac{137}{144}\right)\) | \(e\left(\frac{77}{144}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{133}{144}\right)\) | \(e\left(\frac{31}{72}\right)\) | \(e\left(\frac{143}{144}\right)\) | \(e\left(\frac{115}{144}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1836}(709,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{144}\right)\) | \(e\left(\frac{23}{144}\right)\) | \(e\left(\frac{35}{144}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{139}{144}\right)\) | \(e\left(\frac{1}{72}\right)\) | \(e\left(\frac{65}{144}\right)\) | \(e\left(\frac{13}{144}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1836}(745,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{133}{144}\right)\) | \(e\left(\frac{107}{144}\right)\) | \(e\left(\frac{119}{144}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{127}{144}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{77}{144}\right)\) | \(e\left(\frac{73}{144}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1836}(805,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{144}\right)\) | \(e\left(\frac{109}{144}\right)\) | \(e\left(\frac{97}{144}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{89}{144}\right)\) | \(e\left(\frac{11}{72}\right)\) | \(e\left(\frac{139}{144}\right)\) | \(e\left(\frac{143}{144}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1836}(853,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{144}\right)\) | \(e\left(\frac{35}{144}\right)\) | \(e\left(\frac{47}{144}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{55}{144}\right)\) | \(e\left(\frac{61}{72}\right)\) | \(e\left(\frac{5}{144}\right)\) | \(e\left(\frac{1}{144}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1836}(877,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{144}\right)\) | \(e\left(\frac{25}{144}\right)\) | \(e\left(\frac{13}{144}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{101}{144}\right)\) | \(e\left(\frac{23}{72}\right)\) | \(e\left(\frac{127}{144}\right)\) | \(e\left(\frac{83}{144}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1836}(889,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{144}\right)\) | \(e\left(\frac{47}{144}\right)\) | \(e\left(\frac{59}{144}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{115}{144}\right)\) | \(e\left(\frac{49}{72}\right)\) | \(e\left(\frac{89}{144}\right)\) | \(e\left(\frac{133}{144}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1836}(913,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{144}\right)\) | \(e\left(\frac{55}{144}\right)\) | \(e\left(\frac{115}{144}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{107}{144}\right)\) | \(e\left(\frac{65}{72}\right)\) | \(e\left(\frac{49}{144}\right)\) | \(e\left(\frac{125}{144}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1836}(925,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{144}\right)\) | \(e\left(\frac{113}{144}\right)\) | \(e\left(\frac{53}{144}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{13}{144}\right)\) | \(e\left(\frac{55}{72}\right)\) | \(e\left(\frac{119}{144}\right)\) | \(e\left(\frac{139}{144}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1836}(949,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{144}\right)\) | \(e\left(\frac{139}{144}\right)\) | \(e\left(\frac{55}{144}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{95}{144}\right)\) | \(e\left(\frac{53}{72}\right)\) | \(e\left(\frac{61}{144}\right)\) | \(e\left(\frac{41}{144}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1836}(997,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{144}\right)\) | \(e\left(\frac{101}{144}\right)\) | \(e\left(\frac{41}{144}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{97}{144}\right)\) | \(e\left(\frac{67}{72}\right)\) | \(e\left(\frac{35}{144}\right)\) | \(e\left(\frac{7}{144}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{1836}(1057,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{144}\right)\) | \(e\left(\frac{67}{144}\right)\) | \(e\left(\frac{127}{144}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{23}{144}\right)\) | \(e\left(\frac{53}{72}\right)\) | \(e\left(\frac{133}{144}\right)\) | \(e\left(\frac{113}{144}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1836}(1093,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{144}\right)\) | \(e\left(\frac{79}{144}\right)\) | \(e\left(\frac{139}{144}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{83}{144}\right)\) | \(e\left(\frac{41}{72}\right)\) | \(e\left(\frac{73}{144}\right)\) | \(e\left(\frac{101}{144}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1836}(1129,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{144}\right)\) | \(e\left(\frac{1}{144}\right)\) | \(e\left(\frac{133}{144}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{125}{144}\right)\) | \(e\left(\frac{47}{72}\right)\) | \(e\left(\frac{103}{144}\right)\) | \(e\left(\frac{107}{144}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{1836}(1201,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{144}\right)\) | \(e\left(\frac{133}{144}\right)\) | \(e\left(\frac{121}{144}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{65}{144}\right)\) | \(e\left(\frac{59}{72}\right)\) | \(e\left(\frac{19}{144}\right)\) | \(e\left(\frac{119}{144}\right)\) | \(e\left(\frac{1}{3}\right)\) |