from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4017, base_ring=CyclotomicField(68))
M = H._module
chi = DirichletCharacter(H, M([0,51,38]))
chi.galois_orbit()
[g,chi] = znchar(Mod(31,4017))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4017\) | |
| Conductor: | \(1339\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(68\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 1339.bn | 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: | $\Q(\zeta_{68})$ |
| Fixed field: | Number field defined by a degree 68 polynomial |
First 31 of 32 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4017}(31,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{21}{34}\right)\) |
| \(\chi_{4017}(73,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{5}{34}\right)\) |
| \(\chi_{4017}(346,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{11}{34}\right)\) |
| \(\chi_{4017}(382,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{5}{34}\right)\) |
| \(\chi_{4017}(502,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{31}{34}\right)\) |
| \(\chi_{4017}(655,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{11}{34}\right)\) |
| \(\chi_{4017}(811,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{31}{34}\right)\) |
| \(\chi_{4017}(1555,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{13}{34}\right)\) |
| \(\chi_{4017}(1672,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{29}{34}\right)\) |
| \(\chi_{4017}(1864,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{13}{34}\right)\) |
| \(\chi_{4017}(1981,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{29}{34}\right)\) |
| \(\chi_{4017}(1984,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{27}{34}\right)\) |
| \(\chi_{4017}(2140,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{33}{34}\right)\) |
| \(\chi_{4017}(2257,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{1}{34}\right)\) |
| \(\chi_{4017}(2293,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{27}{34}\right)\) |
| \(\chi_{4017}(2335,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{25}{34}\right)\) |
| \(\chi_{4017}(2449,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{33}{34}\right)\) |
| \(\chi_{4017}(2566,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{1}{34}\right)\) |
| \(\chi_{4017}(2644,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{25}{34}\right)\) |
| \(\chi_{4017}(2803,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{19}{34}\right)\) |
| \(\chi_{4017}(3076,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{15}{34}\right)\) |
| \(\chi_{4017}(3112,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{19}{34}\right)\) |
| \(\chi_{4017}(3232,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{23}{34}\right)\) |
| \(\chi_{4017}(3385,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{15}{34}\right)\) |
| \(\chi_{4017}(3505,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{9}{34}\right)\) |
| \(\chi_{4017}(3541,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{23}{34}\right)\) |
| \(\chi_{4017}(3544,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{7}{34}\right)\) |
| \(\chi_{4017}(3700,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{3}{34}\right)\) |
| \(\chi_{4017}(3739,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{21}{34}\right)\) |
| \(\chi_{4017}(3814,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{9}{34}\right)\) |
| \(\chi_{4017}(3853,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{7}{34}\right)\) |