from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2156, base_ring=CyclotomicField(70))
M = H._module
chi = DirichletCharacter(H, M([0,65,56]))
chi.galois_orbit()
[g,chi] = znchar(Mod(69,2156))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2156\) | |
| Conductor: | \(539\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(70\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 539.bb | 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: | $\Q(\zeta_{35})$ |
| Fixed field: | Number field defined by a degree 70 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2156}(69,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{70}\right)\) | \(e\left(\frac{9}{70}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{31}{70}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{69}{70}\right)\) |
| \(\chi_{2156}(125,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{70}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{39}{70}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{1}{70}\right)\) |
| \(\chi_{2156}(181,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{57}{70}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{33}{70}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{17}{70}\right)\) |
| \(\chi_{2156}(377,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{39}{70}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{59}{70}\right)\) |
| \(\chi_{2156}(405,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{13}{70}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{37}{70}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{3}{70}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{53}{70}\right)\) |
| \(\chi_{2156}(433,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{11}{70}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{51}{70}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{61}{70}\right)\) |
| \(\chi_{2156}(713,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{33}{70}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{13}{70}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{43}{70}\right)\) |
| \(\chi_{2156}(741,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{31}{70}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{51}{70}\right)\) |
| \(\chi_{2156}(797,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{23}{70}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{67}{70}\right)\) |
| \(\chi_{2156}(993,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{70}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{51}{70}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{59}{70}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{39}{70}\right)\) |
| \(\chi_{2156}(1021,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{70}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{23}{70}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{33}{70}\right)\) |
| \(\chi_{2156}(1049,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{70}\right)\) | \(e\left(\frac{51}{70}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{59}{70}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{41}{70}\right)\) |
| \(\chi_{2156}(1105,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{70}\right)\) | \(e\left(\frac{47}{70}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{57}{70}\right)\) |
| \(\chi_{2156}(1301,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{70}\right)\) | \(e\left(\frac{19}{70}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{11}{70}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{29}{70}\right)\) |
| \(\chi_{2156}(1329,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{70}\right)\) | \(e\left(\frac{3}{70}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{57}{70}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{33}{70}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{23}{70}\right)\) |
| \(\chi_{2156}(1357,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{70}\right)\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{19}{70}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{11}{70}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{31}{70}\right)\) |
| \(\chi_{2156}(1413,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{70}\right)\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{13}{70}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{37}{70}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{47}{70}\right)\) |
| \(\chi_{2156}(1609,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{39}{70}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{9}{70}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{19}{70}\right)\) |
| \(\chi_{2156}(1637,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{70}\right)\) | \(e\left(\frac{23}{70}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{13}{70}\right)\) |
| \(\chi_{2156}(1721,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{70}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{47}{70}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{37}{70}\right)\) |
| \(\chi_{2156}(1917,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{70}\right)\) | \(e\left(\frac{59}{70}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{19}{70}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{9}{70}\right)\) |
| \(\chi_{2156}(1945,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{47}{70}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{3}{70}\right)\) |
| \(\chi_{2156}(1973,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{9}{70}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{31}{70}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{11}{70}\right)\) |
| \(\chi_{2156}(2029,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{70}\right)\) | \(e\left(\frac{37}{70}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{3}{70}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{57}{70}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{27}{70}\right)\) |