from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3712, base_ring=CyclotomicField(112))
M = H._module
chi = DirichletCharacter(H, M([0,21,40]))
chi.galois_orbit()
[g,chi] = znchar(Mod(9,3712))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3712\) | |
| Conductor: | \(1856\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(112\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 1856.co | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{112})$ |
| Fixed field: | Number field defined by a degree 112 polynomial (not computed) |
First 31 of 48 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3712}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{112}\right)\) | \(e\left(\frac{5}{112}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{97}{112}\right)\) | \(e\left(\frac{27}{112}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(-i\) | \(e\left(\frac{59}{112}\right)\) | \(e\left(\frac{57}{112}\right)\) |
| \(\chi_{3712}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{112}\right)\) | \(e\left(\frac{67}{112}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{23}{112}\right)\) | \(e\left(\frac{93}{112}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(i\) | \(e\left(\frac{29}{112}\right)\) | \(e\left(\frac{47}{112}\right)\) |
| \(\chi_{3712}(265,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{112}\right)\) | \(e\left(\frac{85}{112}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{81}{112}\right)\) | \(e\left(\frac{11}{112}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(-i\) | \(e\left(\frac{107}{112}\right)\) | \(e\left(\frac{73}{112}\right)\) |
| \(\chi_{3712}(361,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{112}\right)\) | \(e\left(\frac{65}{112}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{29}{112}\right)\) | \(e\left(\frac{15}{112}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(-i\) | \(e\left(\frac{95}{112}\right)\) | \(e\left(\frac{69}{112}\right)\) |
| \(\chi_{3712}(441,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{112}\right)\) | \(e\left(\frac{59}{112}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{47}{112}\right)\) | \(e\left(\frac{5}{112}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(i\) | \(e\left(\frac{69}{112}\right)\) | \(e\left(\frac{23}{112}\right)\) |
| \(\chi_{3712}(457,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{112}\right)\) | \(e\left(\frac{13}{112}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{73}{112}\right)\) | \(e\left(\frac{3}{112}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(-i\) | \(e\left(\frac{19}{112}\right)\) | \(e\left(\frac{81}{112}\right)\) |
| \(\chi_{3712}(473,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{112}\right)\) | \(e\left(\frac{47}{112}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{83}{112}\right)\) | \(e\left(\frac{97}{112}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(i\) | \(e\left(\frac{17}{112}\right)\) | \(e\left(\frac{43}{112}\right)\) |
| \(\chi_{3712}(585,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{111}{112}\right)\) | \(e\left(\frac{109}{112}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{9}{112}\right)\) | \(e\left(\frac{51}{112}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(-i\) | \(e\left(\frac{99}{112}\right)\) | \(e\left(\frac{33}{112}\right)\) |
| \(\chi_{3712}(729,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{112}\right)\) | \(e\left(\frac{15}{112}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{67}{112}\right)\) | \(e\left(\frac{81}{112}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(i\) | \(e\left(\frac{65}{112}\right)\) | \(e\left(\frac{59}{112}\right)\) |
| \(\chi_{3712}(825,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{112}\right)\) | \(e\left(\frac{107}{112}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{15}{112}\right)\) | \(e\left(\frac{85}{112}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(i\) | \(e\left(\frac{53}{112}\right)\) | \(e\left(\frac{55}{112}\right)\) |
| \(\chi_{3712}(905,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{112}\right)\) | \(e\left(\frac{101}{112}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{33}{112}\right)\) | \(e\left(\frac{75}{112}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(-i\) | \(e\left(\frac{27}{112}\right)\) | \(e\left(\frac{9}{112}\right)\) |
| \(\chi_{3712}(921,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{93}{112}\right)\) | \(e\left(\frac{55}{112}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{59}{112}\right)\) | \(e\left(\frac{73}{112}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(i\) | \(e\left(\frac{89}{112}\right)\) | \(e\left(\frac{67}{112}\right)\) |
| \(\chi_{3712}(937,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{112}\right)\) | \(e\left(\frac{89}{112}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{69}{112}\right)\) | \(e\left(\frac{55}{112}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(-i\) | \(e\left(\frac{87}{112}\right)\) | \(e\left(\frac{29}{112}\right)\) |
| \(\chi_{3712}(1049,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{112}\right)\) | \(e\left(\frac{39}{112}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{107}{112}\right)\) | \(e\left(\frac{9}{112}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(i\) | \(e\left(\frac{57}{112}\right)\) | \(e\left(\frac{19}{112}\right)\) |
| \(\chi_{3712}(1193,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{112}\right)\) | \(e\left(\frac{57}{112}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{53}{112}\right)\) | \(e\left(\frac{39}{112}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(-i\) | \(e\left(\frac{23}{112}\right)\) | \(e\left(\frac{45}{112}\right)\) |
| \(\chi_{3712}(1289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{87}{112}\right)\) | \(e\left(\frac{37}{112}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{1}{112}\right)\) | \(e\left(\frac{43}{112}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(-i\) | \(e\left(\frac{11}{112}\right)\) | \(e\left(\frac{41}{112}\right)\) |
| \(\chi_{3712}(1369,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{112}\right)\) | \(e\left(\frac{31}{112}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{19}{112}\right)\) | \(e\left(\frac{33}{112}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(i\) | \(e\left(\frac{97}{112}\right)\) | \(e\left(\frac{107}{112}\right)\) |
| \(\chi_{3712}(1385,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{112}\right)\) | \(e\left(\frac{97}{112}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{45}{112}\right)\) | \(e\left(\frac{31}{112}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(-i\) | \(e\left(\frac{47}{112}\right)\) | \(e\left(\frac{53}{112}\right)\) |
| \(\chi_{3712}(1401,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{112}\right)\) | \(e\left(\frac{19}{112}\right)\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{55}{112}\right)\) | \(e\left(\frac{13}{112}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(i\) | \(e\left(\frac{45}{112}\right)\) | \(e\left(\frac{15}{112}\right)\) |
| \(\chi_{3712}(1513,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{112}\right)\) | \(e\left(\frac{81}{112}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{93}{112}\right)\) | \(e\left(\frac{79}{112}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(-i\) | \(e\left(\frac{15}{112}\right)\) | \(e\left(\frac{5}{112}\right)\) |
| \(\chi_{3712}(1657,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{112}\right)\) | \(e\left(\frac{99}{112}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{39}{112}\right)\) | \(e\left(\frac{109}{112}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(i\) | \(e\left(\frac{93}{112}\right)\) | \(e\left(\frac{31}{112}\right)\) |
| \(\chi_{3712}(1753,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{112}\right)\) | \(e\left(\frac{79}{112}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{99}{112}\right)\) | \(e\left(\frac{1}{112}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(i\) | \(e\left(\frac{81}{112}\right)\) | \(e\left(\frac{27}{112}\right)\) |
| \(\chi_{3712}(1833,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{99}{112}\right)\) | \(e\left(\frac{73}{112}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{5}{112}\right)\) | \(e\left(\frac{103}{112}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(-i\) | \(e\left(\frac{55}{112}\right)\) | \(e\left(\frac{93}{112}\right)\) |
| \(\chi_{3712}(1849,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{112}\right)\) | \(e\left(\frac{27}{112}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{31}{112}\right)\) | \(e\left(\frac{101}{112}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(i\) | \(e\left(\frac{5}{112}\right)\) | \(e\left(\frac{39}{112}\right)\) |
| \(\chi_{3712}(1865,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{95}{112}\right)\) | \(e\left(\frac{61}{112}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{41}{112}\right)\) | \(e\left(\frac{83}{112}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(-i\) | \(e\left(\frac{3}{112}\right)\) | \(e\left(\frac{1}{112}\right)\) |
| \(\chi_{3712}(1977,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{112}\right)\) | \(e\left(\frac{11}{112}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{79}{112}\right)\) | \(e\left(\frac{37}{112}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(i\) | \(e\left(\frac{85}{112}\right)\) | \(e\left(\frac{103}{112}\right)\) |
| \(\chi_{3712}(2121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{112}\right)\) | \(e\left(\frac{29}{112}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{25}{112}\right)\) | \(e\left(\frac{67}{112}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(-i\) | \(e\left(\frac{51}{112}\right)\) | \(e\left(\frac{17}{112}\right)\) |
| \(\chi_{3712}(2217,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{112}\right)\) | \(e\left(\frac{9}{112}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{85}{112}\right)\) | \(e\left(\frac{71}{112}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(-i\) | \(e\left(\frac{39}{112}\right)\) | \(e\left(\frac{13}{112}\right)\) |
| \(\chi_{3712}(2297,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{112}\right)\) | \(e\left(\frac{3}{112}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{103}{112}\right)\) | \(e\left(\frac{61}{112}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(i\) | \(e\left(\frac{13}{112}\right)\) | \(e\left(\frac{79}{112}\right)\) |
| \(\chi_{3712}(2313,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{112}\right)\) | \(e\left(\frac{69}{112}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{17}{112}\right)\) | \(e\left(\frac{59}{112}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(-i\) | \(e\left(\frac{75}{112}\right)\) | \(e\left(\frac{25}{112}\right)\) |
| \(\chi_{3712}(2329,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{112}\right)\) | \(e\left(\frac{103}{112}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{27}{112}\right)\) | \(e\left(\frac{41}{112}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(i\) | \(e\left(\frac{73}{112}\right)\) | \(e\left(\frac{99}{112}\right)\) |