from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8624, base_ring=CyclotomicField(140))
M = H._module
chi = DirichletCharacter(H, M([70,35,10,56]))
chi.galois_orbit()
[g,chi] = znchar(Mod(27,8624))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8624\) | |
| Conductor: | \(8624\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(140\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | $\Q(\zeta_{140})$ |
| Fixed field: | Number field defined by a degree 140 polynomial (not computed) |
First 31 of 48 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8624}(27,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{140}\right)\) | \(e\left(\frac{129}{140}\right)\) | \(e\left(\frac{3}{70}\right)\) | \(e\left(\frac{71}{140}\right)\) | \(e\left(\frac{31}{70}\right)\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{59}{70}\right)\) | \(e\left(\frac{79}{140}\right)\) |
| \(\chi_{8624}(251,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{140}\right)\) | \(e\left(\frac{41}{140}\right)\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{79}{140}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{33}{70}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{11}{140}\right)\) |
| \(\chi_{8624}(531,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{140}\right)\) | \(e\left(\frac{43}{140}\right)\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{117}{140}\right)\) | \(e\left(\frac{57}{70}\right)\) | \(e\left(\frac{9}{70}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{73}{140}\right)\) |
| \(\chi_{8624}(643,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{140}\right)\) | \(e\left(\frac{139}{140}\right)\) | \(e\left(\frac{13}{70}\right)\) | \(e\left(\frac{121}{140}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{47}{70}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{109}{140}\right)\) |
| \(\chi_{8624}(867,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{140}\right)\) | \(e\left(\frac{51}{140}\right)\) | \(e\left(\frac{37}{70}\right)\) | \(e\left(\frac{129}{140}\right)\) | \(e\left(\frac{9}{70}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{51}{70}\right)\) | \(e\left(\frac{41}{140}\right)\) |
| \(\chi_{8624}(1147,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{140}\right)\) | \(e\left(\frac{53}{140}\right)\) | \(e\left(\frac{11}{70}\right)\) | \(e\left(\frac{27}{140}\right)\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{103}{140}\right)\) |
| \(\chi_{8624}(1203,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{140}\right)\) | \(e\left(\frac{87}{140}\right)\) | \(e\left(\frac{59}{70}\right)\) | \(e\left(\frac{113}{140}\right)\) | \(e\left(\frac{3}{70}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{37}{140}\right)\) |
| \(\chi_{8624}(1259,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{93}{140}\right)\) | \(e\left(\frac{9}{140}\right)\) | \(e\left(\frac{23}{70}\right)\) | \(e\left(\frac{31}{140}\right)\) | \(e\left(\frac{51}{70}\right)\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{9}{70}\right)\) | \(e\left(\frac{139}{140}\right)\) |
| \(\chi_{8624}(1483,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{117}{140}\right)\) | \(e\left(\frac{61}{140}\right)\) | \(e\left(\frac{47}{70}\right)\) | \(e\left(\frac{39}{140}\right)\) | \(e\left(\frac{19}{70}\right)\) | \(e\left(\frac{3}{70}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{71}{140}\right)\) |
| \(\chi_{8624}(1819,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{69}{140}\right)\) | \(e\left(\frac{97}{140}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{23}{140}\right)\) | \(e\left(\frac{13}{70}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{67}{140}\right)\) |
| \(\chi_{8624}(1875,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{140}\right)\) | \(e\left(\frac{19}{140}\right)\) | \(e\left(\frac{33}{70}\right)\) | \(e\left(\frac{81}{140}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{19}{70}\right)\) | \(e\left(\frac{29}{140}\right)\) |
| \(\chi_{8624}(2099,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{140}\right)\) | \(e\left(\frac{71}{140}\right)\) | \(e\left(\frac{57}{70}\right)\) | \(e\left(\frac{89}{140}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{23}{70}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{101}{140}\right)\) |
| \(\chi_{8624}(2379,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{140}\right)\) | \(e\left(\frac{73}{140}\right)\) | \(e\left(\frac{31}{70}\right)\) | \(e\left(\frac{127}{140}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{3}{70}\right)\) | \(e\left(\frac{23}{140}\right)\) |
| \(\chi_{8624}(2435,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{140}\right)\) | \(e\left(\frac{107}{140}\right)\) | \(e\left(\frac{9}{70}\right)\) | \(e\left(\frac{73}{140}\right)\) | \(e\left(\frac{23}{70}\right)\) | \(e\left(\frac{11}{70}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{37}{70}\right)\) | \(e\left(\frac{97}{140}\right)\) |
| \(\chi_{8624}(2491,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{140}\right)\) | \(e\left(\frac{29}{140}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{131}{140}\right)\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{37}{70}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{59}{140}\right)\) |
| \(\chi_{8624}(2715,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{140}\right)\) | \(e\left(\frac{81}{140}\right)\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{139}{140}\right)\) | \(e\left(\frac{39}{70}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{11}{70}\right)\) | \(e\left(\frac{131}{140}\right)\) |
| \(\chi_{8624}(2995,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{111}{140}\right)\) | \(e\left(\frac{83}{140}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{37}{140}\right)\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{19}{70}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{13}{70}\right)\) | \(e\left(\frac{53}{140}\right)\) |
| \(\chi_{8624}(3051,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{140}\right)\) | \(e\left(\frac{117}{140}\right)\) | \(e\left(\frac{19}{70}\right)\) | \(e\left(\frac{123}{140}\right)\) | \(e\left(\frac{33}{70}\right)\) | \(e\left(\frac{31}{70}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{47}{70}\right)\) | \(e\left(\frac{127}{140}\right)\) |
| \(\chi_{8624}(3107,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{123}{140}\right)\) | \(e\left(\frac{39}{140}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{41}{140}\right)\) | \(e\left(\frac{11}{70}\right)\) | \(e\left(\frac{57}{70}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{39}{70}\right)\) | \(e\left(\frac{89}{140}\right)\) |
| \(\chi_{8624}(3611,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{140}\right)\) | \(e\left(\frac{93}{140}\right)\) | \(e\left(\frac{51}{70}\right)\) | \(e\left(\frac{87}{140}\right)\) | \(e\left(\frac{37}{70}\right)\) | \(e\left(\frac{39}{70}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{23}{70}\right)\) | \(e\left(\frac{83}{140}\right)\) |
| \(\chi_{8624}(3667,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{99}{140}\right)\) | \(e\left(\frac{127}{140}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{33}{140}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{51}{70}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{57}{70}\right)\) | \(e\left(\frac{17}{140}\right)\) |
| \(\chi_{8624}(3947,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{140}\right)\) | \(e\left(\frac{101}{140}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{99}{140}\right)\) | \(e\left(\frac{59}{70}\right)\) | \(e\left(\frac{13}{70}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{31}{70}\right)\) | \(e\left(\frac{51}{140}\right)\) |
| \(\chi_{8624}(4227,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{140}\right)\) | \(e\left(\frac{103}{140}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{137}{140}\right)\) | \(e\left(\frac{47}{70}\right)\) | \(e\left(\frac{59}{70}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{33}{70}\right)\) | \(e\left(\frac{113}{140}\right)\) |
| \(\chi_{8624}(4283,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{140}\right)\) | \(e\left(\frac{137}{140}\right)\) | \(e\left(\frac{39}{70}\right)\) | \(e\left(\frac{83}{140}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{47}{140}\right)\) |
| \(\chi_{8624}(4339,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{140}\right)\) | \(e\left(\frac{59}{140}\right)\) | \(e\left(\frac{3}{70}\right)\) | \(e\left(\frac{1}{140}\right)\) | \(e\left(\frac{31}{70}\right)\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{59}{70}\right)\) | \(e\left(\frac{9}{140}\right)\) |
| \(\chi_{8624}(4563,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{140}\right)\) | \(e\left(\frac{111}{140}\right)\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{9}{140}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{33}{70}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{81}{140}\right)\) |
| \(\chi_{8624}(4843,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{140}\right)\) | \(e\left(\frac{113}{140}\right)\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{47}{140}\right)\) | \(e\left(\frac{57}{70}\right)\) | \(e\left(\frac{9}{70}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{3}{140}\right)\) |
| \(\chi_{8624}(4955,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{140}\right)\) | \(e\left(\frac{69}{140}\right)\) | \(e\left(\frac{13}{70}\right)\) | \(e\left(\frac{51}{140}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{47}{70}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{39}{140}\right)\) |
| \(\chi_{8624}(5179,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{140}\right)\) | \(e\left(\frac{121}{140}\right)\) | \(e\left(\frac{37}{70}\right)\) | \(e\left(\frac{59}{140}\right)\) | \(e\left(\frac{9}{70}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{51}{70}\right)\) | \(e\left(\frac{111}{140}\right)\) |
| \(\chi_{8624}(5459,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{140}\right)\) | \(e\left(\frac{123}{140}\right)\) | \(e\left(\frac{11}{70}\right)\) | \(e\left(\frac{97}{140}\right)\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{33}{140}\right)\) |
| \(\chi_{8624}(5515,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{129}{140}\right)\) | \(e\left(\frac{17}{140}\right)\) | \(e\left(\frac{59}{70}\right)\) | \(e\left(\frac{43}{140}\right)\) | \(e\left(\frac{3}{70}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{107}{140}\right)\) |