from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4450, base_ring=CyclotomicField(440))
M = H._module
chi = DirichletCharacter(H, M([418,115]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,4450))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4450\) | |
| Conductor: | \(2225\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(440\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 2225.cb | 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: | $\Q(\zeta_{440})$ |
| Fixed field: | Number field defined by a degree 440 polynomial (not computed) |
First 31 of 160 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4450}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{401}{440}\right)\) | \(e\left(\frac{81}{88}\right)\) | \(e\left(\frac{181}{220}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{27}{440}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{109}{440}\right)\) | \(e\left(\frac{183}{220}\right)\) | \(e\left(\frac{153}{440}\right)\) | \(e\left(\frac{323}{440}\right)\) |
| \(\chi_{4450}(63,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{261}{440}\right)\) | \(e\left(\frac{13}{88}\right)\) | \(e\left(\frac{41}{220}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{327}{440}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{49}{440}\right)\) | \(e\left(\frac{163}{220}\right)\) | \(e\left(\frac{93}{440}\right)\) | \(e\left(\frac{343}{440}\right)\) |
| \(\chi_{4450}(117,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{367}{440}\right)\) | \(e\left(\frac{23}{88}\right)\) | \(e\left(\frac{147}{220}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{389}{440}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{283}{440}\right)\) | \(e\left(\frac{21}{220}\right)\) | \(e\left(\frac{151}{440}\right)\) | \(e\left(\frac{221}{440}\right)\) |
| \(\chi_{4450}(137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{391}{440}\right)\) | \(e\left(\frac{7}{88}\right)\) | \(e\left(\frac{171}{220}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{237}{440}\right)\) | \(e\left(\frac{31}{110}\right)\) | \(e\left(\frac{419}{440}\right)\) | \(e\left(\frac{213}{220}\right)\) | \(e\left(\frac{23}{440}\right)\) | \(e\left(\frac{293}{440}\right)\) |
| \(\chi_{4450}(163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{421}{440}\right)\) | \(e\left(\frac{53}{88}\right)\) | \(e\left(\frac{201}{220}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{47}{440}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{369}{440}\right)\) | \(e\left(\frac{123}{220}\right)\) | \(e\left(\frac{413}{440}\right)\) | \(e\left(\frac{383}{440}\right)\) |
| \(\chi_{4450}(213,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{161}{440}\right)\) | \(e\left(\frac{65}{88}\right)\) | \(e\left(\frac{161}{220}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{227}{440}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{69}{440}\right)\) | \(e\left(\frac{23}{220}\right)\) | \(e\left(\frac{113}{440}\right)\) | \(e\left(\frac{43}{440}\right)\) |
| \(\chi_{4450}(383,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{440}\right)\) | \(e\left(\frac{45}{88}\right)\) | \(e\left(\frac{37}{220}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{279}{440}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{393}{440}\right)\) | \(e\left(\frac{131}{220}\right)\) | \(e\left(\frac{261}{440}\right)\) | \(e\left(\frac{111}{440}\right)\) |
| \(\chi_{4450}(397,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{440}\right)\) | \(e\left(\frac{51}{88}\right)\) | \(e\left(\frac{83}{220}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{281}{440}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{287}{440}\right)\) | \(e\left(\frac{169}{220}\right)\) | \(e\left(\frac{419}{440}\right)\) | \(e\left(\frac{249}{440}\right)\) |
| \(\chi_{4450}(417,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{147}{440}\right)\) | \(e\left(\frac{67}{88}\right)\) | \(e\left(\frac{147}{220}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{169}{440}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{63}{440}\right)\) | \(e\left(\frac{21}{220}\right)\) | \(e\left(\frac{371}{440}\right)\) | \(e\left(\frac{1}{440}\right)\) |
| \(\chi_{4450}(483,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{277}{440}\right)\) | \(e\left(\frac{61}{88}\right)\) | \(e\left(\frac{57}{220}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{79}{440}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{433}{440}\right)\) | \(e\left(\frac{71}{220}\right)\) | \(e\left(\frac{301}{440}\right)\) | \(e\left(\frac{391}{440}\right)\) |
| \(\chi_{4450}(503,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{133}{440}\right)\) | \(e\left(\frac{69}{88}\right)\) | \(e\left(\frac{133}{220}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{111}{440}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{57}{440}\right)\) | \(e\left(\frac{19}{220}\right)\) | \(e\left(\frac{189}{440}\right)\) | \(e\left(\frac{399}{440}\right)\) |
| \(\chi_{4450}(527,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{339}{440}\right)\) | \(e\left(\frac{27}{88}\right)\) | \(e\left(\frac{119}{220}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{273}{440}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{271}{440}\right)\) | \(e\left(\frac{17}{220}\right)\) | \(e\left(\frac{227}{440}\right)\) | \(e\left(\frac{137}{440}\right)\) |
| \(\chi_{4450}(537,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{440}\right)\) | \(e\left(\frac{15}{88}\right)\) | \(e\left(\frac{71}{220}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{357}{440}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{219}{440}\right)\) | \(e\left(\frac{73}{220}\right)\) | \(e\left(\frac{263}{440}\right)\) | \(e\left(\frac{213}{440}\right)\) |
| \(\chi_{4450}(553,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{373}{440}\right)\) | \(e\left(\frac{85}{88}\right)\) | \(e\left(\frac{153}{220}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{351}{440}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{97}{440}\right)\) | \(e\left(\frac{179}{220}\right)\) | \(e\left(\frac{229}{440}\right)\) | \(e\left(\frac{239}{440}\right)\) |
| \(\chi_{4450}(563,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{141}{440}\right)\) | \(e\left(\frac{5}{88}\right)\) | \(e\left(\frac{141}{220}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{207}{440}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{249}{440}\right)\) | \(e\left(\frac{83}{220}\right)\) | \(e\left(\frac{293}{440}\right)\) | \(e\left(\frac{423}{440}\right)\) |
| \(\chi_{4450}(567,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{227}{440}\right)\) | \(e\left(\frac{43}{88}\right)\) | \(e\left(\frac{7}{220}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{249}{440}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{223}{440}\right)\) | \(e\left(\frac{1}{220}\right)\) | \(e\left(\frac{91}{440}\right)\) | \(e\left(\frac{241}{440}\right)\) |
| \(\chi_{4450}(577,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{299}{440}\right)\) | \(e\left(\frac{83}{88}\right)\) | \(e\left(\frac{79}{220}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{233}{440}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{191}{440}\right)\) | \(e\left(\frac{137}{220}\right)\) | \(e\left(\frac{147}{440}\right)\) | \(e\left(\frac{17}{440}\right)\) |
| \(\chi_{4450}(617,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{440}\right)\) | \(e\left(\frac{35}{88}\right)\) | \(e\left(\frac{107}{220}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{129}{440}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{423}{440}\right)\) | \(e\left(\frac{141}{220}\right)\) | \(e\left(\frac{291}{440}\right)\) | \(e\left(\frac{321}{440}\right)\) |
| \(\chi_{4450}(637,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{111}{440}\right)\) | \(e\left(\frac{47}{88}\right)\) | \(e\left(\frac{111}{220}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{397}{440}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{299}{440}\right)\) | \(e\left(\frac{173}{220}\right)\) | \(e\left(\frac{343}{440}\right)\) | \(e\left(\frac{333}{440}\right)\) |
| \(\chi_{4450}(647,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{223}{440}\right)\) | \(e\left(\frac{31}{88}\right)\) | \(e\left(\frac{3}{220}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{421}{440}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{347}{440}\right)\) | \(e\left(\frac{189}{220}\right)\) | \(e\left(\frac{39}{440}\right)\) | \(e\left(\frac{229}{440}\right)\) |
| \(\chi_{4450}(653,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{440}\right)\) | \(e\left(\frac{73}{88}\right)\) | \(e\left(\frac{193}{220}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{171}{440}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{397}{440}\right)\) | \(e\left(\frac{59}{220}\right)\) | \(e\left(\frac{89}{440}\right)\) | \(e\left(\frac{139}{440}\right)\) |
| \(\chi_{4450}(683,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{440}\right)\) | \(e\left(\frac{49}{88}\right)\) | \(e\left(\frac{97}{220}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{339}{440}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{293}{440}\right)\) | \(e\left(\frac{171}{220}\right)\) | \(e\left(\frac{161}{440}\right)\) | \(e\left(\frac{291}{440}\right)\) |
| \(\chi_{4450}(763,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{321}{440}\right)\) | \(e\left(\frac{17}{88}\right)\) | \(e\left(\frac{101}{220}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{387}{440}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{389}{440}\right)\) | \(e\left(\frac{203}{220}\right)\) | \(e\left(\frac{433}{440}\right)\) | \(e\left(\frac{83}{440}\right)\) |
| \(\chi_{4450}(777,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{179}{440}\right)\) | \(e\left(\frac{75}{88}\right)\) | \(e\left(\frac{179}{220}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{113}{440}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{391}{440}\right)\) | \(e\left(\frac{57}{220}\right)\) | \(e\left(\frac{347}{440}\right)\) | \(e\left(\frac{97}{440}\right)\) |
| \(\chi_{4450}(787,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{331}{440}\right)\) | \(e\left(\frac{3}{88}\right)\) | \(e\left(\frac{111}{220}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{177}{440}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{79}{440}\right)\) | \(e\left(\frac{173}{220}\right)\) | \(e\left(\frac{123}{440}\right)\) | \(e\left(\frac{113}{440}\right)\) |
| \(\chi_{4450}(847,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{343}{440}\right)\) | \(e\left(\frac{39}{88}\right)\) | \(e\left(\frac{123}{220}\right)\) | \(e\left(\frac{31}{110}\right)\) | \(e\left(\frac{101}{440}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{147}{440}\right)\) | \(e\left(\frac{49}{220}\right)\) | \(e\left(\frac{279}{440}\right)\) | \(e\left(\frac{149}{440}\right)\) |
| \(\chi_{4450}(863,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{440}\right)\) | \(e\left(\frac{1}{88}\right)\) | \(e\left(\frac{81}{220}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{147}{440}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{349}{440}\right)\) | \(e\left(\frac{43}{220}\right)\) | \(e\left(\frac{393}{440}\right)\) | \(e\left(\frac{243}{440}\right)\) |
| \(\chi_{4450}(867,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{307}{440}\right)\) | \(e\left(\frac{19}{88}\right)\) | \(e\left(\frac{87}{220}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{329}{440}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{383}{440}\right)\) | \(e\left(\frac{201}{220}\right)\) | \(e\left(\frac{251}{440}\right)\) | \(e\left(\frac{41}{440}\right)\) |
| \(\chi_{4450}(887,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{291}{440}\right)\) | \(e\left(\frac{59}{88}\right)\) | \(e\left(\frac{71}{220}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{137}{440}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{439}{440}\right)\) | \(e\left(\frac{73}{220}\right)\) | \(e\left(\frac{43}{440}\right)\) | \(e\left(\frac{433}{440}\right)\) |
| \(\chi_{4450}(897,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{383}{440}\right)\) | \(e\left(\frac{71}{88}\right)\) | \(e\left(\frac{163}{220}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{141}{440}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{227}{440}\right)\) | \(e\left(\frac{149}{220}\right)\) | \(e\left(\frac{359}{440}\right)\) | \(e\left(\frac{269}{440}\right)\) |
| \(\chi_{4450}(903,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{313}{440}\right)\) | \(e\left(\frac{81}{88}\right)\) | \(e\left(\frac{93}{220}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{291}{440}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{197}{440}\right)\) | \(e\left(\frac{139}{220}\right)\) | \(e\left(\frac{329}{440}\right)\) | \(e\left(\frac{59}{440}\right)\) |