from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1225, base_ring=CyclotomicField(420))
M = H._module
chi = DirichletCharacter(H, M([21,260]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,1225))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1225\) | |
| Conductor: | \(1225\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(420\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | $\Q(\zeta_{420})$ |
| Fixed field: | Number field defined by a degree 420 polynomial (not computed) |
First 31 of 96 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1225}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{420}\right)\) | \(e\left(\frac{407}{420}\right)\) | \(e\left(\frac{61}{210}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{61}{140}\right)\) | \(e\left(\frac{197}{210}\right)\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{109}{420}\right)\) | \(e\left(\frac{53}{140}\right)\) | \(e\left(\frac{61}{105}\right)\) |
| \(\chi_{1225}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{420}\right)\) | \(e\left(\frac{317}{420}\right)\) | \(e\left(\frac{31}{210}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{31}{140}\right)\) | \(e\left(\frac{107}{210}\right)\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{379}{420}\right)\) | \(e\left(\frac{43}{140}\right)\) | \(e\left(\frac{31}{105}\right)\) |
| \(\chi_{1225}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{420}\right)\) | \(e\left(\frac{383}{420}\right)\) | \(e\left(\frac{109}{210}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{109}{140}\right)\) | \(e\left(\frac{173}{210}\right)\) | \(e\left(\frac{71}{105}\right)\) | \(e\left(\frac{181}{420}\right)\) | \(e\left(\frac{97}{140}\right)\) | \(e\left(\frac{4}{105}\right)\) |
| \(\chi_{1225}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{227}{420}\right)\) | \(e\left(\frac{289}{420}\right)\) | \(e\left(\frac{17}{210}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{87}{140}\right)\) | \(e\left(\frac{79}{210}\right)\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{323}{420}\right)\) | \(e\left(\frac{71}{140}\right)\) | \(e\left(\frac{17}{105}\right)\) |
| \(\chi_{1225}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{163}{420}\right)\) | \(e\left(\frac{41}{420}\right)\) | \(e\left(\frac{163}{210}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{23}{140}\right)\) | \(e\left(\frac{41}{210}\right)\) | \(e\left(\frac{32}{105}\right)\) | \(e\left(\frac{367}{420}\right)\) | \(e\left(\frac{59}{140}\right)\) | \(e\left(\frac{58}{105}\right)\) |
| \(\chi_{1225}(72,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{420}\right)\) | \(e\left(\frac{359}{420}\right)\) | \(e\left(\frac{157}{210}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{17}{140}\right)\) | \(e\left(\frac{149}{210}\right)\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{253}{420}\right)\) | \(e\left(\frac{1}{140}\right)\) | \(e\left(\frac{52}{105}\right)\) |
| \(\chi_{1225}(88,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{419}{420}\right)\) | \(e\left(\frac{193}{420}\right)\) | \(e\left(\frac{209}{210}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{139}{140}\right)\) | \(e\left(\frac{193}{210}\right)\) | \(e\left(\frac{61}{105}\right)\) | \(e\left(\frac{191}{420}\right)\) | \(e\left(\frac{107}{140}\right)\) | \(e\left(\frac{104}{105}\right)\) |
| \(\chi_{1225}(102,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{420}\right)\) | \(e\left(\frac{247}{420}\right)\) | \(e\left(\frac{101}{210}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{101}{140}\right)\) | \(e\left(\frac{37}{210}\right)\) | \(e\left(\frac{34}{105}\right)\) | \(e\left(\frac{29}{420}\right)\) | \(e\left(\frac{113}{140}\right)\) | \(e\left(\frac{101}{105}\right)\) |
| \(\chi_{1225}(123,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{420}\right)\) | \(e\left(\frac{97}{420}\right)\) | \(e\left(\frac{191}{210}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{51}{140}\right)\) | \(e\left(\frac{97}{210}\right)\) | \(e\left(\frac{4}{105}\right)\) | \(e\left(\frac{59}{420}\right)\) | \(e\left(\frac{3}{140}\right)\) | \(e\left(\frac{86}{105}\right)\) |
| \(\chi_{1225}(137,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{209}{420}\right)\) | \(e\left(\frac{403}{420}\right)\) | \(e\left(\frac{209}{210}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{69}{140}\right)\) | \(e\left(\frac{193}{210}\right)\) | \(e\left(\frac{61}{105}\right)\) | \(e\left(\frac{401}{420}\right)\) | \(e\left(\frac{37}{140}\right)\) | \(e\left(\frac{104}{105}\right)\) |
| \(\chi_{1225}(142,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{253}{420}\right)\) | \(e\left(\frac{311}{420}\right)\) | \(e\left(\frac{43}{210}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{113}{140}\right)\) | \(e\left(\frac{101}{210}\right)\) | \(e\left(\frac{2}{105}\right)\) | \(e\left(\frac{397}{420}\right)\) | \(e\left(\frac{89}{140}\right)\) | \(e\left(\frac{43}{105}\right)\) |
| \(\chi_{1225}(158,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{383}{420}\right)\) | \(e\left(\frac{1}{420}\right)\) | \(e\left(\frac{173}{210}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{103}{140}\right)\) | \(e\left(\frac{1}{210}\right)\) | \(e\left(\frac{52}{105}\right)\) | \(e\left(\frac{347}{420}\right)\) | \(e\left(\frac{39}{140}\right)\) | \(e\left(\frac{68}{105}\right)\) |
| \(\chi_{1225}(163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{420}\right)\) | \(e\left(\frac{53}{420}\right)\) | \(e\left(\frac{139}{210}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{139}{140}\right)\) | \(e\left(\frac{53}{210}\right)\) | \(e\left(\frac{26}{105}\right)\) | \(e\left(\frac{331}{420}\right)\) | \(e\left(\frac{107}{140}\right)\) | \(e\left(\frac{34}{105}\right)\) |
| \(\chi_{1225}(172,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{317}{420}\right)\) | \(e\left(\frac{139}{420}\right)\) | \(e\left(\frac{107}{210}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{37}{140}\right)\) | \(e\left(\frac{139}{210}\right)\) | \(e\left(\frac{88}{105}\right)\) | \(e\left(\frac{353}{420}\right)\) | \(e\left(\frac{101}{140}\right)\) | \(e\left(\frac{2}{105}\right)\) |
| \(\chi_{1225}(198,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{271}{420}\right)\) | \(e\left(\frac{197}{420}\right)\) | \(e\left(\frac{61}{210}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{131}{140}\right)\) | \(e\left(\frac{197}{210}\right)\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{319}{420}\right)\) | \(e\left(\frac{123}{140}\right)\) | \(e\left(\frac{61}{105}\right)\) |
| \(\chi_{1225}(212,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{349}{420}\right)\) | \(e\left(\frac{263}{420}\right)\) | \(e\left(\frac{139}{210}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{69}{140}\right)\) | \(e\left(\frac{53}{210}\right)\) | \(e\left(\frac{26}{105}\right)\) | \(e\left(\frac{121}{420}\right)\) | \(e\left(\frac{37}{140}\right)\) | \(e\left(\frac{34}{105}\right)\) |
| \(\chi_{1225}(228,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{347}{420}\right)\) | \(e\left(\frac{229}{420}\right)\) | \(e\left(\frac{137}{210}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{67}{140}\right)\) | \(e\left(\frac{19}{210}\right)\) | \(e\left(\frac{43}{105}\right)\) | \(e\left(\frac{83}{420}\right)\) | \(e\left(\frac{111}{140}\right)\) | \(e\left(\frac{32}{105}\right)\) |
| \(\chi_{1225}(233,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{403}{420}\right)\) | \(e\left(\frac{341}{420}\right)\) | \(e\left(\frac{193}{210}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{123}{140}\right)\) | \(e\left(\frac{131}{210}\right)\) | \(e\left(\frac{92}{105}\right)\) | \(e\left(\frac{307}{420}\right)\) | \(e\left(\frac{139}{140}\right)\) | \(e\left(\frac{88}{105}\right)\) |
| \(\chi_{1225}(242,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{420}\right)\) | \(e\left(\frac{31}{420}\right)\) | \(e\left(\frac{113}{210}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{113}{140}\right)\) | \(e\left(\frac{31}{210}\right)\) | \(e\left(\frac{37}{105}\right)\) | \(e\left(\frac{257}{420}\right)\) | \(e\left(\frac{89}{140}\right)\) | \(e\left(\frac{8}{105}\right)\) |
| \(\chi_{1225}(247,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{397}{420}\right)\) | \(e\left(\frac{239}{420}\right)\) | \(e\left(\frac{187}{210}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{117}{140}\right)\) | \(e\left(\frac{29}{210}\right)\) | \(e\left(\frac{38}{105}\right)\) | \(e\left(\frac{193}{420}\right)\) | \(e\left(\frac{81}{140}\right)\) | \(e\left(\frac{82}{105}\right)\) |
| \(\chi_{1225}(277,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{221}{420}\right)\) | \(e\left(\frac{187}{420}\right)\) | \(e\left(\frac{11}{210}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{81}{140}\right)\) | \(e\left(\frac{187}{210}\right)\) | \(e\left(\frac{64}{105}\right)\) | \(e\left(\frac{209}{420}\right)\) | \(e\left(\frac{13}{140}\right)\) | \(e\left(\frac{11}{105}\right)\) |
| \(\chi_{1225}(298,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{311}{420}\right)\) | \(e\left(\frac{37}{420}\right)\) | \(e\left(\frac{101}{210}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{31}{140}\right)\) | \(e\left(\frac{37}{210}\right)\) | \(e\left(\frac{34}{105}\right)\) | \(e\left(\frac{239}{420}\right)\) | \(e\left(\frac{43}{140}\right)\) | \(e\left(\frac{101}{105}\right)\) |
| \(\chi_{1225}(303,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{247}{420}\right)\) | \(e\left(\frac{209}{420}\right)\) | \(e\left(\frac{37}{210}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{107}{140}\right)\) | \(e\left(\frac{209}{210}\right)\) | \(e\left(\frac{53}{105}\right)\) | \(e\left(\frac{283}{420}\right)\) | \(e\left(\frac{31}{140}\right)\) | \(e\left(\frac{37}{105}\right)\) |
| \(\chi_{1225}(317,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{420}\right)\) | \(e\left(\frac{191}{420}\right)\) | \(e\left(\frac{73}{210}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{73}{140}\right)\) | \(e\left(\frac{191}{210}\right)\) | \(e\left(\frac{62}{105}\right)\) | \(e\left(\frac{337}{420}\right)\) | \(e\left(\frac{29}{140}\right)\) | \(e\left(\frac{73}{105}\right)\) |
| \(\chi_{1225}(333,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{420}\right)\) | \(e\left(\frac{361}{420}\right)\) | \(e\left(\frac{83}{210}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{83}{140}\right)\) | \(e\left(\frac{151}{210}\right)\) | \(e\left(\frac{82}{105}\right)\) | \(e\left(\frac{107}{420}\right)\) | \(e\left(\frac{79}{140}\right)\) | \(e\left(\frac{83}{105}\right)\) |
| \(\chi_{1225}(338,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{379}{420}\right)\) | \(e\left(\frac{353}{420}\right)\) | \(e\left(\frac{169}{210}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{99}{140}\right)\) | \(e\left(\frac{143}{210}\right)\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{271}{420}\right)\) | \(e\left(\frac{47}{140}\right)\) | \(e\left(\frac{64}{105}\right)\) |
| \(\chi_{1225}(347,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{420}\right)\) | \(e\left(\frac{79}{420}\right)\) | \(e\left(\frac{17}{210}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{17}{140}\right)\) | \(e\left(\frac{79}{210}\right)\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{113}{420}\right)\) | \(e\left(\frac{1}{140}\right)\) | \(e\left(\frac{17}{105}\right)\) |
| \(\chi_{1225}(352,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{420}\right)\) | \(e\left(\frac{167}{420}\right)\) | \(e\left(\frac{121}{210}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{121}{140}\right)\) | \(e\left(\frac{167}{210}\right)\) | \(e\left(\frac{74}{105}\right)\) | \(e\left(\frac{409}{420}\right)\) | \(e\left(\frac{73}{140}\right)\) | \(e\left(\frac{16}{105}\right)\) |
| \(\chi_{1225}(387,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{169}{420}\right)\) | \(e\left(\frac{143}{420}\right)\) | \(e\left(\frac{169}{210}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{29}{140}\right)\) | \(e\left(\frac{143}{210}\right)\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{61}{420}\right)\) | \(e\left(\frac{117}{140}\right)\) | \(e\left(\frac{64}{105}\right)\) |
| \(\chi_{1225}(403,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{420}\right)\) | \(e\left(\frac{169}{420}\right)\) | \(e\left(\frac{47}{210}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{47}{140}\right)\) | \(e\left(\frac{169}{210}\right)\) | \(e\left(\frac{73}{105}\right)\) | \(e\left(\frac{263}{420}\right)\) | \(e\left(\frac{11}{140}\right)\) | \(e\left(\frac{47}{105}\right)\) |
| \(\chi_{1225}(408,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{223}{420}\right)\) | \(e\left(\frac{221}{420}\right)\) | \(e\left(\frac{13}{210}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{83}{140}\right)\) | \(e\left(\frac{11}{210}\right)\) | \(e\left(\frac{47}{105}\right)\) | \(e\left(\frac{247}{420}\right)\) | \(e\left(\frac{79}{140}\right)\) | \(e\left(\frac{13}{105}\right)\) |