from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4224, base_ring=CyclotomicField(160))
M = H._module
chi = DirichletCharacter(H, M([80,55,80,16]))
chi.galois_orbit()
[g,chi] = znchar(Mod(35,4224))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4224\) | |
| Conductor: | \(4224\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(160\) | 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: | $\Q(\zeta_{160})$ |
| Fixed field: | Number field defined by a degree 160 polynomial (not computed) |
First 31 of 64 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4224}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{160}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{41}{160}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{113}{160}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{77}{160}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{141}{160}\right)\) |
| \(\chi_{4224}(83,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{160}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{29}{160}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{37}{160}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{113}{160}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{49}{160}\right)\) |
| \(\chi_{4224}(107,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{160}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{63}{160}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{119}{160}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{91}{160}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{123}{160}\right)\) |
| \(\chi_{4224}(227,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{160}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{57}{160}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{1}{160}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{29}{160}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{157}{160}\right)\) |
| \(\chi_{4224}(299,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{129}{160}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{111}{160}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{103}{160}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{107}{160}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{11}{160}\right)\) |
| \(\chi_{4224}(347,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{141}{160}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{99}{160}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{27}{160}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{143}{160}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{79}{160}\right)\) |
| \(\chi_{4224}(371,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{160}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{53}{160}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{29}{160}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{41}{160}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{73}{160}\right)\) |
| \(\chi_{4224}(491,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{160}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{127}{160}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{151}{160}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{59}{160}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{27}{160}\right)\) |
| \(\chi_{4224}(563,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{160}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{101}{160}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{13}{160}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{57}{160}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{121}{160}\right)\) |
| \(\chi_{4224}(611,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{160}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{89}{160}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{97}{160}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{93}{160}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{29}{160}\right)\) |
| \(\chi_{4224}(635,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{117}{160}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{123}{160}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{19}{160}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{71}{160}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{103}{160}\right)\) |
| \(\chi_{4224}(755,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{123}{160}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{117}{160}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{61}{160}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{9}{160}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{137}{160}\right)\) |
| \(\chi_{4224}(827,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{160}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{11}{160}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{3}{160}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{87}{160}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{151}{160}\right)\) |
| \(\chi_{4224}(875,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{81}{160}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{159}{160}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{87}{160}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{123}{160}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{59}{160}\right)\) |
| \(\chi_{4224}(899,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{160}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{113}{160}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{89}{160}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{21}{160}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{53}{160}\right)\) |
| \(\chi_{4224}(1019,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{160}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{27}{160}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{51}{160}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{39}{160}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{160}\right)\) |
| \(\chi_{4224}(1091,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{160}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{1}{160}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{73}{160}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{37}{160}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{101}{160}\right)\) |
| \(\chi_{4224}(1139,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{160}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{149}{160}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{157}{160}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{73}{160}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{9}{160}\right)\) |
| \(\chi_{4224}(1163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{160}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{23}{160}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{79}{160}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{51}{160}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{83}{160}\right)\) |
| \(\chi_{4224}(1283,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{160}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{17}{160}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{121}{160}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{149}{160}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{117}{160}\right)\) |
| \(\chi_{4224}(1355,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{160}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{71}{160}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{63}{160}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{67}{160}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{131}{160}\right)\) |
| \(\chi_{4224}(1403,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{160}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{59}{160}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{147}{160}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{103}{160}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{39}{160}\right)\) |
| \(\chi_{4224}(1427,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{160}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{13}{160}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{149}{160}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{1}{160}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{33}{160}\right)\) |
| \(\chi_{4224}(1547,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{153}{160}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{87}{160}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{111}{160}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{19}{160}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{147}{160}\right)\) |
| \(\chi_{4224}(1619,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{160}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{61}{160}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{133}{160}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{17}{160}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{81}{160}\right)\) |
| \(\chi_{4224}(1667,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{160}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{49}{160}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{57}{160}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{53}{160}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{149}{160}\right)\) |
| \(\chi_{4224}(1691,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{160}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{83}{160}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{139}{160}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{31}{160}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{63}{160}\right)\) |
| \(\chi_{4224}(1811,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{160}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{77}{160}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{21}{160}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{129}{160}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{97}{160}\right)\) |
| \(\chi_{4224}(1883,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{160}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{131}{160}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{123}{160}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{47}{160}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{111}{160}\right)\) |
| \(\chi_{4224}(1931,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{160}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{119}{160}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{47}{160}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{83}{160}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{19}{160}\right)\) |
| \(\chi_{4224}(1955,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{160}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{73}{160}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{49}{160}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{141}{160}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{13}{160}\right)\) |