from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4001, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([57]))
chi.galois_orbit()
[g,chi] = znchar(Mod(121,4001))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4001\) | |
| Conductor: | \(4001\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(80\) | 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: | $\Q(\zeta_{80})$ |
| Fixed field: | Number field defined by a degree 80 polynomial |
First 31 of 32 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4001}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(-i\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{5}{16}\right)\) |
| \(\chi_{4001}(439,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(-i\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{13}{16}\right)\) |
| \(\chi_{4001}(468,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(i\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{16}\right)\) |
| \(\chi_{4001}(496,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(i\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{16}\right)\) |
| \(\chi_{4001}(627,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(i\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{15}{16}\right)\) |
| \(\chi_{4001}(635,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(i\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{15}{16}\right)\) |
| \(\chi_{4001}(720,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(i\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{16}\right)\) |
| \(\chi_{4001}(752,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(-i\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{16}\right)\) |
| \(\chi_{4001}(882,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(i\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{16}\right)\) |
| \(\chi_{4001}(1278,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(i\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{16}\right)\) |
| \(\chi_{4001}(1289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(-i\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{16}\right)\) |
| \(\chi_{4001}(1438,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(-i\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{16}\right)\) |
| \(\chi_{4001}(1479,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(-i\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{5}{16}\right)\) |
| \(\chi_{4001}(1613,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(-i\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{9}{16}\right)\) |
| \(\chi_{4001}(1725,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(-i\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{5}{16}\right)\) |
| \(\chi_{4001}(1793,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(i\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{16}\right)\) |
| \(\chi_{4001}(2208,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(i\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{15}{16}\right)\) |
| \(\chi_{4001}(2276,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(-i\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{13}{16}\right)\) |
| \(\chi_{4001}(2388,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(-i\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{16}\right)\) |
| \(\chi_{4001}(2522,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(-i\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{13}{16}\right)\) |
| \(\chi_{4001}(2563,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(-i\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{16}\right)\) |
| \(\chi_{4001}(2712,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(-i\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{9}{16}\right)\) |
| \(\chi_{4001}(2723,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(i\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{11}{16}\right)\) |
| \(\chi_{4001}(3119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(i\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{15}{16}\right)\) |
| \(\chi_{4001}(3249,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(-i\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{16}\right)\) |
| \(\chi_{4001}(3281,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(i\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{11}{16}\right)\) |
| \(\chi_{4001}(3366,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(i\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{16}\right)\) |
| \(\chi_{4001}(3374,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(i\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{16}\right)\) |
| \(\chi_{4001}(3505,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(i\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{16}\right)\) |
| \(\chi_{4001}(3533,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(i\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{16}\right)\) |
| \(\chi_{4001}(3562,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(-i\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{5}{16}\right)\) |