from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5610, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([40,60,64,35]))
chi.galois_orbit()
[g,chi] = znchar(Mod(113,5610))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(5610\) | |
| Conductor: | \(2805\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(80\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 2805.ez | 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: | $\Q(\zeta_{80})$ |
| Fixed field: | Number field defined by a degree 80 polynomial |
First 31 of 32 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(7\) | \(13\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{5610}(113,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{2}{5}\right)\) |
| \(\chi_{5610}(377,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{2}{5}\right)\) |
| \(\chi_{5610}(533,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{5}\right)\) |
| \(\chi_{5610}(653,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{5}\right)\) |
| \(\chi_{5610}(1043,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{4}{5}\right)\) |
| \(\chi_{5610}(1127,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{5}\right)\) |
| \(\chi_{5610}(1457,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{5}\right)\) |
| \(\chi_{5610}(1523,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{5}\right)\) |
| \(\chi_{5610}(1637,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{4}{5}\right)\) |
| \(\chi_{5610}(1643,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{5}\right)\) |
| \(\chi_{5610}(1697,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{2}{5}\right)\) |
| \(\chi_{5610}(1907,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{5}\right)\) |
| \(\chi_{5610}(1967,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{4}{5}\right)\) |
| \(\chi_{5610}(2033,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{4}{5}\right)\) |
| \(\chi_{5610}(2183,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{5}\right)\) |
| \(\chi_{5610}(2693,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{4}{5}\right)\) |
| \(\chi_{5610}(3083,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{2}{5}\right)\) |
| \(\chi_{5610}(3173,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{5}\right)\) |
| \(\chi_{5610}(3227,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{5}\right)\) |
| \(\chi_{5610}(3437,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{5}\right)\) |
| \(\chi_{5610}(3677,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{2}{5}\right)\) |
| \(\chi_{5610}(3683,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{4}{5}\right)\) |
| \(\chi_{5610}(3947,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{4}{5}\right)\) |
| \(\chi_{5610}(4007,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{2}{5}\right)\) |
| \(\chi_{5610}(4073,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{2}{5}\right)\) |
| \(\chi_{5610}(4613,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{5}\right)\) |
| \(\chi_{5610}(4733,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{2}{5}\right)\) |
| \(\chi_{5610}(4757,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{5}\right)\) |
| \(\chi_{5610}(5207,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{5}\right)\) |
| \(\chi_{5610}(5267,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{4}{5}\right)\) |
| \(\chi_{5610}(5537,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{5}\right)\) |