from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5610, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([40,40,32,45]))
chi.galois_orbit()
[g,chi] = znchar(Mod(269,5610))
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.fj | 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 | \(-1\) | \(1\) | \(7\) | \(13\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{5610}(269,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{9}{20}\right)\) |
| \(\chi_{5610}(449,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{11}{20}\right)\) |
| \(\chi_{5610}(719,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{17}{20}\right)\) |
| \(\chi_{5610}(779,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{20}\right)\) |
| \(\chi_{5610}(839,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{20}\right)\) |
| \(\chi_{5610}(929,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{19}{20}\right)\) |
| \(\chi_{5610}(1049,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{17}{20}\right)\) |
| \(\chi_{5610}(1439,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{20}\right)\) |
| \(\chi_{5610}(1499,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{20}\right)\) |
| \(\chi_{5610}(1829,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{20}\right)\) |
| \(\chi_{5610}(2249,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{20}\right)\) |
| \(\chi_{5610}(2369,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{20}\right)\) |
| \(\chi_{5610}(2489,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{20}\right)\) |
| \(\chi_{5610}(2579,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{20}\right)\) |
| \(\chi_{5610}(2759,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{20}\right)\) |
| \(\chi_{5610}(2819,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{20}\right)\) |
| \(\chi_{5610}(3029,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{17}{20}\right)\) |
| \(\chi_{5610}(3089,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{20}\right)\) |
| \(\chi_{5610}(3359,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{20}\right)\) |
| \(\chi_{5610}(3479,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{20}\right)\) |
| \(\chi_{5610}(3899,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{19}{20}\right)\) |
| \(\chi_{5610}(4019,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{20}\right)\) |
| \(\chi_{5610}(4349,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{17}{20}\right)\) |
| \(\chi_{5610}(4409,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{20}\right)\) |
| \(\chi_{5610}(4559,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{9}{20}\right)\) |
| \(\chi_{5610}(4799,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{13}{20}\right)\) |
| \(\chi_{5610}(4889,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{19}{20}\right)\) |
| \(\chi_{5610}(5009,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{20}\right)\) |
| \(\chi_{5610}(5069,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{20}\right)\) |
| \(\chi_{5610}(5129,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{13}{20}\right)\) |
| \(\chi_{5610}(5399,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{11}{20}\right)\) |