from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1309, base_ring=CyclotomicField(240))
M = H._module
chi = DirichletCharacter(H, M([40,192,15]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,1309))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1309\) | |
| Conductor: | \(1309\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(240\) | 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: | $\Q(\zeta_{240})$ |
| Fixed field: | Number field defined by a degree 240 polynomial (not computed) |
First 31 of 64 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1309}(3,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{120}\right)\) | \(e\left(\frac{151}{240}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{83}{240}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{31}{120}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{11}{20}\right)\) |
| \(\chi_{1309}(5,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{120}\right)\) | \(e\left(\frac{83}{240}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{79}{240}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{83}{120}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{3}{20}\right)\) |
| \(\chi_{1309}(31,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{127}{240}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{11}{240}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{7}{20}\right)\) |
| \(\chi_{1309}(75,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{77}{240}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{1}{240}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{17}{20}\right)\) |
| \(\chi_{1309}(80,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{120}\right)\) | \(e\left(\frac{91}{240}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{23}{240}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{11}{20}\right)\) |
| \(\chi_{1309}(82,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{120}\right)\) | \(e\left(\frac{143}{240}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{139}{240}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{3}{20}\right)\) |
| \(\chi_{1309}(108,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{217}{240}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{221}{240}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{17}{20}\right)\) |
| \(\chi_{1309}(124,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{131}{240}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{223}{240}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{11}{120}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{11}{20}\right)\) |
| \(\chi_{1309}(159,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{233}{240}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{109}{240}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{113}{120}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{13}{20}\right)\) |
| \(\chi_{1309}(180,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{149}{240}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{217}{240}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{9}{20}\right)\) |
| \(\chi_{1309}(192,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{163}{240}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{239}{240}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{3}{20}\right)\) |
| \(\chi_{1309}(201,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{120}\right)\) | \(e\left(\frac{191}{240}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{43}{240}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{11}{20}\right)\) |
| \(\chi_{1309}(262,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{157}{240}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{161}{240}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{37}{120}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{17}{20}\right)\) |
| \(\chi_{1309}(269,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{223}{240}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{59}{240}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{3}{20}\right)\) |
| \(\chi_{1309}(278,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{41}{240}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{13}{240}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{41}{120}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{1}{20}\right)\) |
| \(\chi_{1309}(311,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{120}\right)\) | \(e\left(\frac{211}{240}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{143}{240}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{11}{20}\right)\) |
| \(\chi_{1309}(313,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{120}\right)\) | \(e\left(\frac{173}{240}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{49}{240}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{53}{120}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{13}{20}\right)\) |
| \(\chi_{1309}(334,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{209}{240}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{37}{240}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{89}{120}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{9}{20}\right)\) |
| \(\chi_{1309}(346,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{73}{240}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{29}{240}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{13}{20}\right)\) |
| \(\chi_{1309}(367,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{120}\right)\) | \(e\left(\frac{229}{240}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{137}{240}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{9}{20}\right)\) |
| \(\chi_{1309}(388,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{120}\right)\) | \(e\left(\frac{31}{240}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{203}{240}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{31}{120}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{11}{20}\right)\) |
| \(\chi_{1309}(411,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{120}\right)\) | \(e\left(\frac{119}{240}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{67}{240}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{19}{20}\right)\) |
| \(\chi_{1309}(432,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{120}\right)\) | \(e\left(\frac{221}{240}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{193}{240}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{1}{20}\right)\) |
| \(\chi_{1309}(465,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{120}\right)\) | \(e\left(\frac{121}{240}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{173}{240}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{1}{120}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{1}{20}\right)\) |
| \(\chi_{1309}(488,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{59}{240}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{7}{240}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{19}{20}\right)\) |
| \(\chi_{1309}(500,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{13}{240}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{209}{240}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{13}{20}\right)\) |
| \(\chi_{1309}(521,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{120}\right)\) | \(e\left(\frac{49}{240}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{197}{240}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{49}{120}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{9}{20}\right)\) |
| \(\chi_{1309}(537,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{197}{240}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{121}{240}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{17}{20}\right)\) |
| \(\chi_{1309}(598,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{120}\right)\) | \(e\left(\frac{199}{240}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{227}{240}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{79}{120}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{19}{20}\right)\) |
| \(\chi_{1309}(619,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{61}{240}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{113}{240}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{61}{120}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{1}{20}\right)\) |
| \(\chi_{1309}(675,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{139}{240}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{167}{240}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{19}{120}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{19}{20}\right)\) |