from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7248, base_ring=CyclotomicField(300))
M = H._module
chi = DirichletCharacter(H, M([0,75,0,32]))
chi.galois_orbit()
[g,chi] = znchar(Mod(37,7248))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(7248\) | |
| Conductor: | \(2416\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(300\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 2416.cq | 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: | $\Q(\zeta_{300})$ |
| Fixed field: | Number field defined by a degree 300 polynomial (not computed) |
First 31 of 80 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{7248}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{300}\right)\) | \(e\left(\frac{97}{150}\right)\) | \(e\left(\frac{83}{300}\right)\) | \(e\left(\frac{37}{300}\right)\) | \(e\left(\frac{14}{75}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{59}{150}\right)\) | \(e\left(\frac{91}{100}\right)\) | \(e\left(\frac{74}{75}\right)\) |
| \(\chi_{7248}(349,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{300}\right)\) | \(e\left(\frac{17}{150}\right)\) | \(e\left(\frac{13}{300}\right)\) | \(e\left(\frac{107}{300}\right)\) | \(e\left(\frac{4}{75}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{49}{150}\right)\) | \(e\left(\frac{1}{100}\right)\) | \(e\left(\frac{64}{75}\right)\) |
| \(\chi_{7248}(397,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{173}{300}\right)\) | \(e\left(\frac{109}{150}\right)\) | \(e\left(\frac{101}{300}\right)\) | \(e\left(\frac{139}{300}\right)\) | \(e\left(\frac{8}{75}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{23}{150}\right)\) | \(e\left(\frac{77}{100}\right)\) | \(e\left(\frac{53}{75}\right)\) |
| \(\chi_{7248}(493,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{300}\right)\) | \(e\left(\frac{49}{150}\right)\) | \(e\left(\frac{161}{300}\right)\) | \(e\left(\frac{79}{300}\right)\) | \(e\left(\frac{38}{75}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{53}{150}\right)\) | \(e\left(\frac{97}{100}\right)\) | \(e\left(\frac{8}{75}\right)\) |
| \(\chi_{7248}(541,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{221}{300}\right)\) | \(e\left(\frac{43}{150}\right)\) | \(e\left(\frac{77}{300}\right)\) | \(e\left(\frac{103}{300}\right)\) | \(e\left(\frac{41}{75}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{71}{150}\right)\) | \(e\left(\frac{29}{100}\right)\) | \(e\left(\frac{56}{75}\right)\) |
| \(\chi_{7248}(589,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{257}{300}\right)\) | \(e\left(\frac{31}{150}\right)\) | \(e\left(\frac{209}{300}\right)\) | \(e\left(\frac{151}{300}\right)\) | \(e\left(\frac{47}{75}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{107}{150}\right)\) | \(e\left(\frac{93}{100}\right)\) | \(e\left(\frac{2}{75}\right)\) |
| \(\chi_{7248}(829,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{289}{300}\right)\) | \(e\left(\frac{137}{150}\right)\) | \(e\left(\frac{193}{300}\right)\) | \(e\left(\frac{227}{300}\right)\) | \(e\left(\frac{19}{75}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{139}{150}\right)\) | \(e\left(\frac{61}{100}\right)\) | \(e\left(\frac{4}{75}\right)\) |
| \(\chi_{7248}(949,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{300}\right)\) | \(e\left(\frac{11}{150}\right)\) | \(e\left(\frac{79}{300}\right)\) | \(e\left(\frac{281}{300}\right)\) | \(e\left(\frac{7}{75}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{67}{150}\right)\) | \(e\left(\frac{83}{100}\right)\) | \(e\left(\frac{37}{75}\right)\) |
| \(\chi_{7248}(1045,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{300}\right)\) | \(e\left(\frac{107}{150}\right)\) | \(e\left(\frac{223}{300}\right)\) | \(e\left(\frac{197}{300}\right)\) | \(e\left(\frac{34}{75}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{79}{150}\right)\) | \(e\left(\frac{71}{100}\right)\) | \(e\left(\frac{19}{75}\right)\) |
| \(\chi_{7248}(1093,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{223}{300}\right)\) | \(e\left(\frac{59}{150}\right)\) | \(e\left(\frac{151}{300}\right)\) | \(e\left(\frac{89}{300}\right)\) | \(e\left(\frac{58}{75}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{73}{150}\right)\) | \(e\left(\frac{27}{100}\right)\) | \(e\left(\frac{28}{75}\right)\) |
| \(\chi_{7248}(1213,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{300}\right)\) | \(e\left(\frac{79}{150}\right)\) | \(e\left(\frac{281}{300}\right)\) | \(e\left(\frac{259}{300}\right)\) | \(e\left(\frac{23}{75}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{113}{150}\right)\) | \(e\left(\frac{37}{100}\right)\) | \(e\left(\frac{68}{75}\right)\) |
| \(\chi_{7248}(1381,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{211}{300}\right)\) | \(e\left(\frac{113}{150}\right)\) | \(e\left(\frac{7}{300}\right)\) | \(e\left(\frac{173}{300}\right)\) | \(e\left(\frac{31}{75}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{61}{150}\right)\) | \(e\left(\frac{39}{100}\right)\) | \(e\left(\frac{46}{75}\right)\) |
| \(\chi_{7248}(1549,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{253}{300}\right)\) | \(e\left(\frac{149}{150}\right)\) | \(e\left(\frac{61}{300}\right)\) | \(e\left(\frac{179}{300}\right)\) | \(e\left(\frac{13}{75}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{103}{150}\right)\) | \(e\left(\frac{97}{100}\right)\) | \(e\left(\frac{58}{75}\right)\) |
| \(\chi_{7248}(1741,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{133}{300}\right)\) | \(e\left(\frac{89}{150}\right)\) | \(e\left(\frac{121}{300}\right)\) | \(e\left(\frac{119}{300}\right)\) | \(e\left(\frac{43}{75}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{133}{150}\right)\) | \(e\left(\frac{17}{100}\right)\) | \(e\left(\frac{13}{75}\right)\) |
| \(\chi_{7248}(1837,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{300}\right)\) | \(e\left(\frac{83}{150}\right)\) | \(e\left(\frac{37}{300}\right)\) | \(e\left(\frac{143}{300}\right)\) | \(e\left(\frac{46}{75}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{150}\right)\) | \(e\left(\frac{49}{100}\right)\) | \(e\left(\frac{61}{75}\right)\) |
| \(\chi_{7248}(1861,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{91}{300}\right)\) | \(e\left(\frac{53}{150}\right)\) | \(e\left(\frac{67}{300}\right)\) | \(e\left(\frac{113}{300}\right)\) | \(e\left(\frac{61}{75}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{91}{150}\right)\) | \(e\left(\frac{59}{100}\right)\) | \(e\left(\frac{1}{75}\right)\) |
| \(\chi_{7248}(1909,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{287}{300}\right)\) | \(e\left(\frac{121}{150}\right)\) | \(e\left(\frac{119}{300}\right)\) | \(e\left(\frac{241}{300}\right)\) | \(e\left(\frac{2}{75}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{137}{150}\right)\) | \(e\left(\frac{63}{100}\right)\) | \(e\left(\frac{32}{75}\right)\) |
| \(\chi_{7248}(1933,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{300}\right)\) | \(e\left(\frac{71}{150}\right)\) | \(e\left(\frac{169}{300}\right)\) | \(e\left(\frac{191}{300}\right)\) | \(e\left(\frac{52}{75}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{37}{150}\right)\) | \(e\left(\frac{13}{100}\right)\) | \(e\left(\frac{7}{75}\right)\) |
| \(\chi_{7248}(1957,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{299}{300}\right)\) | \(e\left(\frac{67}{150}\right)\) | \(e\left(\frac{263}{300}\right)\) | \(e\left(\frac{157}{300}\right)\) | \(e\left(\frac{29}{75}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{149}{150}\right)\) | \(e\left(\frac{51}{100}\right)\) | \(e\left(\frac{14}{75}\right)\) |
| \(\chi_{7248}(1981,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{293}{300}\right)\) | \(e\left(\frac{19}{150}\right)\) | \(e\left(\frac{41}{300}\right)\) | \(e\left(\frac{199}{300}\right)\) | \(e\left(\frac{53}{75}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{143}{150}\right)\) | \(e\left(\frac{57}{100}\right)\) | \(e\left(\frac{23}{75}\right)\) |
| \(\chi_{7248}(2005,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{300}\right)\) | \(e\left(\frac{131}{150}\right)\) | \(e\left(\frac{259}{300}\right)\) | \(e\left(\frac{101}{300}\right)\) | \(e\left(\frac{22}{75}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{7}{150}\right)\) | \(e\left(\frac{43}{100}\right)\) | \(e\left(\frac{52}{75}\right)\) |
| \(\chi_{7248}(2053,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{300}\right)\) | \(e\left(\frac{23}{150}\right)\) | \(e\left(\frac{247}{300}\right)\) | \(e\left(\frac{233}{300}\right)\) | \(e\left(\frac{1}{75}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{31}{150}\right)\) | \(e\left(\frac{19}{100}\right)\) | \(e\left(\frac{16}{75}\right)\) |
| \(\chi_{7248}(2101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{259}{300}\right)\) | \(e\left(\frac{47}{150}\right)\) | \(e\left(\frac{283}{300}\right)\) | \(e\left(\frac{137}{300}\right)\) | \(e\left(\frac{64}{75}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{109}{150}\right)\) | \(e\left(\frac{91}{100}\right)\) | \(e\left(\frac{49}{75}\right)\) |
| \(\chi_{7248}(2125,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{281}{300}\right)\) | \(e\left(\frac{73}{150}\right)\) | \(e\left(\frac{197}{300}\right)\) | \(e\left(\frac{283}{300}\right)\) | \(e\left(\frac{26}{75}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{131}{150}\right)\) | \(e\left(\frac{69}{100}\right)\) | \(e\left(\frac{41}{75}\right)\) |
| \(\chi_{7248}(2365,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{161}{300}\right)\) | \(e\left(\frac{13}{150}\right)\) | \(e\left(\frac{257}{300}\right)\) | \(e\left(\frac{223}{300}\right)\) | \(e\left(\frac{56}{75}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{11}{150}\right)\) | \(e\left(\frac{89}{100}\right)\) | \(e\left(\frac{71}{75}\right)\) |
| \(\chi_{7248}(2437,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{227}{300}\right)\) | \(e\left(\frac{91}{150}\right)\) | \(e\left(\frac{299}{300}\right)\) | \(e\left(\frac{61}{300}\right)\) | \(e\left(\frac{17}{75}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{77}{150}\right)\) | \(e\left(\frac{23}{100}\right)\) | \(e\left(\frac{47}{75}\right)\) |
| \(\chi_{7248}(2461,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{300}\right)\) | \(e\left(\frac{133}{150}\right)\) | \(e\left(\frac{137}{300}\right)\) | \(e\left(\frac{43}{300}\right)\) | \(e\left(\frac{71}{75}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{101}{150}\right)\) | \(e\left(\frac{49}{100}\right)\) | \(e\left(\frac{11}{75}\right)\) |
| \(\chi_{7248}(2485,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{179}{300}\right)\) | \(e\left(\frac{7}{150}\right)\) | \(e\left(\frac{23}{300}\right)\) | \(e\left(\frac{97}{300}\right)\) | \(e\left(\frac{59}{75}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{29}{150}\right)\) | \(e\left(\frac{71}{100}\right)\) | \(e\left(\frac{44}{75}\right)\) |
| \(\chi_{7248}(2629,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{300}\right)\) | \(e\left(\frac{41}{150}\right)\) | \(e\left(\frac{199}{300}\right)\) | \(e\left(\frac{161}{300}\right)\) | \(e\left(\frac{67}{75}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{127}{150}\right)\) | \(e\left(\frac{23}{100}\right)\) | \(e\left(\frac{22}{75}\right)\) |
| \(\chi_{7248}(2749,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{197}{300}\right)\) | \(e\left(\frac{1}{150}\right)\) | \(e\left(\frac{89}{300}\right)\) | \(e\left(\frac{271}{300}\right)\) | \(e\left(\frac{62}{75}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{47}{150}\right)\) | \(e\left(\frac{53}{100}\right)\) | \(e\left(\frac{17}{75}\right)\) |
| \(\chi_{7248}(2773,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{300}\right)\) | \(e\left(\frac{77}{150}\right)\) | \(e\left(\frac{103}{300}\right)\) | \(e\left(\frac{17}{300}\right)\) | \(e\left(\frac{49}{75}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{19}{150}\right)\) | \(e\left(\frac{31}{100}\right)\) | \(e\left(\frac{34}{75}\right)\) |