from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3307, base_ring=CyclotomicField(3306))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,3307))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3307\) | |
| Conductor: | \(3307\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(3306\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | $\Q(\zeta_{1653})$ |
| Fixed field: | Number field defined by a degree 3306 polynomial (not computed) |
First 31 of 1008 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3307}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{3306}\right)\) | \(e\left(\frac{1129}{3306}\right)\) | \(e\left(\frac{1}{1653}\right)\) | \(e\left(\frac{2653}{3306}\right)\) | \(e\left(\frac{565}{1653}\right)\) | \(e\left(\frac{143}{1653}\right)\) | \(e\left(\frac{1}{1102}\right)\) | \(e\left(\frac{1129}{1653}\right)\) | \(e\left(\frac{1327}{1653}\right)\) | \(e\left(\frac{1057}{1653}\right)\) |
| \(\chi_{3307}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1129}{3306}\right)\) | \(e\left(\frac{1831}{3306}\right)\) | \(e\left(\frac{1129}{1653}\right)\) | \(e\left(\frac{1}{3306}\right)\) | \(e\left(\frac{1480}{1653}\right)\) | \(e\left(\frac{1106}{1653}\right)\) | \(e\left(\frac{27}{1102}\right)\) | \(e\left(\frac{178}{1653}\right)\) | \(e\left(\frac{565}{1653}\right)\) | \(e\left(\frac{1540}{1653}\right)\) |
| \(\chi_{3307}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2653}{3306}\right)\) | \(e\left(\frac{1}{3306}\right)\) | \(e\left(\frac{1000}{1653}\right)\) | \(e\left(\frac{3241}{3306}\right)\) | \(e\left(\frac{1327}{1653}\right)\) | \(e\left(\frac{842}{1653}\right)\) | \(e\left(\frac{449}{1102}\right)\) | \(e\left(\frac{1}{1653}\right)\) | \(e\left(\frac{1294}{1653}\right)\) | \(e\left(\frac{733}{1653}\right)\) |
| \(\chi_{3307}(14,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{287}{3306}\right)\) | \(e\left(\frac{35}{3306}\right)\) | \(e\left(\frac{287}{1653}\right)\) | \(e\left(\frac{1031}{3306}\right)\) | \(e\left(\frac{161}{1653}\right)\) | \(e\left(\frac{1369}{1653}\right)\) | \(e\left(\frac{287}{1102}\right)\) | \(e\left(\frac{35}{1653}\right)\) | \(e\left(\frac{659}{1653}\right)\) | \(e\left(\frac{860}{1653}\right)\) |
| \(\chi_{3307}(21,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1415}{3306}\right)\) | \(e\left(\frac{737}{3306}\right)\) | \(e\left(\frac{1415}{1653}\right)\) | \(e\left(\frac{1685}{3306}\right)\) | \(e\left(\frac{1076}{1653}\right)\) | \(e\left(\frac{679}{1653}\right)\) | \(e\left(\frac{313}{1102}\right)\) | \(e\left(\frac{737}{1653}\right)\) | \(e\left(\frac{1550}{1653}\right)\) | \(e\left(\frac{1343}{1653}\right)\) |
| \(\chi_{3307}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{391}{3306}\right)\) | \(e\left(\frac{1741}{3306}\right)\) | \(e\left(\frac{391}{1653}\right)\) | \(e\left(\frac{2545}{3306}\right)\) | \(e\left(\frac{1066}{1653}\right)\) | \(e\left(\frac{1364}{1653}\right)\) | \(e\left(\frac{391}{1102}\right)\) | \(e\left(\frac{88}{1653}\right)\) | \(e\left(\frac{1468}{1653}\right)\) | \(e\left(\frac{37}{1653}\right)\) |
| \(\chi_{3307}(32,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{3306}\right)\) | \(e\left(\frac{2339}{3306}\right)\) | \(e\left(\frac{5}{1653}\right)\) | \(e\left(\frac{41}{3306}\right)\) | \(e\left(\frac{1172}{1653}\right)\) | \(e\left(\frac{715}{1653}\right)\) | \(e\left(\frac{5}{1102}\right)\) | \(e\left(\frac{686}{1653}\right)\) | \(e\left(\frac{23}{1653}\right)\) | \(e\left(\frac{326}{1653}\right)\) |
| \(\chi_{3307}(34,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1427}{3306}\right)\) | \(e\left(\frac{1061}{3306}\right)\) | \(e\left(\frac{1427}{1653}\right)\) | \(e\left(\frac{461}{3306}\right)\) | \(e\left(\frac{1244}{1653}\right)\) | \(e\left(\frac{742}{1653}\right)\) | \(e\left(\frac{325}{1102}\right)\) | \(e\left(\frac{1061}{1653}\right)\) | \(e\left(\frac{944}{1653}\right)\) | \(e\left(\frac{803}{1653}\right)\) |
| \(\chi_{3307}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2939}{3306}\right)\) | \(e\left(\frac{2213}{3306}\right)\) | \(e\left(\frac{1286}{1653}\right)\) | \(e\left(\frac{1619}{3306}\right)\) | \(e\left(\frac{923}{1653}\right)\) | \(e\left(\frac{415}{1653}\right)\) | \(e\left(\frac{735}{1102}\right)\) | \(e\left(\frac{560}{1653}\right)\) | \(e\left(\frac{626}{1653}\right)\) | \(e\left(\frac{536}{1653}\right)\) |
| \(\chi_{3307}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2383}{3306}\right)\) | \(e\left(\frac{2629}{3306}\right)\) | \(e\left(\frac{730}{1653}\right)\) | \(e\left(\frac{1027}{3306}\right)\) | \(e\left(\frac{853}{1653}\right)\) | \(e\left(\frac{251}{1653}\right)\) | \(e\left(\frac{179}{1102}\right)\) | \(e\left(\frac{976}{1653}\right)\) | \(e\left(\frac{52}{1653}\right)\) | \(e\left(\frac{1312}{1653}\right)\) |
| \(\chi_{3307}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{3306}\right)\) | \(e\left(\frac{1549}{3306}\right)\) | \(e\left(\frac{139}{1653}\right)\) | \(e\left(\frac{1801}{3306}\right)\) | \(e\left(\frac{844}{1653}\right)\) | \(e\left(\frac{41}{1653}\right)\) | \(e\left(\frac{139}{1102}\right)\) | \(e\left(\frac{1549}{1653}\right)\) | \(e\left(\frac{970}{1653}\right)\) | \(e\left(\frac{1459}{1653}\right)\) |
| \(\chi_{3307}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1133}{3306}\right)\) | \(e\left(\frac{3041}{3306}\right)\) | \(e\left(\frac{1133}{1653}\right)\) | \(e\left(\frac{695}{3306}\right)\) | \(e\left(\frac{434}{1653}\right)\) | \(e\left(\frac{25}{1653}\right)\) | \(e\left(\frac{31}{1102}\right)\) | \(e\left(\frac{1388}{1653}\right)\) | \(e\left(\frac{914}{1653}\right)\) | \(e\left(\frac{809}{1653}\right)\) |
| \(\chi_{3307}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2555}{3306}\right)\) | \(e\left(\frac{1763}{3306}\right)\) | \(e\left(\frac{902}{1653}\right)\) | \(e\left(\frac{1115}{3306}\right)\) | \(e\left(\frac{506}{1653}\right)\) | \(e\left(\frac{52}{1653}\right)\) | \(e\left(\frac{351}{1102}\right)\) | \(e\left(\frac{110}{1653}\right)\) | \(e\left(\frac{182}{1653}\right)\) | \(e\left(\frac{1286}{1653}\right)\) |
| \(\chi_{3307}(52,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1409}{3306}\right)\) | \(e\left(\frac{575}{3306}\right)\) | \(e\left(\frac{1409}{1653}\right)\) | \(e\left(\frac{2297}{3306}\right)\) | \(e\left(\frac{992}{1653}\right)\) | \(e\left(\frac{1474}{1653}\right)\) | \(e\left(\frac{307}{1102}\right)\) | \(e\left(\frac{575}{1653}\right)\) | \(e\left(\frac{200}{1653}\right)\) | \(e\left(\frac{1613}{1653}\right)\) |
| \(\chi_{3307}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{767}{3306}\right)\) | \(e\left(\frac{3077}{3306}\right)\) | \(e\left(\frac{767}{1653}\right)\) | \(e\left(\frac{1661}{3306}\right)\) | \(e\left(\frac{269}{1653}\right)\) | \(e\left(\frac{583}{1653}\right)\) | \(e\left(\frac{767}{1102}\right)\) | \(e\left(\frac{1424}{1653}\right)\) | \(e\left(\frac{1214}{1653}\right)\) | \(e\left(\frac{749}{1653}\right)\) |
| \(\chi_{3307}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{289}{3306}\right)\) | \(e\left(\frac{2293}{3306}\right)\) | \(e\left(\frac{289}{1653}\right)\) | \(e\left(\frac{3031}{3306}\right)\) | \(e\left(\frac{1291}{1653}\right)\) | \(e\left(\frac{2}{1653}\right)\) | \(e\left(\frac{289}{1102}\right)\) | \(e\left(\frac{640}{1653}\right)\) | \(e\left(\frac{7}{1653}\right)\) | \(e\left(\frac{1321}{1653}\right)\) |
| \(\chi_{3307}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2213}{3306}\right)\) | \(e\left(\frac{2447}{3306}\right)\) | \(e\left(\frac{560}{1653}\right)\) | \(e\left(\frac{2939}{3306}\right)\) | \(e\left(\frac{677}{1653}\right)\) | \(e\left(\frac{736}{1653}\right)\) | \(e\left(\frac{9}{1102}\right)\) | \(e\left(\frac{794}{1653}\right)\) | \(e\left(\frac{923}{1653}\right)\) | \(e\left(\frac{146}{1653}\right)\) |
| \(\chi_{3307}(76,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3281}{3306}\right)\) | \(e\left(\frac{1529}{3306}\right)\) | \(e\left(\frac{1628}{1653}\right)\) | \(e\left(\frac{3101}{3306}\right)\) | \(e\left(\frac{752}{1653}\right)\) | \(e\left(\frac{1384}{1653}\right)\) | \(e\left(\frac{1077}{1102}\right)\) | \(e\left(\frac{1529}{1653}\right)\) | \(e\left(\frac{1538}{1653}\right)\) | \(e\left(\frac{23}{1653}\right)\) |
| \(\chi_{3307}(78,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2537}{3306}\right)\) | \(e\left(\frac{1277}{3306}\right)\) | \(e\left(\frac{884}{1653}\right)\) | \(e\left(\frac{2951}{3306}\right)\) | \(e\left(\frac{254}{1653}\right)\) | \(e\left(\frac{784}{1653}\right)\) | \(e\left(\frac{333}{1102}\right)\) | \(e\left(\frac{1277}{1653}\right)\) | \(e\left(\frac{1091}{1653}\right)\) | \(e\left(\frac{443}{1653}\right)\) |
| \(\chi_{3307}(80,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2657}{3306}\right)\) | \(e\left(\frac{1211}{3306}\right)\) | \(e\left(\frac{1004}{1653}\right)\) | \(e\left(\frac{629}{3306}\right)\) | \(e\left(\frac{281}{1653}\right)\) | \(e\left(\frac{1414}{1653}\right)\) | \(e\left(\frac{453}{1102}\right)\) | \(e\left(\frac{1211}{1653}\right)\) | \(e\left(\frac{1643}{1653}\right)\) | \(e\left(\frac{2}{1653}\right)\) |
| \(\chi_{3307}(83,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2057}{3306}\right)\) | \(e\left(\frac{1541}{3306}\right)\) | \(e\left(\frac{404}{1653}\right)\) | \(e\left(\frac{2321}{3306}\right)\) | \(e\left(\frac{146}{1653}\right)\) | \(e\left(\frac{1570}{1653}\right)\) | \(e\left(\frac{955}{1102}\right)\) | \(e\left(\frac{1541}{1653}\right)\) | \(e\left(\frac{536}{1653}\right)\) | \(e\left(\frac{554}{1653}\right)\) |
| \(\chi_{3307}(84,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1417}{3306}\right)\) | \(e\left(\frac{2995}{3306}\right)\) | \(e\left(\frac{1417}{1653}\right)\) | \(e\left(\frac{379}{3306}\right)\) | \(e\left(\frac{553}{1653}\right)\) | \(e\left(\frac{965}{1653}\right)\) | \(e\left(\frac{315}{1102}\right)\) | \(e\left(\frac{1342}{1653}\right)\) | \(e\left(\frac{898}{1653}\right)\) | \(e\left(\frac{151}{1653}\right)\) |
| \(\chi_{3307}(85,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{773}{3306}\right)\) | \(e\left(\frac{3239}{3306}\right)\) | \(e\left(\frac{773}{1653}\right)\) | \(e\left(\frac{1049}{3306}\right)\) | \(e\left(\frac{353}{1653}\right)\) | \(e\left(\frac{1441}{1653}\right)\) | \(e\left(\frac{773}{1102}\right)\) | \(e\left(\frac{1586}{1653}\right)\) | \(e\left(\frac{911}{1653}\right)\) | \(e\left(\frac{479}{1653}\right)\) |
| \(\chi_{3307}(87,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1679}{3306}\right)\) | \(e\left(\frac{1253}{3306}\right)\) | \(e\left(\frac{26}{1653}\right)\) | \(e\left(\frac{1205}{3306}\right)\) | \(e\left(\frac{1466}{1653}\right)\) | \(e\left(\frac{412}{1653}\right)\) | \(e\left(\frac{577}{1102}\right)\) | \(e\left(\frac{1253}{1653}\right)\) | \(e\left(\frac{1442}{1653}\right)\) | \(e\left(\frac{1034}{1653}\right)\) |
| \(\chi_{3307}(89,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3155}{3306}\right)\) | \(e\left(\frac{1433}{3306}\right)\) | \(e\left(\frac{1502}{1653}\right)\) | \(e\left(\frac{2729}{3306}\right)\) | \(e\left(\frac{641}{1653}\right)\) | \(e\left(\frac{1549}{1653}\right)\) | \(e\left(\frac{951}{1102}\right)\) | \(e\left(\frac{1433}{1653}\right)\) | \(e\left(\frac{1289}{1653}\right)\) | \(e\left(\frac{734}{1653}\right)\) |
| \(\chi_{3307}(91,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1693}{3306}\right)\) | \(e\left(\frac{529}{3306}\right)\) | \(e\left(\frac{40}{1653}\right)\) | \(e\left(\frac{1981}{3306}\right)\) | \(e\left(\frac{1111}{1653}\right)\) | \(e\left(\frac{761}{1653}\right)\) | \(e\left(\frac{591}{1102}\right)\) | \(e\left(\frac{529}{1653}\right)\) | \(e\left(\frac{184}{1653}\right)\) | \(e\left(\frac{955}{1653}\right)\) |
| \(\chi_{3307}(101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1471}{3306}\right)\) | \(e\left(\frac{1147}{3306}\right)\) | \(e\left(\frac{1471}{1653}\right)\) | \(e\left(\frac{1483}{3306}\right)\) | \(e\left(\frac{1309}{1653}\right)\) | \(e\left(\frac{422}{1653}\right)\) | \(e\left(\frac{369}{1102}\right)\) | \(e\left(\frac{1147}{1653}\right)\) | \(e\left(\frac{1477}{1653}\right)\) | \(e\left(\frac{1027}{1653}\right)\) |
| \(\chi_{3307}(108,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{3306}\right)\) | \(e\left(\frac{1139}{3306}\right)\) | \(e\left(\frac{83}{1653}\right)\) | \(e\left(\frac{2003}{3306}\right)\) | \(e\left(\frac{611}{1653}\right)\) | \(e\left(\frac{298}{1653}\right)\) | \(e\left(\frac{83}{1102}\right)\) | \(e\left(\frac{1139}{1653}\right)\) | \(e\left(\frac{1043}{1653}\right)\) | \(e\left(\frac{122}{1653}\right)\) |
| \(\chi_{3307}(114,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1103}{3306}\right)\) | \(e\left(\frac{2231}{3306}\right)\) | \(e\left(\frac{1103}{1653}\right)\) | \(e\left(\frac{449}{3306}\right)\) | \(e\left(\frac{14}{1653}\right)\) | \(e\left(\frac{694}{1653}\right)\) | \(e\left(\frac{1}{1102}\right)\) | \(e\left(\frac{578}{1653}\right)\) | \(e\left(\frac{776}{1653}\right)\) | \(e\left(\frac{506}{1653}\right)\) |
| \(\chi_{3307}(117,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{359}{3306}\right)\) | \(e\left(\frac{1979}{3306}\right)\) | \(e\left(\frac{359}{1653}\right)\) | \(e\left(\frac{299}{3306}\right)\) | \(e\left(\frac{1169}{1653}\right)\) | \(e\left(\frac{94}{1653}\right)\) | \(e\left(\frac{359}{1102}\right)\) | \(e\left(\frac{326}{1653}\right)\) | \(e\left(\frac{329}{1653}\right)\) | \(e\left(\frac{926}{1653}\right)\) |
| \(\chi_{3307}(120,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{479}{3306}\right)\) | \(e\left(\frac{1913}{3306}\right)\) | \(e\left(\frac{479}{1653}\right)\) | \(e\left(\frac{1283}{3306}\right)\) | \(e\left(\frac{1196}{1653}\right)\) | \(e\left(\frac{724}{1653}\right)\) | \(e\left(\frac{479}{1102}\right)\) | \(e\left(\frac{260}{1653}\right)\) | \(e\left(\frac{881}{1653}\right)\) | \(e\left(\frac{485}{1653}\right)\) |