from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3721, base_ring=CyclotomicField(1830))
M = H._module
chi = DirichletCharacter(H, M([1444]))
chi.galois_orbit()
[g,chi] = znchar(Mod(12,3721))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3721\) | |
| Conductor: | \(3721\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(915\) | 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: | $\Q(\zeta_{915})$ |
| Fixed field: | Number field defined by a degree 915 polynomial (not computed) |
First 31 of 480 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3721}(12,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{722}{915}\right)\) | \(e\left(\frac{79}{305}\right)\) | \(e\left(\frac{529}{915}\right)\) | \(e\left(\frac{674}{915}\right)\) | \(e\left(\frac{44}{915}\right)\) | \(e\left(\frac{443}{915}\right)\) | \(e\left(\frac{112}{305}\right)\) | \(e\left(\frac{158}{305}\right)\) | \(e\left(\frac{481}{915}\right)\) | \(e\left(\frac{35}{61}\right)\) |
| \(\chi_{3721}(15,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{787}{915}\right)\) | \(e\left(\frac{84}{305}\right)\) | \(e\left(\frac{659}{915}\right)\) | \(e\left(\frac{319}{915}\right)\) | \(e\left(\frac{124}{915}\right)\) | \(e\left(\frac{583}{915}\right)\) | \(e\left(\frac{177}{305}\right)\) | \(e\left(\frac{168}{305}\right)\) | \(e\left(\frac{191}{915}\right)\) | \(e\left(\frac{21}{61}\right)\) |
| \(\chi_{3721}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{915}\right)\) | \(e\left(\frac{47}{305}\right)\) | \(e\left(\frac{2}{915}\right)\) | \(e\left(\frac{262}{915}\right)\) | \(e\left(\frac{142}{915}\right)\) | \(e\left(\frac{889}{915}\right)\) | \(e\left(\frac{1}{305}\right)\) | \(e\left(\frac{94}{305}\right)\) | \(e\left(\frac{263}{915}\right)\) | \(e\left(\frac{27}{61}\right)\) |
| \(\chi_{3721}(22,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{559}{915}\right)\) | \(e\left(\frac{43}{305}\right)\) | \(e\left(\frac{203}{915}\right)\) | \(e\left(\frac{58}{915}\right)\) | \(e\left(\frac{688}{915}\right)\) | \(e\left(\frac{106}{915}\right)\) | \(e\left(\frac{254}{305}\right)\) | \(e\left(\frac{86}{305}\right)\) | \(e\left(\frac{617}{915}\right)\) | \(e\left(\frac{26}{61}\right)\) |
| \(\chi_{3721}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{915}\right)\) | \(e\left(\frac{57}{305}\right)\) | \(e\left(\frac{262}{915}\right)\) | \(e\left(\frac{467}{915}\right)\) | \(e\left(\frac{302}{915}\right)\) | \(e\left(\frac{254}{915}\right)\) | \(e\left(\frac{131}{305}\right)\) | \(e\left(\frac{114}{305}\right)\) | \(e\left(\frac{598}{915}\right)\) | \(e\left(\frac{60}{61}\right)\) |
| \(\chi_{3721}(42,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{915}\right)\) | \(e\left(\frac{143}{305}\right)\) | \(e\left(\frac{58}{915}\right)\) | \(e\left(\frac{278}{915}\right)\) | \(e\left(\frac{458}{915}\right)\) | \(e\left(\frac{161}{915}\right)\) | \(e\left(\frac{29}{305}\right)\) | \(e\left(\frac{286}{305}\right)\) | \(e\left(\frac{307}{915}\right)\) | \(e\left(\frac{51}{61}\right)\) |
| \(\chi_{3721}(56,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{223}{915}\right)\) | \(e\left(\frac{111}{305}\right)\) | \(e\left(\frac{446}{915}\right)\) | \(e\left(\frac{781}{915}\right)\) | \(e\left(\frac{556}{915}\right)\) | \(e\left(\frac{607}{915}\right)\) | \(e\left(\frac{223}{305}\right)\) | \(e\left(\frac{222}{305}\right)\) | \(e\left(\frac{89}{915}\right)\) | \(e\left(\frac{43}{61}\right)\) |
| \(\chi_{3721}(57,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{915}\right)\) | \(e\left(\frac{241}{305}\right)\) | \(e\left(\frac{166}{915}\right)\) | \(e\left(\frac{701}{915}\right)\) | \(e\left(\frac{806}{915}\right)\) | \(e\left(\frac{587}{915}\right)\) | \(e\left(\frac{83}{305}\right)\) | \(e\left(\frac{177}{305}\right)\) | \(e\left(\frac{784}{915}\right)\) | \(e\left(\frac{45}{61}\right)\) |
| \(\chi_{3721}(73,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{542}{915}\right)\) | \(e\left(\frac{159}{305}\right)\) | \(e\left(\frac{169}{915}\right)\) | \(e\left(\frac{179}{915}\right)\) | \(e\left(\frac{104}{915}\right)\) | \(e\left(\frac{548}{915}\right)\) | \(e\left(\frac{237}{305}\right)\) | \(e\left(\frac{13}{305}\right)\) | \(e\left(\frac{721}{915}\right)\) | \(e\left(\frac{55}{61}\right)\) |
| \(\chi_{3721}(76,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{277}{915}\right)\) | \(e\left(\frac{209}{305}\right)\) | \(e\left(\frac{554}{915}\right)\) | \(e\left(\frac{289}{915}\right)\) | \(e\left(\frac{904}{915}\right)\) | \(e\left(\frac{118}{915}\right)\) | \(e\left(\frac{277}{305}\right)\) | \(e\left(\frac{113}{305}\right)\) | \(e\left(\frac{566}{915}\right)\) | \(e\left(\frac{37}{61}\right)\) |
| \(\chi_{3721}(77,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{781}{915}\right)\) | \(e\left(\frac{107}{305}\right)\) | \(e\left(\frac{647}{915}\right)\) | \(e\left(\frac{577}{915}\right)\) | \(e\left(\frac{187}{915}\right)\) | \(e\left(\frac{739}{915}\right)\) | \(e\left(\frac{171}{305}\right)\) | \(e\left(\frac{214}{305}\right)\) | \(e\left(\frac{443}{915}\right)\) | \(e\left(\frac{42}{61}\right)\) |
| \(\chi_{3721}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{544}{915}\right)\) | \(e\left(\frac{253}{305}\right)\) | \(e\left(\frac{173}{915}\right)\) | \(e\left(\frac{703}{915}\right)\) | \(e\left(\frac{388}{915}\right)\) | \(e\left(\frac{496}{915}\right)\) | \(e\left(\frac{239}{305}\right)\) | \(e\left(\frac{201}{305}\right)\) | \(e\left(\frac{332}{915}\right)\) | \(e\left(\frac{48}{61}\right)\) |
| \(\chi_{3721}(86,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{191}{915}\right)\) | \(e\left(\frac{132}{305}\right)\) | \(e\left(\frac{382}{915}\right)\) | \(e\left(\frac{632}{915}\right)\) | \(e\left(\frac{587}{915}\right)\) | \(e\left(\frac{524}{915}\right)\) | \(e\left(\frac{191}{305}\right)\) | \(e\left(\frac{264}{305}\right)\) | \(e\left(\frac{823}{915}\right)\) | \(e\left(\frac{33}{61}\right)\) |
| \(\chi_{3721}(103,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{239}{915}\right)\) | \(e\left(\frac{253}{305}\right)\) | \(e\left(\frac{478}{915}\right)\) | \(e\left(\frac{398}{915}\right)\) | \(e\left(\frac{83}{915}\right)\) | \(e\left(\frac{191}{915}\right)\) | \(e\left(\frac{239}{305}\right)\) | \(e\left(\frac{201}{305}\right)\) | \(e\left(\frac{637}{915}\right)\) | \(e\left(\frac{48}{61}\right)\) |
| \(\chi_{3721}(117,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{838}{915}\right)\) | \(e\left(\frac{41}{305}\right)\) | \(e\left(\frac{761}{915}\right)\) | \(e\left(\frac{871}{915}\right)\) | \(e\left(\frac{46}{915}\right)\) | \(e\left(\frac{172}{915}\right)\) | \(e\left(\frac{228}{305}\right)\) | \(e\left(\frac{82}{305}\right)\) | \(e\left(\frac{794}{915}\right)\) | \(e\left(\frac{56}{61}\right)\) |
| \(\chi_{3721}(118,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{623}{915}\right)\) | \(e\left(\frac{1}{305}\right)\) | \(e\left(\frac{331}{915}\right)\) | \(e\left(\frac{356}{915}\right)\) | \(e\left(\frac{626}{915}\right)\) | \(e\left(\frac{272}{915}\right)\) | \(e\left(\frac{13}{305}\right)\) | \(e\left(\frac{2}{305}\right)\) | \(e\left(\frac{64}{915}\right)\) | \(e\left(\frac{46}{61}\right)\) |
| \(\chi_{3721}(134,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{362}{915}\right)\) | \(e\left(\frac{239}{305}\right)\) | \(e\left(\frac{724}{915}\right)\) | \(e\left(\frac{599}{915}\right)\) | \(e\left(\frac{164}{915}\right)\) | \(e\left(\frac{653}{915}\right)\) | \(e\left(\frac{57}{305}\right)\) | \(e\left(\frac{173}{305}\right)\) | \(e\left(\frac{46}{915}\right)\) | \(e\left(\frac{14}{61}\right)\) |
| \(\chi_{3721}(137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{682}{915}\right)\) | \(e\left(\frac{29}{305}\right)\) | \(e\left(\frac{449}{915}\right)\) | \(e\left(\frac{259}{915}\right)\) | \(e\left(\frac{769}{915}\right)\) | \(e\left(\frac{568}{915}\right)\) | \(e\left(\frac{72}{305}\right)\) | \(e\left(\frac{58}{305}\right)\) | \(e\left(\frac{26}{915}\right)\) | \(e\left(\frac{53}{61}\right)\) |
| \(\chi_{3721}(138,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{646}{915}\right)\) | \(e\left(\frac{167}{305}\right)\) | \(e\left(\frac{377}{915}\right)\) | \(e\left(\frac{892}{915}\right)\) | \(e\left(\frac{232}{915}\right)\) | \(e\left(\frac{589}{915}\right)\) | \(e\left(\frac{36}{305}\right)\) | \(e\left(\frac{29}{305}\right)\) | \(e\left(\frac{623}{915}\right)\) | \(e\left(\frac{57}{61}\right)\) |
| \(\chi_{3721}(144,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{529}{915}\right)\) | \(e\left(\frac{158}{305}\right)\) | \(e\left(\frac{143}{915}\right)\) | \(e\left(\frac{433}{915}\right)\) | \(e\left(\frac{88}{915}\right)\) | \(e\left(\frac{886}{915}\right)\) | \(e\left(\frac{224}{305}\right)\) | \(e\left(\frac{11}{305}\right)\) | \(e\left(\frac{47}{915}\right)\) | \(e\left(\frac{9}{61}\right)\) |
| \(\chi_{3721}(147,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{251}{915}\right)\) | \(e\left(\frac{207}{305}\right)\) | \(e\left(\frac{502}{915}\right)\) | \(e\left(\frac{797}{915}\right)\) | \(e\left(\frac{872}{915}\right)\) | \(e\left(\frac{794}{915}\right)\) | \(e\left(\frac{251}{305}\right)\) | \(e\left(\frac{109}{305}\right)\) | \(e\left(\frac{133}{915}\right)\) | \(e\left(\frac{6}{61}\right)\) |
| \(\chi_{3721}(164,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{449}{915}\right)\) | \(e\left(\frac{58}{305}\right)\) | \(e\left(\frac{898}{915}\right)\) | \(e\left(\frac{518}{915}\right)\) | \(e\left(\frac{623}{915}\right)\) | \(e\left(\frac{221}{915}\right)\) | \(e\left(\frac{144}{305}\right)\) | \(e\left(\frac{116}{305}\right)\) | \(e\left(\frac{52}{915}\right)\) | \(e\left(\frac{45}{61}\right)\) |
| \(\chi_{3721}(178,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{538}{915}\right)\) | \(e\left(\frac{276}{305}\right)\) | \(e\left(\frac{161}{915}\right)\) | \(e\left(\frac{46}{915}\right)\) | \(e\left(\frac{451}{915}\right)\) | \(e\left(\frac{652}{915}\right)\) | \(e\left(\frac{233}{305}\right)\) | \(e\left(\frac{247}{305}\right)\) | \(e\left(\frac{584}{915}\right)\) | \(e\left(\frac{8}{61}\right)\) |
| \(\chi_{3721}(179,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{248}{915}\right)\) | \(e\left(\frac{66}{305}\right)\) | \(e\left(\frac{496}{915}\right)\) | \(e\left(\frac{11}{915}\right)\) | \(e\left(\frac{446}{915}\right)\) | \(e\left(\frac{872}{915}\right)\) | \(e\left(\frac{248}{305}\right)\) | \(e\left(\frac{132}{305}\right)\) | \(e\left(\frac{259}{915}\right)\) | \(e\left(\frac{47}{61}\right)\) |
| \(\chi_{3721}(195,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{182}{915}\right)\) | \(e\left(\frac{14}{305}\right)\) | \(e\left(\frac{364}{915}\right)\) | \(e\left(\frac{104}{915}\right)\) | \(e\left(\frac{224}{915}\right)\) | \(e\left(\frac{758}{915}\right)\) | \(e\left(\frac{182}{305}\right)\) | \(e\left(\frac{28}{305}\right)\) | \(e\left(\frac{286}{915}\right)\) | \(e\left(\frac{34}{61}\right)\) |
| \(\chi_{3721}(198,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{172}{915}\right)\) | \(e\left(\frac{154}{305}\right)\) | \(e\left(\frac{344}{915}\right)\) | \(e\left(\frac{229}{915}\right)\) | \(e\left(\frac{634}{915}\right)\) | \(e\left(\frac{103}{915}\right)\) | \(e\left(\frac{172}{305}\right)\) | \(e\left(\frac{3}{305}\right)\) | \(e\left(\frac{401}{915}\right)\) | \(e\left(\frac{8}{61}\right)\) |
| \(\chi_{3721}(199,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{511}{915}\right)\) | \(e\left(\frac{227}{305}\right)\) | \(e\left(\frac{107}{915}\right)\) | \(e\left(\frac{292}{915}\right)\) | \(e\left(\frac{277}{915}\right)\) | \(e\left(\frac{439}{915}\right)\) | \(e\left(\frac{206}{305}\right)\) | \(e\left(\frac{149}{305}\right)\) | \(e\left(\frac{803}{915}\right)\) | \(e\left(\frac{11}{61}\right)\) |
| \(\chi_{3721}(205,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{514}{915}\right)\) | \(e\left(\frac{63}{305}\right)\) | \(e\left(\frac{113}{915}\right)\) | \(e\left(\frac{163}{915}\right)\) | \(e\left(\frac{703}{915}\right)\) | \(e\left(\frac{361}{915}\right)\) | \(e\left(\frac{209}{305}\right)\) | \(e\left(\frac{126}{305}\right)\) | \(e\left(\frac{677}{915}\right)\) | \(e\left(\frac{31}{61}\right)\) |
| \(\chi_{3721}(208,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{311}{915}\right)\) | \(e\left(\frac{282}{305}\right)\) | \(e\left(\frac{622}{915}\right)\) | \(e\left(\frac{47}{915}\right)\) | \(e\left(\frac{242}{915}\right)\) | \(e\left(\frac{149}{915}\right)\) | \(e\left(\frac{6}{305}\right)\) | \(e\left(\frac{259}{305}\right)\) | \(e\left(\frac{358}{915}\right)\) | \(e\left(\frac{40}{61}\right)\) |
| \(\chi_{3721}(225,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{659}{915}\right)\) | \(e\left(\frac{168}{305}\right)\) | \(e\left(\frac{403}{915}\right)\) | \(e\left(\frac{638}{915}\right)\) | \(e\left(\frac{248}{915}\right)\) | \(e\left(\frac{251}{915}\right)\) | \(e\left(\frac{49}{305}\right)\) | \(e\left(\frac{31}{305}\right)\) | \(e\left(\frac{382}{915}\right)\) | \(e\left(\frac{42}{61}\right)\) |
| \(\chi_{3721}(239,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{238}{915}\right)\) | \(e\left(\frac{206}{305}\right)\) | \(e\left(\frac{476}{915}\right)\) | \(e\left(\frac{136}{915}\right)\) | \(e\left(\frac{856}{915}\right)\) | \(e\left(\frac{217}{915}\right)\) | \(e\left(\frac{238}{305}\right)\) | \(e\left(\frac{107}{305}\right)\) | \(e\left(\frac{374}{915}\right)\) | \(e\left(\frac{21}{61}\right)\) |