from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(987696, base_ring=CyclotomicField(2166))
M = H._module
chi = DirichletCharacter(H, M([0,0,722,328]))
chi.galois_orbit()
[g,chi] = znchar(Mod(49,987696))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(987696\) | |
| Conductor: | \(61731\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1083\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 61731.cs | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{1083})$ |
| Fixed field: | Number field defined by a degree 1083 polynomial (not computed) |
First 31 of 684 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{987696}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{922}{1083}\right)\) | \(e\left(\frac{907}{1083}\right)\) | \(e\left(\frac{217}{1083}\right)\) | \(e\left(\frac{119}{361}\right)\) | \(e\left(\frac{470}{1083}\right)\) | \(e\left(\frac{166}{361}\right)\) | \(e\left(\frac{761}{1083}\right)\) | \(e\left(\frac{926}{1083}\right)\) | \(e\left(\frac{26}{1083}\right)\) | \(e\left(\frac{746}{1083}\right)\) |
| \(\chi_{987696}(2401,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{761}{1083}\right)\) | \(e\left(\frac{731}{1083}\right)\) | \(e\left(\frac{434}{1083}\right)\) | \(e\left(\frac{238}{361}\right)\) | \(e\left(\frac{940}{1083}\right)\) | \(e\left(\frac{332}{361}\right)\) | \(e\left(\frac{439}{1083}\right)\) | \(e\left(\frac{769}{1083}\right)\) | \(e\left(\frac{52}{1083}\right)\) | \(e\left(\frac{409}{1083}\right)\) |
| \(\chi_{987696}(2785,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{394}{1083}\right)\) | \(e\left(\frac{202}{1083}\right)\) | \(e\left(\frac{34}{1083}\right)\) | \(e\left(\frac{7}{361}\right)\) | \(e\left(\frac{962}{1083}\right)\) | \(e\left(\frac{31}{361}\right)\) | \(e\left(\frac{788}{1083}\right)\) | \(e\left(\frac{734}{1083}\right)\) | \(e\left(\frac{44}{1083}\right)\) | \(e\left(\frac{596}{1083}\right)\) |
| \(\chi_{987696}(5137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1055}{1083}\right)\) | \(e\left(\frac{299}{1083}\right)\) | \(e\left(\frac{179}{1083}\right)\) | \(e\left(\frac{5}{361}\right)\) | \(e\left(\frac{223}{1083}\right)\) | \(e\left(\frac{280}{361}\right)\) | \(e\left(\frac{1027}{1083}\right)\) | \(e\left(\frac{679}{1083}\right)\) | \(e\left(\frac{805}{1083}\right)\) | \(e\left(\frac{271}{1083}\right)\) |
| \(\chi_{987696}(5521,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{1083}\right)\) | \(e\left(\frac{979}{1083}\right)\) | \(e\left(\frac{79}{1083}\right)\) | \(e\left(\frac{218}{361}\right)\) | \(e\left(\frac{770}{1083}\right)\) | \(e\left(\frac{295}{361}\right)\) | \(e\left(\frac{302}{1083}\right)\) | \(e\left(\frac{941}{1083}\right)\) | \(e\left(\frac{803}{1083}\right)\) | \(e\left(\frac{47}{1083}\right)\) |
| \(\chi_{987696}(8257,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{1083}\right)\) | \(e\left(\frac{1072}{1083}\right)\) | \(e\left(\frac{352}{1083}\right)\) | \(e\left(\frac{30}{361}\right)\) | \(e\left(\frac{977}{1083}\right)\) | \(e\left(\frac{236}{361}\right)\) | \(e\left(\frac{386}{1083}\right)\) | \(e\left(\frac{464}{1083}\right)\) | \(e\left(\frac{137}{1083}\right)\) | \(e\left(\frac{182}{1083}\right)\) |
| \(\chi_{987696}(10609,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{218}{1083}\right)\) | \(e\left(\frac{689}{1083}\right)\) | \(e\left(\frac{695}{1083}\right)\) | \(e\left(\frac{90}{361}\right)\) | \(e\left(\frac{43}{1083}\right)\) | \(e\left(\frac{347}{361}\right)\) | \(e\left(\frac{436}{1083}\right)\) | \(e\left(\frac{670}{1083}\right)\) | \(e\left(\frac{772}{1083}\right)\) | \(e\left(\frac{907}{1083}\right)\) |
| \(\chi_{987696}(10993,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{520}{1083}\right)\) | \(e\left(\frac{481}{1083}\right)\) | \(e\left(\frac{853}{1083}\right)\) | \(e\left(\frac{165}{361}\right)\) | \(e\left(\frac{500}{1083}\right)\) | \(e\left(\frac{215}{361}\right)\) | \(e\left(\frac{1040}{1083}\right)\) | \(e\left(\frac{386}{1083}\right)\) | \(e\left(\frac{212}{1083}\right)\) | \(e\left(\frac{1001}{1083}\right)\) |
| \(\chi_{987696}(13345,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{170}{1083}\right)\) | \(e\left(\frac{428}{1083}\right)\) | \(e\left(\frac{383}{1083}\right)\) | \(e\left(\frac{47}{361}\right)\) | \(e\left(\frac{580}{1083}\right)\) | \(e\left(\frac{105}{361}\right)\) | \(e\left(\frac{340}{1083}\right)\) | \(e\left(\frac{751}{1083}\right)\) | \(e\left(\frac{1069}{1083}\right)\) | \(e\left(\frac{598}{1083}\right)\) |
| \(\chi_{987696}(13729,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{1083}\right)\) | \(e\left(\frac{289}{1083}\right)\) | \(e\left(\frac{499}{1083}\right)\) | \(e\left(\frac{262}{361}\right)\) | \(e\left(\frac{422}{1083}\right)\) | \(e\left(\frac{232}{361}\right)\) | \(e\left(\frac{98}{1083}\right)\) | \(e\left(\frac{707}{1083}\right)\) | \(e\left(\frac{1028}{1083}\right)\) | \(e\left(\frac{338}{1083}\right)\) |
| \(\chi_{987696}(16081,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{1083}\right)\) | \(e\left(\frac{224}{1083}\right)\) | \(e\left(\frac{413}{1083}\right)\) | \(e\left(\frac{308}{361}\right)\) | \(e\left(\frac{91}{1083}\right)\) | \(e\left(\frac{281}{361}\right)\) | \(e\left(\frac{16}{1083}\right)\) | \(e\left(\frac{889}{1083}\right)\) | \(e\left(\frac{853}{1083}\right)\) | \(e\left(\frac{232}{1083}\right)\) |
| \(\chi_{987696}(16465,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{946}{1083}\right)\) | \(e\left(\frac{496}{1083}\right)\) | \(e\left(\frac{373}{1083}\right)\) | \(e\left(\frac{321}{361}\right)\) | \(e\left(\frac{743}{1083}\right)\) | \(e\left(\frac{287}{361}\right)\) | \(e\left(\frac{809}{1083}\right)\) | \(e\left(\frac{344}{1083}\right)\) | \(e\left(\frac{419}{1083}\right)\) | \(e\left(\frac{359}{1083}\right)\) |
| \(\chi_{987696}(18817,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{815}{1083}\right)\) | \(e\left(\frac{77}{1083}\right)\) | \(e\left(\frac{785}{1083}\right)\) | \(e\left(\frac{151}{361}\right)\) | \(e\left(\frac{742}{1083}\right)\) | \(e\left(\frac{153}{361}\right)\) | \(e\left(\frac{547}{1083}\right)\) | \(e\left(\frac{1}{1083}\right)\) | \(e\left(\frac{124}{1083}\right)\) | \(e\left(\frac{892}{1083}\right)\) |
| \(\chi_{987696}(21553,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{425}{1083}\right)\) | \(e\left(\frac{1070}{1083}\right)\) | \(e\left(\frac{416}{1083}\right)\) | \(e\left(\frac{298}{361}\right)\) | \(e\left(\frac{367}{1083}\right)\) | \(e\left(\frac{82}{361}\right)\) | \(e\left(\frac{850}{1083}\right)\) | \(e\left(\frac{253}{1083}\right)\) | \(e\left(\frac{1048}{1083}\right)\) | \(e\left(\frac{412}{1083}\right)\) |
| \(\chi_{987696}(21937,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{346}{1083}\right)\) | \(e\left(\frac{1024}{1083}\right)\) | \(e\left(\frac{805}{1083}\right)\) | \(e\left(\frac{325}{361}\right)\) | \(e\left(\frac{416}{1083}\right)\) | \(e\left(\frac{150}{361}\right)\) | \(e\left(\frac{692}{1083}\right)\) | \(e\left(\frac{815}{1083}\right)\) | \(e\left(\frac{341}{1083}\right)\) | \(e\left(\frac{287}{1083}\right)\) |
| \(\chi_{987696}(24289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1004}{1083}\right)\) | \(e\left(\frac{1037}{1083}\right)\) | \(e\left(\frac{389}{1083}\right)\) | \(e\left(\frac{27}{361}\right)\) | \(e\left(\frac{49}{1083}\right)\) | \(e\left(\frac{68}{361}\right)\) | \(e\left(\frac{925}{1083}\right)\) | \(e\left(\frac{562}{1083}\right)\) | \(e\left(\frac{376}{1083}\right)\) | \(e\left(\frac{958}{1083}\right)\) |
| \(\chi_{987696}(24673,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1015}{1083}\right)\) | \(e\left(\frac{262}{1083}\right)\) | \(e\left(\frac{280}{1083}\right)\) | \(e\left(\frac{270}{361}\right)\) | \(e\left(\frac{851}{1083}\right)\) | \(e\left(\frac{319}{361}\right)\) | \(e\left(\frac{947}{1083}\right)\) | \(e\left(\frac{566}{1083}\right)\) | \(e\left(\frac{872}{1083}\right)\) | \(e\left(\frac{194}{1083}\right)\) |
| \(\chi_{987696}(27025,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{386}{1083}\right)\) | \(e\left(\frac{1061}{1083}\right)\) | \(e\left(\frac{704}{1083}\right)\) | \(e\left(\frac{60}{361}\right)\) | \(e\left(\frac{871}{1083}\right)\) | \(e\left(\frac{111}{361}\right)\) | \(e\left(\frac{772}{1083}\right)\) | \(e\left(\frac{928}{1083}\right)\) | \(e\left(\frac{274}{1083}\right)\) | \(e\left(\frac{364}{1083}\right)\) |
| \(\chi_{987696}(27409,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{886}{1083}\right)\) | \(e\left(\frac{982}{1083}\right)\) | \(e\left(\frac{1066}{1083}\right)\) | \(e\left(\frac{177}{361}\right)\) | \(e\left(\frac{602}{1083}\right)\) | \(e\left(\frac{165}{361}\right)\) | \(e\left(\frac{689}{1083}\right)\) | \(e\left(\frac{716}{1083}\right)\) | \(e\left(\frac{1061}{1083}\right)\) | \(e\left(\frac{785}{1083}\right)\) |
| \(\chi_{987696}(29761,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{737}{1083}\right)\) | \(e\left(\frac{59}{1083}\right)\) | \(e\left(\frac{278}{1083}\right)\) | \(e\left(\frac{36}{361}\right)\) | \(e\left(\frac{667}{1083}\right)\) | \(e\left(\frac{211}{361}\right)\) | \(e\left(\frac{391}{1083}\right)\) | \(e\left(\frac{268}{1083}\right)\) | \(e\left(\frac{742}{1083}\right)\) | \(e\left(\frac{796}{1083}\right)\) |
| \(\chi_{987696}(30145,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1042}{1083}\right)\) | \(e\left(\frac{1018}{1083}\right)\) | \(e\left(\frac{997}{1083}\right)\) | \(e\left(\frac{46}{361}\right)\) | \(e\left(\frac{752}{1083}\right)\) | \(e\left(\frac{49}{361}\right)\) | \(e\left(\frac{1001}{1083}\right)\) | \(e\left(\frac{182}{1083}\right)\) | \(e\left(\frac{908}{1083}\right)\) | \(e\left(\frac{977}{1083}\right)\) |
| \(\chi_{987696}(32497,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{974}{1083}\right)\) | \(e\left(\frac{197}{1083}\right)\) | \(e\left(\frac{194}{1083}\right)\) | \(e\left(\frac{316}{361}\right)\) | \(e\left(\frac{520}{1083}\right)\) | \(e\left(\frac{7}{361}\right)\) | \(e\left(\frac{865}{1083}\right)\) | \(e\left(\frac{748}{1083}\right)\) | \(e\left(\frac{697}{1083}\right)\) | \(e\left(\frac{88}{1083}\right)\) |
| \(\chi_{987696}(32881,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{400}{1083}\right)\) | \(e\left(\frac{370}{1083}\right)\) | \(e\left(\frac{73}{1083}\right)\) | \(e\left(\frac{238}{361}\right)\) | \(e\left(\frac{218}{1083}\right)\) | \(e\left(\frac{332}{361}\right)\) | \(e\left(\frac{800}{1083}\right)\) | \(e\left(\frac{47}{1083}\right)\) | \(e\left(\frac{413}{1083}\right)\) | \(e\left(\frac{770}{1083}\right)\) |
| \(\chi_{987696}(35233,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{1083}\right)\) | \(e\left(\frac{392}{1083}\right)\) | \(e\left(\frac{452}{1083}\right)\) | \(e\left(\frac{178}{361}\right)\) | \(e\left(\frac{430}{1083}\right)\) | \(e\left(\frac{221}{361}\right)\) | \(e\left(\frac{28}{1083}\right)\) | \(e\left(\frac{202}{1083}\right)\) | \(e\left(\frac{139}{1083}\right)\) | \(e\left(\frac{406}{1083}\right)\) |
| \(\chi_{987696}(35617,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{1083}\right)\) | \(e\left(\frac{121}{1083}\right)\) | \(e\left(\frac{460}{1083}\right)\) | \(e\left(\frac{31}{361}\right)\) | \(e\left(\frac{83}{1083}\right)\) | \(e\left(\frac{292}{361}\right)\) | \(e\left(\frac{86}{1083}\right)\) | \(e\left(\frac{311}{1083}\right)\) | \(e\left(\frac{659}{1083}\right)\) | \(e\left(\frac{164}{1083}\right)\) |
| \(\chi_{987696}(37969,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{1083}\right)\) | \(e\left(\frac{644}{1083}\right)\) | \(e\left(\frac{1052}{1083}\right)\) | \(e\left(\frac{344}{361}\right)\) | \(e\left(\frac{397}{1083}\right)\) | \(e\left(\frac{131}{361}\right)\) | \(e\left(\frac{46}{1083}\right)\) | \(e\left(\frac{796}{1083}\right)\) | \(e\left(\frac{151}{1083}\right)\) | \(e\left(\frac{667}{1083}\right)\) |
| \(\chi_{987696}(38353,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1054}{1083}\right)\) | \(e\left(\frac{271}{1083}\right)\) | \(e\left(\frac{1075}{1083}\right)\) | \(e\left(\frac{147}{361}\right)\) | \(e\left(\frac{347}{1083}\right)\) | \(e\left(\frac{290}{361}\right)\) | \(e\left(\frac{1025}{1083}\right)\) | \(e\left(\frac{974}{1083}\right)\) | \(e\left(\frac{563}{1083}\right)\) | \(e\left(\frac{242}{1083}\right)\) |
| \(\chi_{987696}(40705,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1001}{1083}\right)\) | \(e\left(\frac{953}{1083}\right)\) | \(e\left(\frac{911}{1083}\right)\) | \(e\left(\frac{92}{361}\right)\) | \(e\left(\frac{421}{1083}\right)\) | \(e\left(\frac{98}{361}\right)\) | \(e\left(\frac{919}{1083}\right)\) | \(e\left(\frac{364}{1083}\right)\) | \(e\left(\frac{733}{1083}\right)\) | \(e\left(\frac{871}{1083}\right)\) |
| \(\chi_{987696}(41089,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{184}{1083}\right)\) | \(e\left(\frac{820}{1083}\right)\) | \(e\left(\frac{835}{1083}\right)\) | \(e\left(\frac{225}{361}\right)\) | \(e\left(\frac{1010}{1083}\right)\) | \(e\left(\frac{326}{361}\right)\) | \(e\left(\frac{368}{1083}\right)\) | \(e\left(\frac{953}{1083}\right)\) | \(e\left(\frac{125}{1083}\right)\) | \(e\left(\frac{1004}{1083}\right)\) |
| \(\chi_{987696}(43441,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{782}{1083}\right)\) | \(e\left(\frac{236}{1083}\right)\) | \(e\left(\frac{29}{1083}\right)\) | \(e\left(\frac{144}{361}\right)\) | \(e\left(\frac{502}{1083}\right)\) | \(e\left(\frac{122}{361}\right)\) | \(e\left(\frac{481}{1083}\right)\) | \(e\left(\frac{1072}{1083}\right)\) | \(e\left(\frac{802}{1083}\right)\) | \(e\left(\frac{1018}{1083}\right)\) |
| \(\chi_{987696}(43825,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{682}{1083}\right)\) | \(e\left(\frac{685}{1083}\right)\) | \(e\left(\frac{823}{1083}\right)\) | \(e\left(\frac{265}{361}\right)\) | \(e\left(\frac{989}{1083}\right)\) | \(e\left(\frac{39}{361}\right)\) | \(e\left(\frac{281}{1083}\right)\) | \(e\left(\frac{248}{1083}\right)\) | \(e\left(\frac{428}{1083}\right)\) | \(e\left(\frac{284}{1083}\right)\) |