from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1049, base_ring=CyclotomicField(1048))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,1049))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1049\) | |
| Conductor: | \(1049\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1048\) | 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: | $\Q(\zeta_{1048})$ |
| Fixed field: | Number field defined by a degree 1048 polynomial (not computed) |
First 31 of 520 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1049}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{149}{262}\right)\) | \(e\left(\frac{1}{1048}\right)\) | \(e\left(\frac{18}{131}\right)\) | \(e\left(\frac{49}{524}\right)\) | \(e\left(\frac{597}{1048}\right)\) | \(e\left(\frac{255}{1048}\right)\) | \(e\left(\frac{185}{262}\right)\) | \(e\left(\frac{1}{524}\right)\) | \(e\left(\frac{347}{524}\right)\) | \(e\left(\frac{209}{524}\right)\) |
| \(\chi_{1049}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{135}{262}\right)\) | \(e\left(\frac{597}{1048}\right)\) | \(e\left(\frac{4}{131}\right)\) | \(e\left(\frac{433}{524}\right)\) | \(e\left(\frac{89}{1048}\right)\) | \(e\left(\frac{275}{1048}\right)\) | \(e\left(\frac{143}{262}\right)\) | \(e\left(\frac{73}{524}\right)\) | \(e\left(\frac{179}{524}\right)\) | \(e\left(\frac{61}{524}\right)\) |
| \(\chi_{1049}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{262}\right)\) | \(e\left(\frac{255}{1048}\right)\) | \(e\left(\frac{5}{131}\right)\) | \(e\left(\frac{443}{524}\right)\) | \(e\left(\frac{275}{1048}\right)\) | \(e\left(\frac{49}{1048}\right)\) | \(e\left(\frac{15}{262}\right)\) | \(e\left(\frac{255}{524}\right)\) | \(e\left(\frac{453}{524}\right)\) | \(e\left(\frac{371}{524}\right)\) |
| \(\chi_{1049}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{262}\right)\) | \(e\left(\frac{145}{1048}\right)\) | \(e\left(\frac{121}{131}\right)\) | \(e\left(\frac{293}{524}\right)\) | \(e\left(\frac{629}{1048}\right)\) | \(e\left(\frac{295}{1048}\right)\) | \(e\left(\frac{101}{262}\right)\) | \(e\left(\frac{145}{524}\right)\) | \(e\left(\frac{11}{524}\right)\) | \(e\left(\frac{437}{524}\right)\) |
| \(\chi_{1049}(14,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{253}{262}\right)\) | \(e\left(\frac{851}{1048}\right)\) | \(e\left(\frac{122}{131}\right)\) | \(e\left(\frac{303}{524}\right)\) | \(e\left(\frac{815}{1048}\right)\) | \(e\left(\frac{69}{1048}\right)\) | \(e\left(\frac{235}{262}\right)\) | \(e\left(\frac{327}{524}\right)\) | \(e\left(\frac{285}{524}\right)\) | \(e\left(\frac{223}{524}\right)\) |
| \(\chi_{1049}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{262}\right)\) | \(e\left(\frac{99}{1048}\right)\) | \(e\left(\frac{79}{131}\right)\) | \(e\left(\frac{135}{524}\right)\) | \(e\left(\frac{415}{1048}\right)\) | \(e\left(\frac{93}{1048}\right)\) | \(e\left(\frac{237}{262}\right)\) | \(e\left(\frac{99}{524}\right)\) | \(e\left(\frac{293}{524}\right)\) | \(e\left(\frac{255}{524}\right)\) |
| \(\chi_{1049}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{133}{262}\right)\) | \(e\left(\frac{757}{1048}\right)\) | \(e\left(\frac{2}{131}\right)\) | \(e\left(\frac{413}{524}\right)\) | \(e\left(\frac{241}{1048}\right)\) | \(e\left(\frac{203}{1048}\right)\) | \(e\left(\frac{137}{262}\right)\) | \(e\left(\frac{233}{524}\right)\) | \(e\left(\frac{155}{524}\right)\) | \(e\left(\frac{489}{524}\right)\) |
| \(\chi_{1049}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{262}\right)\) | \(e\left(\frac{227}{1048}\right)\) | \(e\left(\frac{25}{131}\right)\) | \(e\left(\frac{119}{524}\right)\) | \(e\left(\frac{327}{1048}\right)\) | \(e\left(\frac{245}{1048}\right)\) | \(e\left(\frac{75}{262}\right)\) | \(e\left(\frac{227}{524}\right)\) | \(e\left(\frac{169}{524}\right)\) | \(e\left(\frac{283}{524}\right)\) |
| \(\chi_{1049}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{262}\right)\) | \(e\left(\frac{741}{1048}\right)\) | \(e\left(\frac{107}{131}\right)\) | \(e\left(\frac{153}{524}\right)\) | \(e\left(\frac{121}{1048}\right)\) | \(e\left(\frac{315}{1048}\right)\) | \(e\left(\frac{59}{262}\right)\) | \(e\left(\frac{217}{524}\right)\) | \(e\left(\frac{367}{524}\right)\) | \(e\left(\frac{289}{524}\right)\) |
| \(\chi_{1049}(27,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{185}{262}\right)\) | \(e\left(\frac{3}{1048}\right)\) | \(e\left(\frac{54}{131}\right)\) | \(e\left(\frac{147}{524}\right)\) | \(e\left(\frac{743}{1048}\right)\) | \(e\left(\frac{765}{1048}\right)\) | \(e\left(\frac{31}{262}\right)\) | \(e\left(\frac{3}{524}\right)\) | \(e\left(\frac{517}{524}\right)\) | \(e\left(\frac{103}{524}\right)\) |
| \(\chi_{1049}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{239}{262}\right)\) | \(e\left(\frac{399}{1048}\right)\) | \(e\left(\frac{108}{131}\right)\) | \(e\left(\frac{163}{524}\right)\) | \(e\left(\frac{307}{1048}\right)\) | \(e\left(\frac{89}{1048}\right)\) | \(e\left(\frac{193}{262}\right)\) | \(e\left(\frac{399}{524}\right)\) | \(e\left(\frac{117}{524}\right)\) | \(e\left(\frac{75}{524}\right)\) |
| \(\chi_{1049}(30,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{262}\right)\) | \(e\left(\frac{695}{1048}\right)\) | \(e\left(\frac{65}{131}\right)\) | \(e\left(\frac{519}{524}\right)\) | \(e\left(\frac{955}{1048}\right)\) | \(e\left(\frac{113}{1048}\right)\) | \(e\left(\frac{195}{262}\right)\) | \(e\left(\frac{171}{524}\right)\) | \(e\left(\frac{125}{524}\right)\) | \(e\left(\frac{107}{524}\right)\) |
| \(\chi_{1049}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{262}\right)\) | \(e\left(\frac{841}{1048}\right)\) | \(e\left(\frac{73}{131}\right)\) | \(e\left(\frac{337}{524}\right)\) | \(e\left(\frac{85}{1048}\right)\) | \(e\left(\frac{663}{1048}\right)\) | \(e\left(\frac{219}{262}\right)\) | \(e\left(\frac{317}{524}\right)\) | \(e\left(\frac{483}{524}\right)\) | \(e\left(\frac{229}{524}\right)\) |
| \(\chi_{1049}(33,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{75}{262}\right)\) | \(e\left(\frac{419}{1048}\right)\) | \(e\left(\frac{75}{131}\right)\) | \(e\left(\frac{95}{524}\right)\) | \(e\left(\frac{719}{1048}\right)\) | \(e\left(\frac{997}{1048}\right)\) | \(e\left(\frac{225}{262}\right)\) | \(e\left(\frac{419}{524}\right)\) | \(e\left(\frac{245}{524}\right)\) | \(e\left(\frac{63}{524}\right)\) |
| \(\chi_{1049}(34,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{119}{262}\right)\) | \(e\left(\frac{305}{1048}\right)\) | \(e\left(\frac{119}{131}\right)\) | \(e\left(\frac{273}{524}\right)\) | \(e\left(\frac{781}{1048}\right)\) | \(e\left(\frac{223}{1048}\right)\) | \(e\left(\frac{95}{262}\right)\) | \(e\left(\frac{305}{524}\right)\) | \(e\left(\frac{511}{524}\right)\) | \(e\left(\frac{341}{524}\right)\) |
| \(\chi_{1049}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{197}{262}\right)\) | \(e\left(\frac{353}{1048}\right)\) | \(e\left(\frac{66}{131}\right)\) | \(e\left(\frac{5}{524}\right)\) | \(e\left(\frac{93}{1048}\right)\) | \(e\left(\frac{935}{1048}\right)\) | \(e\left(\frac{67}{262}\right)\) | \(e\left(\frac{353}{524}\right)\) | \(e\left(\frac{399}{524}\right)\) | \(e\left(\frac{417}{524}\right)\) |
| \(\chi_{1049}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{185}{262}\right)\) | \(e\left(\frac{265}{1048}\right)\) | \(e\left(\frac{54}{131}\right)\) | \(e\left(\frac{409}{524}\right)\) | \(e\left(\frac{1005}{1048}\right)\) | \(e\left(\frac{503}{1048}\right)\) | \(e\left(\frac{31}{262}\right)\) | \(e\left(\frac{265}{524}\right)\) | \(e\left(\frac{255}{524}\right)\) | \(e\left(\frac{365}{524}\right)\) |
| \(\chi_{1049}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{262}\right)\) | \(e\left(\frac{141}{1048}\right)\) | \(e\left(\frac{49}{131}\right)\) | \(e\left(\frac{97}{524}\right)\) | \(e\left(\frac{337}{1048}\right)\) | \(e\left(\frac{323}{1048}\right)\) | \(e\left(\frac{147}{262}\right)\) | \(e\left(\frac{141}{524}\right)\) | \(e\left(\frac{195}{524}\right)\) | \(e\left(\frac{125}{524}\right)\) |
| \(\chi_{1049}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{247}{262}\right)\) | \(e\left(\frac{21}{1048}\right)\) | \(e\left(\frac{116}{131}\right)\) | \(e\left(\frac{505}{524}\right)\) | \(e\left(\frac{1009}{1048}\right)\) | \(e\left(\frac{115}{1048}\right)\) | \(e\left(\frac{217}{262}\right)\) | \(e\left(\frac{21}{524}\right)\) | \(e\left(\frac{475}{524}\right)\) | \(e\left(\frac{197}{524}\right)\) |
| \(\chi_{1049}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{262}\right)\) | \(e\left(\frac{823}{1048}\right)\) | \(e\left(\frac{11}{131}\right)\) | \(e\left(\frac{503}{524}\right)\) | \(e\left(\frac{867}{1048}\right)\) | \(e\left(\frac{265}{1048}\right)\) | \(e\left(\frac{33}{262}\right)\) | \(e\left(\frac{299}{524}\right)\) | \(e\left(\frac{1}{524}\right)\) | \(e\left(\frac{135}{524}\right)\) |
| \(\chi_{1049}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{167}{262}\right)\) | \(e\left(\frac{657}{1048}\right)\) | \(e\left(\frac{36}{131}\right)\) | \(e\left(\frac{229}{524}\right)\) | \(e\left(\frac{277}{1048}\right)\) | \(e\left(\frac{903}{1048}\right)\) | \(e\left(\frac{239}{262}\right)\) | \(e\left(\frac{133}{524}\right)\) | \(e\left(\frac{39}{524}\right)\) | \(e\left(\frac{25}{524}\right)\) |
| \(\chi_{1049}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{93}{262}\right)\) | \(e\left(\frac{289}{1048}\right)\) | \(e\left(\frac{93}{131}\right)\) | \(e\left(\frac{13}{524}\right)\) | \(e\left(\frac{661}{1048}\right)\) | \(e\left(\frac{335}{1048}\right)\) | \(e\left(\frac{17}{262}\right)\) | \(e\left(\frac{289}{524}\right)\) | \(e\left(\frac{199}{524}\right)\) | \(e\left(\frac{141}{524}\right)\) |
| \(\chi_{1049}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{171}{262}\right)\) | \(e\left(\frac{599}{1048}\right)\) | \(e\left(\frac{40}{131}\right)\) | \(e\left(\frac{7}{524}\right)\) | \(e\left(\frac{235}{1048}\right)\) | \(e\left(\frac{785}{1048}\right)\) | \(e\left(\frac{251}{262}\right)\) | \(e\left(\frac{75}{524}\right)\) | \(e\left(\frac{349}{524}\right)\) | \(e\left(\frac{479}{524}\right)\) |
| \(\chi_{1049}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{225}{262}\right)\) | \(e\left(\frac{995}{1048}\right)\) | \(e\left(\frac{94}{131}\right)\) | \(e\left(\frac{23}{524}\right)\) | \(e\left(\frac{847}{1048}\right)\) | \(e\left(\frac{109}{1048}\right)\) | \(e\left(\frac{151}{262}\right)\) | \(e\left(\frac{471}{524}\right)\) | \(e\left(\frac{473}{524}\right)\) | \(e\left(\frac{451}{524}\right)\) |
| \(\chi_{1049}(57,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{262}\right)\) | \(e\left(\frac{153}{1048}\right)\) | \(e\left(\frac{3}{131}\right)\) | \(e\left(\frac{161}{524}\right)\) | \(e\left(\frac{165}{1048}\right)\) | \(e\left(\frac{239}{1048}\right)\) | \(e\left(\frac{9}{262}\right)\) | \(e\left(\frac{153}{524}\right)\) | \(e\left(\frac{167}{524}\right)\) | \(e\left(\frac{13}{524}\right)\) |
| \(\chi_{1049}(60,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{262}\right)\) | \(e\left(\frac{243}{1048}\right)\) | \(e\left(\frac{51}{131}\right)\) | \(e\left(\frac{379}{524}\right)\) | \(e\left(\frac{447}{1048}\right)\) | \(e\left(\frac{133}{1048}\right)\) | \(e\left(\frac{153}{262}\right)\) | \(e\left(\frac{243}{524}\right)\) | \(e\left(\frac{481}{524}\right)\) | \(e\left(\frac{483}{524}\right)\) |
| \(\chi_{1049}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{262}\right)\) | \(e\left(\frac{389}{1048}\right)\) | \(e\left(\frac{59}{131}\right)\) | \(e\left(\frac{197}{524}\right)\) | \(e\left(\frac{625}{1048}\right)\) | \(e\left(\frac{683}{1048}\right)\) | \(e\left(\frac{177}{262}\right)\) | \(e\left(\frac{389}{524}\right)\) | \(e\left(\frac{315}{524}\right)\) | \(e\left(\frac{81}{524}\right)\) |
| \(\chi_{1049}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{262}\right)\) | \(e\left(\frac{257}{1048}\right)\) | \(e\left(\frac{41}{131}\right)\) | \(e\left(\frac{17}{524}\right)\) | \(e\left(\frac{421}{1048}\right)\) | \(e\left(\frac{559}{1048}\right)\) | \(e\left(\frac{123}{262}\right)\) | \(e\left(\frac{257}{524}\right)\) | \(e\left(\frac{99}{524}\right)\) | \(e\left(\frac{265}{524}\right)\) |
| \(\chi_{1049}(66,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{262}\right)\) | \(e\left(\frac{1015}{1048}\right)\) | \(e\left(\frac{61}{131}\right)\) | \(e\left(\frac{479}{524}\right)\) | \(e\left(\frac{211}{1048}\right)\) | \(e\left(\frac{1017}{1048}\right)\) | \(e\left(\frac{183}{262}\right)\) | \(e\left(\frac{491}{524}\right)\) | \(e\left(\frac{77}{524}\right)\) | \(e\left(\frac{439}{524}\right)\) |
| \(\chi_{1049}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{81}{262}\right)\) | \(e\left(\frac{987}{1048}\right)\) | \(e\left(\frac{81}{131}\right)\) | \(e\left(\frac{155}{524}\right)\) | \(e\left(\frac{263}{1048}\right)\) | \(e\left(\frac{165}{1048}\right)\) | \(e\left(\frac{243}{262}\right)\) | \(e\left(\frac{463}{524}\right)\) | \(e\left(\frac{317}{524}\right)\) | \(e\left(\frac{351}{524}\right)\) |
| \(\chi_{1049}(68,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{105}{262}\right)\) | \(e\left(\frac{901}{1048}\right)\) | \(e\left(\frac{105}{131}\right)\) | \(e\left(\frac{133}{524}\right)\) | \(e\left(\frac{273}{1048}\right)\) | \(e\left(\frac{243}{1048}\right)\) | \(e\left(\frac{53}{262}\right)\) | \(e\left(\frac{377}{524}\right)\) | \(e\left(\frac{343}{524}\right)\) | \(e\left(\frac{193}{524}\right)\) |