from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2672, base_ring=CyclotomicField(332))
M = H._module
chi = DirichletCharacter(H, M([0,83,92]))
chi.galois_orbit()
[g,chi] = znchar(Mod(21,2672))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2672\) | |
| Conductor: | \(2672\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(332\) | 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: | $\Q(\zeta_{332})$ |
| Fixed field: | Number field defined by a degree 332 polynomial (not computed) |
First 31 of 164 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2672}(21,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{265}{332}\right)\) | \(e\left(\frac{175}{332}\right)\) | \(e\left(\frac{33}{166}\right)\) | \(e\left(\frac{99}{166}\right)\) | \(e\left(\frac{3}{332}\right)\) | \(e\left(\frac{97}{332}\right)\) | \(e\left(\frac{27}{83}\right)\) | \(e\left(\frac{57}{83}\right)\) | \(e\left(\frac{273}{332}\right)\) | \(e\left(\frac{331}{332}\right)\) |
| \(\chi_{2672}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{332}\right)\) | \(e\left(\frac{217}{332}\right)\) | \(e\left(\frac{21}{166}\right)\) | \(e\left(\frac{63}{166}\right)\) | \(e\left(\frac{17}{332}\right)\) | \(e\left(\frac{107}{332}\right)\) | \(e\left(\frac{70}{83}\right)\) | \(e\left(\frac{74}{83}\right)\) | \(e\left(\frac{219}{332}\right)\) | \(e\left(\frac{105}{332}\right)\) |
| \(\chi_{2672}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{207}{332}\right)\) | \(e\left(\frac{49}{332}\right)\) | \(e\left(\frac{69}{166}\right)\) | \(e\left(\frac{41}{166}\right)\) | \(e\left(\frac{293}{332}\right)\) | \(e\left(\frac{67}{332}\right)\) | \(e\left(\frac{64}{83}\right)\) | \(e\left(\frac{6}{83}\right)\) | \(e\left(\frac{103}{332}\right)\) | \(e\left(\frac{13}{332}\right)\) |
| \(\chi_{2672}(77,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{307}{332}\right)\) | \(e\left(\frac{209}{332}\right)\) | \(e\left(\frac{47}{166}\right)\) | \(e\left(\frac{141}{166}\right)\) | \(e\left(\frac{125}{332}\right)\) | \(e\left(\frac{279}{332}\right)\) | \(e\left(\frac{46}{83}\right)\) | \(e\left(\frac{51}{83}\right)\) | \(e\left(\frac{87}{332}\right)\) | \(e\left(\frac{69}{332}\right)\) |
| \(\chi_{2672}(85,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{332}\right)\) | \(e\left(\frac{191}{332}\right)\) | \(e\left(\frac{147}{166}\right)\) | \(e\left(\frac{109}{166}\right)\) | \(e\left(\frac{119}{332}\right)\) | \(e\left(\frac{85}{332}\right)\) | \(e\left(\frac{75}{83}\right)\) | \(e\left(\frac{20}{83}\right)\) | \(e\left(\frac{205}{332}\right)\) | \(e\left(\frac{71}{332}\right)\) |
| \(\chi_{2672}(93,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{147}{332}\right)\) | \(e\left(\frac{285}{332}\right)\) | \(e\left(\frac{49}{166}\right)\) | \(e\left(\frac{147}{166}\right)\) | \(e\left(\frac{261}{332}\right)\) | \(e\left(\frac{139}{332}\right)\) | \(e\left(\frac{25}{83}\right)\) | \(e\left(\frac{62}{83}\right)\) | \(e\left(\frac{179}{332}\right)\) | \(e\left(\frac{245}{332}\right)\) |
| \(\chi_{2672}(133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{332}\right)\) | \(e\left(\frac{103}{332}\right)\) | \(e\left(\frac{101}{166}\right)\) | \(e\left(\frac{137}{166}\right)\) | \(e\left(\frac{311}{332}\right)\) | \(e\left(\frac{317}{332}\right)\) | \(e\left(\frac{60}{83}\right)\) | \(e\left(\frac{16}{83}\right)\) | \(e\left(\frac{81}{332}\right)\) | \(e\left(\frac{7}{332}\right)\) |
| \(\chi_{2672}(141,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{75}{332}\right)\) | \(e\left(\frac{37}{332}\right)\) | \(e\left(\frac{25}{166}\right)\) | \(e\left(\frac{75}{166}\right)\) | \(e\left(\frac{289}{332}\right)\) | \(e\left(\frac{159}{332}\right)\) | \(e\left(\frac{28}{83}\right)\) | \(e\left(\frac{13}{83}\right)\) | \(e\left(\frac{71}{332}\right)\) | \(e\left(\frac{125}{332}\right)\) |
| \(\chi_{2672}(157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{155}{332}\right)\) | \(e\left(\frac{165}{332}\right)\) | \(e\left(\frac{107}{166}\right)\) | \(e\left(\frac{155}{166}\right)\) | \(e\left(\frac{221}{332}\right)\) | \(e\left(\frac{63}{332}\right)\) | \(e\left(\frac{80}{83}\right)\) | \(e\left(\frac{49}{83}\right)\) | \(e\left(\frac{191}{332}\right)\) | \(e\left(\frac{37}{332}\right)\) |
| \(\chi_{2672}(173,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{332}\right)\) | \(e\left(\frac{185}{332}\right)\) | \(e\left(\frac{125}{166}\right)\) | \(e\left(\frac{43}{166}\right)\) | \(e\left(\frac{117}{332}\right)\) | \(e\left(\frac{131}{332}\right)\) | \(e\left(\frac{57}{83}\right)\) | \(e\left(\frac{65}{83}\right)\) | \(e\left(\frac{23}{332}\right)\) | \(e\left(\frac{293}{332}\right)\) |
| \(\chi_{2672}(181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{332}\right)\) | \(e\left(\frac{67}{332}\right)\) | \(e\left(\frac{135}{166}\right)\) | \(e\left(\frac{73}{166}\right)\) | \(e\left(\frac{299}{332}\right)\) | \(e\left(\frac{261}{332}\right)\) | \(e\left(\frac{35}{83}\right)\) | \(e\left(\frac{37}{83}\right)\) | \(e\left(\frac{317}{332}\right)\) | \(e\left(\frac{11}{332}\right)\) |
| \(\chi_{2672}(189,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{251}{332}\right)\) | \(e\left(\frac{53}{332}\right)\) | \(e\left(\frac{139}{166}\right)\) | \(e\left(\frac{85}{166}\right)\) | \(e\left(\frac{73}{332}\right)\) | \(e\left(\frac{147}{332}\right)\) | \(e\left(\frac{76}{83}\right)\) | \(e\left(\frac{59}{83}\right)\) | \(e\left(\frac{3}{332}\right)\) | \(e\left(\frac{197}{332}\right)\) |
| \(\chi_{2672}(205,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{247}{332}\right)\) | \(e\left(\frac{113}{332}\right)\) | \(e\left(\frac{27}{166}\right)\) | \(e\left(\frac{81}{166}\right)\) | \(e\left(\frac{93}{332}\right)\) | \(e\left(\frac{19}{332}\right)\) | \(e\left(\frac{7}{83}\right)\) | \(e\left(\frac{24}{83}\right)\) | \(e\left(\frac{163}{332}\right)\) | \(e\left(\frac{301}{332}\right)\) |
| \(\chi_{2672}(221,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{195}{332}\right)\) | \(e\left(\frac{229}{332}\right)\) | \(e\left(\frac{65}{166}\right)\) | \(e\left(\frac{29}{166}\right)\) | \(e\left(\frac{21}{332}\right)\) | \(e\left(\frac{15}{332}\right)\) | \(e\left(\frac{23}{83}\right)\) | \(e\left(\frac{67}{83}\right)\) | \(e\left(\frac{251}{332}\right)\) | \(e\left(\frac{325}{332}\right)\) |
| \(\chi_{2672}(229,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{332}\right)\) | \(e\left(\frac{11}{332}\right)\) | \(e\left(\frac{151}{166}\right)\) | \(e\left(\frac{121}{166}\right)\) | \(e\left(\frac{59}{332}\right)\) | \(e\left(\frac{137}{332}\right)\) | \(e\left(\frac{33}{83}\right)\) | \(e\left(\frac{42}{83}\right)\) | \(e\left(\frac{57}{332}\right)\) | \(e\left(\frac{91}{332}\right)\) |
| \(\chi_{2672}(261,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{332}\right)\) | \(e\left(\frac{95}{332}\right)\) | \(e\left(\frac{127}{166}\right)\) | \(e\left(\frac{49}{166}\right)\) | \(e\left(\frac{87}{332}\right)\) | \(e\left(\frac{157}{332}\right)\) | \(e\left(\frac{36}{83}\right)\) | \(e\left(\frac{76}{83}\right)\) | \(e\left(\frac{281}{332}\right)\) | \(e\left(\frac{303}{332}\right)\) |
| \(\chi_{2672}(293,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{225}{332}\right)\) | \(e\left(\frac{111}{332}\right)\) | \(e\left(\frac{75}{166}\right)\) | \(e\left(\frac{59}{166}\right)\) | \(e\left(\frac{203}{332}\right)\) | \(e\left(\frac{145}{332}\right)\) | \(e\left(\frac{1}{83}\right)\) | \(e\left(\frac{39}{83}\right)\) | \(e\left(\frac{213}{332}\right)\) | \(e\left(\frac{43}{332}\right)\) |
| \(\chi_{2672}(317,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{87}{332}\right)\) | \(e\left(\frac{189}{332}\right)\) | \(e\left(\frac{29}{166}\right)\) | \(e\left(\frac{87}{166}\right)\) | \(e\left(\frac{229}{332}\right)\) | \(e\left(\frac{211}{332}\right)\) | \(e\left(\frac{69}{83}\right)\) | \(e\left(\frac{35}{83}\right)\) | \(e\left(\frac{255}{332}\right)\) | \(e\left(\frac{145}{332}\right)\) |
| \(\chi_{2672}(341,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{189}{332}\right)\) | \(e\left(\frac{319}{332}\right)\) | \(e\left(\frac{63}{166}\right)\) | \(e\left(\frac{23}{166}\right)\) | \(e\left(\frac{51}{332}\right)\) | \(e\left(\frac{321}{332}\right)\) | \(e\left(\frac{44}{83}\right)\) | \(e\left(\frac{56}{83}\right)\) | \(e\left(\frac{325}{332}\right)\) | \(e\left(\frac{315}{332}\right)\) |
| \(\chi_{2672}(365,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{332}\right)\) | \(e\left(\frac{97}{332}\right)\) | \(e\left(\frac{79}{166}\right)\) | \(e\left(\frac{71}{166}\right)\) | \(e\left(\frac{309}{332}\right)\) | \(e\left(\frac{31}{332}\right)\) | \(e\left(\frac{42}{83}\right)\) | \(e\left(\frac{61}{83}\right)\) | \(e\left(\frac{231}{332}\right)\) | \(e\left(\frac{229}{332}\right)\) |
| \(\chi_{2672}(381,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{331}{332}\right)\) | \(e\left(\frac{181}{332}\right)\) | \(e\left(\frac{55}{166}\right)\) | \(e\left(\frac{165}{166}\right)\) | \(e\left(\frac{5}{332}\right)\) | \(e\left(\frac{51}{332}\right)\) | \(e\left(\frac{45}{83}\right)\) | \(e\left(\frac{12}{83}\right)\) | \(e\left(\frac{123}{332}\right)\) | \(e\left(\frac{109}{332}\right)\) |
| \(\chi_{2672}(397,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{175}{332}\right)\) | \(e\left(\frac{197}{332}\right)\) | \(e\left(\frac{3}{166}\right)\) | \(e\left(\frac{9}{166}\right)\) | \(e\left(\frac{121}{332}\right)\) | \(e\left(\frac{39}{332}\right)\) | \(e\left(\frac{10}{83}\right)\) | \(e\left(\frac{58}{83}\right)\) | \(e\left(\frac{55}{332}\right)\) | \(e\left(\frac{181}{332}\right)\) |
| \(\chi_{2672}(421,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{305}{332}\right)\) | \(e\left(\frac{239}{332}\right)\) | \(e\left(\frac{157}{166}\right)\) | \(e\left(\frac{139}{166}\right)\) | \(e\left(\frac{135}{332}\right)\) | \(e\left(\frac{49}{332}\right)\) | \(e\left(\frac{53}{83}\right)\) | \(e\left(\frac{75}{83}\right)\) | \(e\left(\frac{1}{332}\right)\) | \(e\left(\frac{287}{332}\right)\) |
| \(\chi_{2672}(461,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{255}{332}\right)\) | \(e\left(\frac{325}{332}\right)\) | \(e\left(\frac{85}{166}\right)\) | \(e\left(\frac{89}{166}\right)\) | \(e\left(\frac{53}{332}\right)\) | \(e\left(\frac{275}{332}\right)\) | \(e\left(\frac{62}{83}\right)\) | \(e\left(\frac{11}{83}\right)\) | \(e\left(\frac{175}{332}\right)\) | \(e\left(\frac{93}{332}\right)\) |
| \(\chi_{2672}(509,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{332}\right)\) | \(e\left(\frac{157}{332}\right)\) | \(e\left(\frac{133}{166}\right)\) | \(e\left(\frac{67}{166}\right)\) | \(e\left(\frac{329}{332}\right)\) | \(e\left(\frac{235}{332}\right)\) | \(e\left(\frac{56}{83}\right)\) | \(e\left(\frac{26}{83}\right)\) | \(e\left(\frac{59}{332}\right)\) | \(e\left(\frac{1}{332}\right)\) |
| \(\chi_{2672}(517,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{117}{332}\right)\) | \(e\left(\frac{71}{332}\right)\) | \(e\left(\frac{39}{166}\right)\) | \(e\left(\frac{117}{166}\right)\) | \(e\left(\frac{79}{332}\right)\) | \(e\left(\frac{9}{332}\right)\) | \(e\left(\frac{47}{83}\right)\) | \(e\left(\frac{7}{83}\right)\) | \(e\left(\frac{217}{332}\right)\) | \(e\left(\frac{195}{332}\right)\) |
| \(\chi_{2672}(525,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{143}{332}\right)\) | \(e\left(\frac{13}{332}\right)\) | \(e\left(\frac{103}{166}\right)\) | \(e\left(\frac{143}{166}\right)\) | \(e\left(\frac{281}{332}\right)\) | \(e\left(\frac{11}{332}\right)\) | \(e\left(\frac{39}{83}\right)\) | \(e\left(\frac{27}{83}\right)\) | \(e\left(\frac{7}{332}\right)\) | \(e\left(\frac{17}{332}\right)\) |
| \(\chi_{2672}(533,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{332}\right)\) | \(e\left(\frac{151}{332}\right)\) | \(e\left(\frac{111}{166}\right)\) | \(e\left(\frac{1}{166}\right)\) | \(e\left(\frac{327}{332}\right)\) | \(e\left(\frac{281}{332}\right)\) | \(e\left(\frac{38}{83}\right)\) | \(e\left(\frac{71}{83}\right)\) | \(e\left(\frac{209}{332}\right)\) | \(e\left(\frac{223}{332}\right)\) |
| \(\chi_{2672}(549,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{332}\right)\) | \(e\left(\frac{259}{332}\right)\) | \(e\left(\frac{9}{166}\right)\) | \(e\left(\frac{27}{166}\right)\) | \(e\left(\frac{31}{332}\right)\) | \(e\left(\frac{117}{332}\right)\) | \(e\left(\frac{30}{83}\right)\) | \(e\left(\frac{8}{83}\right)\) | \(e\left(\frac{165}{332}\right)\) | \(e\left(\frac{211}{332}\right)\) |
| \(\chi_{2672}(557,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{332}\right)\) | \(e\left(\frac{61}{332}\right)\) | \(e\left(\frac{113}{166}\right)\) | \(e\left(\frac{7}{166}\right)\) | \(e\left(\frac{297}{332}\right)\) | \(e\left(\frac{307}{332}\right)\) | \(e\left(\frac{17}{83}\right)\) | \(e\left(\frac{82}{83}\right)\) | \(e\left(\frac{135}{332}\right)\) | \(e\left(\frac{233}{332}\right)\) |
| \(\chi_{2672}(565,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{217}{332}\right)\) | \(e\left(\frac{231}{332}\right)\) | \(e\left(\frac{17}{166}\right)\) | \(e\left(\frac{51}{166}\right)\) | \(e\left(\frac{243}{332}\right)\) | \(e\left(\frac{221}{332}\right)\) | \(e\left(\frac{29}{83}\right)\) | \(e\left(\frac{52}{83}\right)\) | \(e\left(\frac{201}{332}\right)\) | \(e\left(\frac{251}{332}\right)\) |