from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2009, base_ring=CyclotomicField(840))
M = H._module
chi = DirichletCharacter(H, M([220,567]))
chi.galois_orbit()
[g,chi] = znchar(Mod(12,2009))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2009\) | |
| Conductor: | \(2009\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(840\) | 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: | $\Q(\zeta_{840})$ |
| Fixed field: | Number field defined by a degree 840 polynomial (not computed) |
First 31 of 192 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2009}(12,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{420}\right)\) | \(e\left(\frac{65}{168}\right)\) | \(e\left(\frac{151}{210}\right)\) | \(e\left(\frac{187}{420}\right)\) | \(e\left(\frac{209}{280}\right)\) | \(e\left(\frac{11}{140}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{169}{210}\right)\) | \(e\left(\frac{421}{840}\right)\) | \(e\left(\frac{89}{840}\right)\) |
| \(\chi_{2009}(17,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{389}{420}\right)\) | \(e\left(\frac{163}{168}\right)\) | \(e\left(\frac{179}{210}\right)\) | \(e\left(\frac{173}{420}\right)\) | \(e\left(\frac{251}{280}\right)\) | \(e\left(\frac{109}{140}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{71}{210}\right)\) | \(e\left(\frac{239}{840}\right)\) | \(e\left(\frac{691}{840}\right)\) |
| \(\chi_{2009}(24,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{420}\right)\) | \(e\left(\frac{127}{168}\right)\) | \(e\left(\frac{149}{210}\right)\) | \(e\left(\frac{293}{420}\right)\) | \(e\left(\frac{31}{280}\right)\) | \(e\left(\frac{9}{140}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{11}{210}\right)\) | \(e\left(\frac{179}{840}\right)\) | \(e\left(\frac{391}{840}\right)\) |
| \(\chi_{2009}(26,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{241}{420}\right)\) | \(e\left(\frac{131}{168}\right)\) | \(e\left(\frac{31}{210}\right)\) | \(e\left(\frac{37}{420}\right)\) | \(e\left(\frac{99}{280}\right)\) | \(e\left(\frac{101}{140}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{139}{210}\right)\) | \(e\left(\frac{391}{840}\right)\) | \(e\left(\frac{779}{840}\right)\) |
| \(\chi_{2009}(47,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{313}{420}\right)\) | \(e\left(\frac{83}{168}\right)\) | \(e\left(\frac{103}{210}\right)\) | \(e\left(\frac{1}{420}\right)\) | \(e\left(\frac{67}{280}\right)\) | \(e\left(\frac{33}{140}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{157}{210}\right)\) | \(e\left(\frac{703}{840}\right)\) | \(e\left(\frac{827}{840}\right)\) |
| \(\chi_{2009}(52,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{239}{420}\right)\) | \(e\left(\frac{25}{168}\right)\) | \(e\left(\frac{29}{210}\right)\) | \(e\left(\frac{143}{420}\right)\) | \(e\left(\frac{201}{280}\right)\) | \(e\left(\frac{99}{140}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{191}{210}\right)\) | \(e\left(\frac{149}{840}\right)\) | \(e\left(\frac{241}{840}\right)\) |
| \(\chi_{2009}(54,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{420}\right)\) | \(e\left(\frac{53}{168}\right)\) | \(e\left(\frac{43}{210}\right)\) | \(e\left(\frac{31}{420}\right)\) | \(e\left(\frac{117}{280}\right)\) | \(e\left(\frac{43}{140}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{37}{210}\right)\) | \(e\left(\frac{793}{840}\right)\) | \(e\left(\frac{437}{840}\right)\) |
| \(\chi_{2009}(75,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{367}{420}\right)\) | \(e\left(\frac{89}{168}\right)\) | \(e\left(\frac{157}{210}\right)\) | \(e\left(\frac{79}{420}\right)\) | \(e\left(\frac{113}{280}\right)\) | \(e\left(\frac{87}{140}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{13}{210}\right)\) | \(e\left(\frac{517}{840}\right)\) | \(e\left(\frac{233}{840}\right)\) |
| \(\chi_{2009}(89,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{247}{420}\right)\) | \(e\left(\frac{29}{168}\right)\) | \(e\left(\frac{37}{210}\right)\) | \(e\left(\frac{139}{420}\right)\) | \(e\left(\frac{213}{280}\right)\) | \(e\left(\frac{107}{140}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{193}{210}\right)\) | \(e\left(\frac{697}{840}\right)\) | \(e\left(\frac{293}{840}\right)\) |
| \(\chi_{2009}(94,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{311}{420}\right)\) | \(e\left(\frac{145}{168}\right)\) | \(e\left(\frac{101}{210}\right)\) | \(e\left(\frac{107}{420}\right)\) | \(e\left(\frac{169}{280}\right)\) | \(e\left(\frac{31}{140}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{209}{210}\right)\) | \(e\left(\frac{461}{840}\right)\) | \(e\left(\frac{289}{840}\right)\) |
| \(\chi_{2009}(101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{197}{420}\right)\) | \(e\left(\frac{67}{168}\right)\) | \(e\left(\frac{197}{210}\right)\) | \(e\left(\frac{269}{420}\right)\) | \(e\left(\frac{243}{280}\right)\) | \(e\left(\frac{57}{140}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{23}{210}\right)\) | \(e\left(\frac{527}{840}\right)\) | \(e\left(\frac{283}{840}\right)\) |
| \(\chi_{2009}(108,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{420}\right)\) | \(e\left(\frac{115}{168}\right)\) | \(e\left(\frac{41}{210}\right)\) | \(e\left(\frac{137}{420}\right)\) | \(e\left(\frac{219}{280}\right)\) | \(e\left(\frac{41}{140}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{89}{210}\right)\) | \(e\left(\frac{551}{840}\right)\) | \(e\left(\frac{739}{840}\right)\) |
| \(\chi_{2009}(110,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{403}{420}\right)\) | \(e\left(\frac{65}{168}\right)\) | \(e\left(\frac{193}{210}\right)\) | \(e\left(\frac{271}{420}\right)\) | \(e\left(\frac{97}{280}\right)\) | \(e\left(\frac{123}{140}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{127}{210}\right)\) | \(e\left(\frac{253}{840}\right)\) | \(e\left(\frac{257}{840}\right)\) |
| \(\chi_{2009}(136,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{383}{420}\right)\) | \(e\left(\frac{13}{168}\right)\) | \(e\left(\frac{173}{210}\right)\) | \(e\left(\frac{71}{420}\right)\) | \(e\left(\frac{277}{280}\right)\) | \(e\left(\frac{103}{140}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{17}{210}\right)\) | \(e\left(\frac{353}{840}\right)\) | \(e\left(\frac{757}{840}\right)\) |
| \(\chi_{2009}(138,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{420}\right)\) | \(e\left(\frac{71}{168}\right)\) | \(e\left(\frac{121}{210}\right)\) | \(e\left(\frac{97}{420}\right)\) | \(e\left(\frac{199}{280}\right)\) | \(e\left(\frac{121}{140}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{109}{210}\right)\) | \(e\left(\frac{571}{840}\right)\) | \(e\left(\frac{839}{840}\right)\) |
| \(\chi_{2009}(145,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{397}{420}\right)\) | \(e\left(\frac{167}{168}\right)\) | \(e\left(\frac{187}{210}\right)\) | \(e\left(\frac{169}{420}\right)\) | \(e\left(\frac{263}{280}\right)\) | \(e\left(\frac{117}{140}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{73}{210}\right)\) | \(e\left(\frac{787}{840}\right)\) | \(e\left(\frac{743}{840}\right)\) |
| \(\chi_{2009}(152,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{211}{420}\right)\) | \(e\left(\frac{53}{168}\right)\) | \(e\left(\frac{1}{210}\right)\) | \(e\left(\frac{367}{420}\right)\) | \(e\left(\frac{229}{280}\right)\) | \(e\left(\frac{71}{140}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{79}{210}\right)\) | \(e\left(\frac{121}{840}\right)\) | \(e\left(\frac{269}{840}\right)\) |
| \(\chi_{2009}(157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{167}{420}\right)\) | \(e\left(\frac{73}{168}\right)\) | \(e\left(\frac{167}{210}\right)\) | \(e\left(\frac{179}{420}\right)\) | \(e\left(\frac{233}{280}\right)\) | \(e\left(\frac{27}{140}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{173}{210}\right)\) | \(e\left(\frac{677}{840}\right)\) | \(e\left(\frac{193}{840}\right)\) |
| \(\chi_{2009}(171,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{420}\right)\) | \(e\left(\frac{85}{168}\right)\) | \(e\left(\frac{107}{210}\right)\) | \(e\left(\frac{419}{420}\right)\) | \(e\left(\frac{213}{280}\right)\) | \(e\left(\frac{107}{140}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{53}{210}\right)\) | \(e\left(\frac{137}{840}\right)\) | \(e\left(\frac{13}{840}\right)\) |
| \(\chi_{2009}(192,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{143}{420}\right)\) | \(e\left(\frac{145}{168}\right)\) | \(e\left(\frac{143}{210}\right)\) | \(e\left(\frac{191}{420}\right)\) | \(e\left(\frac{57}{280}\right)\) | \(e\left(\frac{3}{140}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{167}{210}\right)\) | \(e\left(\frac{293}{840}\right)\) | \(e\left(\frac{457}{840}\right)\) |
| \(\chi_{2009}(194,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{420}\right)\) | \(e\left(\frac{125}{168}\right)\) | \(e\left(\frac{19}{210}\right)\) | \(e\left(\frac{43}{420}\right)\) | \(e\left(\frac{221}{280}\right)\) | \(e\left(\frac{19}{140}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{31}{210}\right)\) | \(e\left(\frac{409}{840}\right)\) | \(e\left(\frac{701}{840}\right)\) |
| \(\chi_{2009}(199,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{420}\right)\) | \(e\left(\frac{151}{168}\right)\) | \(e\left(\frac{113}{210}\right)\) | \(e\left(\frac{101}{420}\right)\) | \(e\left(\frac{47}{280}\right)\) | \(e\left(\frac{113}{140}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{107}{210}\right)\) | \(e\left(\frac{443}{840}\right)\) | \(e\left(\frac{367}{840}\right)\) |
| \(\chi_{2009}(220,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{401}{420}\right)\) | \(e\left(\frac{127}{168}\right)\) | \(e\left(\frac{191}{210}\right)\) | \(e\left(\frac{377}{420}\right)\) | \(e\left(\frac{199}{280}\right)\) | \(e\left(\frac{121}{140}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{179}{210}\right)\) | \(e\left(\frac{11}{840}\right)\) | \(e\left(\frac{559}{840}\right)\) |
| \(\chi_{2009}(222,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{409}{420}\right)\) | \(e\left(\frac{131}{168}\right)\) | \(e\left(\frac{199}{210}\right)\) | \(e\left(\frac{373}{420}\right)\) | \(e\left(\frac{211}{280}\right)\) | \(e\left(\frac{129}{140}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{181}{210}\right)\) | \(e\left(\frac{559}{840}\right)\) | \(e\left(\frac{611}{840}\right)\) |
| \(\chi_{2009}(229,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{349}{420}\right)\) | \(e\left(\frac{143}{168}\right)\) | \(e\left(\frac{139}{210}\right)\) | \(e\left(\frac{193}{420}\right)\) | \(e\left(\frac{191}{280}\right)\) | \(e\left(\frac{69}{140}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{61}{210}\right)\) | \(e\left(\frac{19}{840}\right)\) | \(e\left(\frac{431}{840}\right)\) |
| \(\chi_{2009}(234,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{420}\right)\) | \(e\left(\frac{13}{168}\right)\) | \(e\left(\frac{131}{210}\right)\) | \(e\left(\frac{407}{420}\right)\) | \(e\left(\frac{109}{280}\right)\) | \(e\left(\frac{131}{140}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{59}{210}\right)\) | \(e\left(\frac{521}{840}\right)\) | \(e\left(\frac{589}{840}\right)\) |
| \(\chi_{2009}(257,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{319}{420}\right)\) | \(e\left(\frac{65}{168}\right)\) | \(e\left(\frac{109}{210}\right)\) | \(e\left(\frac{103}{420}\right)\) | \(e\left(\frac{41}{280}\right)\) | \(e\left(\frac{39}{140}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{1}{210}\right)\) | \(e\left(\frac{589}{840}\right)\) | \(e\left(\frac{761}{840}\right)\) |
| \(\chi_{2009}(299,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{211}{420}\right)\) | \(e\left(\frac{137}{168}\right)\) | \(e\left(\frac{1}{210}\right)\) | \(e\left(\frac{367}{420}\right)\) | \(e\left(\frac{89}{280}\right)\) | \(e\left(\frac{71}{140}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{79}{210}\right)\) | \(e\left(\frac{541}{840}\right)\) | \(e\left(\frac{689}{840}\right)\) |
| \(\chi_{2009}(304,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{209}{420}\right)\) | \(e\left(\frac{115}{168}\right)\) | \(e\left(\frac{209}{210}\right)\) | \(e\left(\frac{53}{420}\right)\) | \(e\left(\frac{51}{280}\right)\) | \(e\left(\frac{69}{140}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{131}{210}\right)\) | \(e\left(\frac{719}{840}\right)\) | \(e\left(\frac{571}{840}\right)\) |
| \(\chi_{2009}(306,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{277}{420}\right)\) | \(e\left(\frac{107}{168}\right)\) | \(e\left(\frac{67}{210}\right)\) | \(e\left(\frac{229}{420}\right)\) | \(e\left(\frac{83}{280}\right)\) | \(e\left(\frac{137}{140}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{43}{210}\right)\) | \(e\left(\frac{127}{840}\right)\) | \(e\left(\frac{803}{840}\right)\) |
| \(\chi_{2009}(311,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{389}{420}\right)\) | \(e\left(\frac{79}{168}\right)\) | \(e\left(\frac{179}{210}\right)\) | \(e\left(\frac{173}{420}\right)\) | \(e\left(\frac{111}{280}\right)\) | \(e\left(\frac{109}{140}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{71}{210}\right)\) | \(e\left(\frac{659}{840}\right)\) | \(e\left(\frac{271}{840}\right)\) |