from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7360, base_ring=CyclotomicField(176))
M = H._module
chi = DirichletCharacter(H, M([0,99,44,168]))
chi.galois_orbit()
[g,chi] = znchar(Mod(37,7360))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(7360\) | |
| Conductor: | \(7360\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(176\) | 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: | $\Q(\zeta_{176})$ |
| Fixed field: | Number field defined by a degree 176 polynomial (not computed) |
First 31 of 80 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{7360}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{125}{176}\right)\) | \(e\left(\frac{1}{88}\right)\) | \(e\left(\frac{37}{88}\right)\) | \(e\left(\frac{71}{176}\right)\) | \(e\left(\frac{97}{176}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{133}{176}\right)\) | \(e\left(\frac{127}{176}\right)\) | \(e\left(\frac{23}{176}\right)\) | \(e\left(\frac{153}{176}\right)\) |
| \(\chi_{7360}(333,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{176}\right)\) | \(e\left(\frac{79}{88}\right)\) | \(e\left(\frac{19}{88}\right)\) | \(e\left(\frac{65}{176}\right)\) | \(e\left(\frac{7}{176}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{35}{176}\right)\) | \(e\left(\frac{89}{176}\right)\) | \(e\left(\frac{145}{176}\right)\) | \(e\left(\frac{31}{176}\right)\) |
| \(\chi_{7360}(493,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{176}\right)\) | \(e\left(\frac{63}{88}\right)\) | \(e\left(\frac{43}{88}\right)\) | \(e\left(\frac{73}{176}\right)\) | \(e\left(\frac{127}{176}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{107}{176}\right)\) | \(e\left(\frac{81}{176}\right)\) | \(e\left(\frac{41}{176}\right)\) | \(e\left(\frac{135}{176}\right)\) |
| \(\chi_{7360}(517,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{176}\right)\) | \(e\left(\frac{57}{88}\right)\) | \(e\left(\frac{85}{88}\right)\) | \(e\left(\frac{175}{176}\right)\) | \(e\left(\frac{73}{176}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{13}{176}\right)\) | \(e\left(\frac{23}{176}\right)\) | \(e\left(\frac{79}{176}\right)\) | \(e\left(\frac{97}{176}\right)\) |
| \(\chi_{7360}(573,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{176}\right)\) | \(e\left(\frac{75}{88}\right)\) | \(e\left(\frac{47}{88}\right)\) | \(e\left(\frac{45}{176}\right)\) | \(e\left(\frac{59}{176}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{119}{176}\right)\) | \(e\left(\frac{21}{176}\right)\) | \(e\left(\frac{141}{176}\right)\) | \(e\left(\frac{35}{176}\right)\) |
| \(\chi_{7360}(677,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{176}\right)\) | \(e\left(\frac{41}{88}\right)\) | \(e\left(\frac{21}{88}\right)\) | \(e\left(\frac{7}{176}\right)\) | \(e\left(\frac{17}{176}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{85}{176}\right)\) | \(e\left(\frac{15}{176}\right)\) | \(e\left(\frac{151}{176}\right)\) | \(e\left(\frac{25}{176}\right)\) |
| \(\chi_{7360}(733,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{167}{176}\right)\) | \(e\left(\frac{83}{88}\right)\) | \(e\left(\frac{79}{88}\right)\) | \(e\left(\frac{85}{176}\right)\) | \(e\left(\frac{131}{176}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{127}{176}\right)\) | \(e\left(\frac{157}{176}\right)\) | \(e\left(\frac{149}{176}\right)\) | \(e\left(\frac{27}{176}\right)\) |
| \(\chi_{7360}(757,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{176}\right)\) | \(e\left(\frac{53}{88}\right)\) | \(e\left(\frac{25}{88}\right)\) | \(e\left(\frac{155}{176}\right)\) | \(e\left(\frac{125}{176}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{97}{176}\right)\) | \(e\left(\frac{131}{176}\right)\) | \(e\left(\frac{75}{176}\right)\) | \(e\left(\frac{101}{176}\right)\) |
| \(\chi_{7360}(893,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{176}\right)\) | \(e\left(\frac{51}{88}\right)\) | \(e\left(\frac{39}{88}\right)\) | \(e\left(\frac{13}{176}\right)\) | \(e\left(\frac{107}{176}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{7}{176}\right)\) | \(e\left(\frac{53}{176}\right)\) | \(e\left(\frac{29}{176}\right)\) | \(e\left(\frac{147}{176}\right)\) |
| \(\chi_{7360}(917,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{145}{176}\right)\) | \(e\left(\frac{61}{88}\right)\) | \(e\left(\frac{57}{88}\right)\) | \(e\left(\frac{19}{176}\right)\) | \(e\left(\frac{21}{176}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{105}{176}\right)\) | \(e\left(\frac{91}{176}\right)\) | \(e\left(\frac{83}{176}\right)\) | \(e\left(\frac{93}{176}\right)\) |
| \(\chi_{7360}(973,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{155}{176}\right)\) | \(e\left(\frac{47}{88}\right)\) | \(e\left(\frac{67}{88}\right)\) | \(e\left(\frac{81}{176}\right)\) | \(e\left(\frac{71}{176}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{3}{176}\right)\) | \(e\left(\frac{73}{176}\right)\) | \(e\left(\frac{113}{176}\right)\) | \(e\left(\frac{63}{176}\right)\) |
| \(\chi_{7360}(1077,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{105}{176}\right)\) | \(e\left(\frac{29}{88}\right)\) | \(e\left(\frac{17}{88}\right)\) | \(e\left(\frac{123}{176}\right)\) | \(e\left(\frac{173}{176}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{161}{176}\right)\) | \(e\left(\frac{163}{176}\right)\) | \(e\left(\frac{139}{176}\right)\) | \(e\left(\frac{37}{176}\right)\) |
| \(\chi_{7360}(1157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{133}{176}\right)\) | \(e\left(\frac{25}{88}\right)\) | \(e\left(\frac{45}{88}\right)\) | \(e\left(\frac{15}{176}\right)\) | \(e\left(\frac{137}{176}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{157}{176}\right)\) | \(e\left(\frac{7}{176}\right)\) | \(e\left(\frac{47}{176}\right)\) | \(e\left(\frac{129}{176}\right)\) |
| \(\chi_{7360}(1213,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{159}{176}\right)\) | \(e\left(\frac{59}{88}\right)\) | \(e\left(\frac{71}{88}\right)\) | \(e\left(\frac{141}{176}\right)\) | \(e\left(\frac{91}{176}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{103}{176}\right)\) | \(e\left(\frac{101}{176}\right)\) | \(e\left(\frac{125}{176}\right)\) | \(e\left(\frac{51}{176}\right)\) |
| \(\chi_{7360}(1293,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{139}{176}\right)\) | \(e\left(\frac{87}{88}\right)\) | \(e\left(\frac{51}{88}\right)\) | \(e\left(\frac{17}{176}\right)\) | \(e\left(\frac{167}{176}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{131}{176}\right)\) | \(e\left(\frac{137}{176}\right)\) | \(e\left(\frac{65}{176}\right)\) | \(e\left(\frac{111}{176}\right)\) |
| \(\chi_{7360}(1397,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{176}\right)\) | \(e\left(\frac{37}{88}\right)\) | \(e\left(\frac{49}{88}\right)\) | \(e\left(\frac{75}{176}\right)\) | \(e\left(\frac{157}{176}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{81}{176}\right)\) | \(e\left(\frac{35}{176}\right)\) | \(e\left(\frac{59}{176}\right)\) | \(e\left(\frac{117}{176}\right)\) |
| \(\chi_{7360}(1477,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{117}{176}\right)\) | \(e\left(\frac{65}{88}\right)\) | \(e\left(\frac{29}{88}\right)\) | \(e\left(\frac{127}{176}\right)\) | \(e\left(\frac{57}{176}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{109}{176}\right)\) | \(e\left(\frac{71}{176}\right)\) | \(e\left(\frac{175}{176}\right)\) | \(e\left(\frac{1}{176}\right)\) |
| \(\chi_{7360}(1533,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{176}\right)\) | \(e\left(\frac{27}{88}\right)\) | \(e\left(\frac{31}{88}\right)\) | \(e\left(\frac{157}{176}\right)\) | \(e\left(\frac{155}{176}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{71}{176}\right)\) | \(e\left(\frac{85}{176}\right)\) | \(e\left(\frac{93}{176}\right)\) | \(e\left(\frac{83}{176}\right)\) |
| \(\chi_{7360}(1693,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{176}\right)\) | \(e\left(\frac{67}{88}\right)\) | \(e\left(\frac{15}{88}\right)\) | \(e\left(\frac{5}{176}\right)\) | \(e\left(\frac{163}{176}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{111}{176}\right)\) | \(e\left(\frac{61}{176}\right)\) | \(e\left(\frac{133}{176}\right)\) | \(e\left(\frac{43}{176}\right)\) |
| \(\chi_{7360}(1717,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{176}\right)\) | \(e\left(\frac{5}{88}\right)\) | \(e\left(\frac{9}{88}\right)\) | \(e\left(\frac{91}{176}\right)\) | \(e\left(\frac{45}{176}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{49}{176}\right)\) | \(e\left(\frac{19}{176}\right)\) | \(e\left(\frac{27}{176}\right)\) | \(e\left(\frac{149}{176}\right)\) |
| \(\chi_{7360}(1877,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{176}\right)\) | \(e\left(\frac{45}{88}\right)\) | \(e\left(\frac{81}{88}\right)\) | \(e\left(\frac{115}{176}\right)\) | \(e\left(\frac{53}{176}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{89}{176}\right)\) | \(e\left(\frac{171}{176}\right)\) | \(e\left(\frac{67}{176}\right)\) | \(e\left(\frac{109}{176}\right)\) |
| \(\chi_{7360}(2173,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{176}\right)\) | \(e\left(\frac{35}{88}\right)\) | \(e\left(\frac{63}{88}\right)\) | \(e\left(\frac{109}{176}\right)\) | \(e\left(\frac{139}{176}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{167}{176}\right)\) | \(e\left(\frac{133}{176}\right)\) | \(e\left(\frac{13}{176}\right)\) | \(e\left(\frac{163}{176}\right)\) |
| \(\chi_{7360}(2333,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{87}{176}\right)\) | \(e\left(\frac{19}{88}\right)\) | \(e\left(\frac{87}{88}\right)\) | \(e\left(\frac{117}{176}\right)\) | \(e\left(\frac{83}{176}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{63}{176}\right)\) | \(e\left(\frac{125}{176}\right)\) | \(e\left(\frac{85}{176}\right)\) | \(e\left(\frac{91}{176}\right)\) |
| \(\chi_{7360}(2357,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{176}\right)\) | \(e\left(\frac{13}{88}\right)\) | \(e\left(\frac{41}{88}\right)\) | \(e\left(\frac{43}{176}\right)\) | \(e\left(\frac{29}{176}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{145}{176}\right)\) | \(e\left(\frac{67}{176}\right)\) | \(e\left(\frac{123}{176}\right)\) | \(e\left(\frac{53}{176}\right)\) |
| \(\chi_{7360}(2413,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{176}\right)\) | \(e\left(\frac{31}{88}\right)\) | \(e\left(\frac{3}{88}\right)\) | \(e\left(\frac{89}{176}\right)\) | \(e\left(\frac{15}{176}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{75}{176}\right)\) | \(e\left(\frac{65}{176}\right)\) | \(e\left(\frac{9}{176}\right)\) | \(e\left(\frac{167}{176}\right)\) |
| \(\chi_{7360}(2517,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{176}\right)\) | \(e\left(\frac{85}{88}\right)\) | \(e\left(\frac{65}{88}\right)\) | \(e\left(\frac{51}{176}\right)\) | \(e\left(\frac{149}{176}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{41}{176}\right)\) | \(e\left(\frac{59}{176}\right)\) | \(e\left(\frac{19}{176}\right)\) | \(e\left(\frac{157}{176}\right)\) |
| \(\chi_{7360}(2573,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{123}{176}\right)\) | \(e\left(\frac{39}{88}\right)\) | \(e\left(\frac{35}{88}\right)\) | \(e\left(\frac{129}{176}\right)\) | \(e\left(\frac{87}{176}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{83}{176}\right)\) | \(e\left(\frac{25}{176}\right)\) | \(e\left(\frac{17}{176}\right)\) | \(e\left(\frac{159}{176}\right)\) |
| \(\chi_{7360}(2597,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{157}{176}\right)\) | \(e\left(\frac{9}{88}\right)\) | \(e\left(\frac{69}{88}\right)\) | \(e\left(\frac{23}{176}\right)\) | \(e\left(\frac{81}{176}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{53}{176}\right)\) | \(e\left(\frac{175}{176}\right)\) | \(e\left(\frac{119}{176}\right)\) | \(e\left(\frac{57}{176}\right)\) |
| \(\chi_{7360}(2733,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{176}\right)\) | \(e\left(\frac{7}{88}\right)\) | \(e\left(\frac{83}{88}\right)\) | \(e\left(\frac{57}{176}\right)\) | \(e\left(\frac{63}{176}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{139}{176}\right)\) | \(e\left(\frac{97}{176}\right)\) | \(e\left(\frac{73}{176}\right)\) | \(e\left(\frac{103}{176}\right)\) |
| \(\chi_{7360}(2757,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{176}\right)\) | \(e\left(\frac{17}{88}\right)\) | \(e\left(\frac{13}{88}\right)\) | \(e\left(\frac{63}{176}\right)\) | \(e\left(\frac{153}{176}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{61}{176}\right)\) | \(e\left(\frac{135}{176}\right)\) | \(e\left(\frac{127}{176}\right)\) | \(e\left(\frac{49}{176}\right)\) |
| \(\chi_{7360}(2813,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{111}{176}\right)\) | \(e\left(\frac{3}{88}\right)\) | \(e\left(\frac{23}{88}\right)\) | \(e\left(\frac{125}{176}\right)\) | \(e\left(\frac{27}{176}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{135}{176}\right)\) | \(e\left(\frac{117}{176}\right)\) | \(e\left(\frac{157}{176}\right)\) | \(e\left(\frac{19}{176}\right)\) |