from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7497, base_ring=CyclotomicField(336))
M = H._module
chi = DirichletCharacter(H, M([56,320,147]))
chi.galois_orbit()
[g,chi] = znchar(Mod(11,7497))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(7497\) | |
| Conductor: | \(7497\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(336\) | 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: | $\Q(\zeta_{336})$ |
| Fixed field: | Number field defined by a degree 336 polynomial (not computed) |
First 31 of 96 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(19\) | \(20\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{7497}(11,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{215}{336}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{233}{336}\right)\) | \(e\left(\frac{109}{336}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{251}{336}\right)\) |
| \(\chi_{7497}(23,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{31}{336}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{193}{336}\right)\) | \(e\left(\frac{197}{336}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{19}{336}\right)\) |
| \(\chi_{7497}(74,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{191}{336}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{257}{336}\right)\) | \(e\left(\frac{325}{336}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{323}{336}\right)\) |
| \(\chi_{7497}(326,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{305}{336}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{143}{336}\right)\) | \(e\left(\frac{139}{336}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{317}{336}\right)\) |
| \(\chi_{7497}(401,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{163}{336}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{61}{336}\right)\) | \(e\left(\frac{17}{336}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{295}{336}\right)\) |
| \(\chi_{7497}(452,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{131}{336}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{317}{336}\right)\) | \(e\left(\frac{193}{336}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{167}{336}\right)\) |
| \(\chi_{7497}(464,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{325}{336}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{235}{336}\right)\) | \(e\left(\frac{71}{336}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{145}{336}\right)\) |
| \(\chi_{7497}(515,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{149}{336}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{299}{336}\right)\) | \(e\left(\frac{199}{336}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{113}{336}\right)\) |
| \(\chi_{7497}(590,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{61}{336}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{163}{336}\right)\) | \(e\left(\frac{95}{336}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{265}{336}\right)\) |
| \(\chi_{7497}(641,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{269}{336}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{179}{336}\right)\) | \(e\left(\frac{127}{336}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{89}{336}\right)\) |
| \(\chi_{7497}(653,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{139}{336}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{85}{336}\right)\) | \(e\left(\frac{233}{336}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{31}{336}\right)\) |
| \(\chi_{7497}(779,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{169}{336}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{55}{336}\right)\) | \(e\left(\frac{131}{336}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{277}{336}\right)\) |
| \(\chi_{7497}(830,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{281}{336}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{167}{336}\right)\) | \(e\left(\frac{19}{336}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{53}{336}\right)\) |
| \(\chi_{7497}(1031,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{103}{336}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{121}{336}\right)\) | \(e\left(\frac{221}{336}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{139}{336}\right)\) |
| \(\chi_{7497}(1082,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{311}{336}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{137}{336}\right)\) | \(e\left(\frac{253}{336}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{299}{336}\right)\) |
| \(\chi_{7497}(1094,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{223}{336}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{1}{336}\right)\) | \(e\left(\frac{149}{336}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{115}{336}\right)\) |
| \(\chi_{7497}(1346,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{241}{336}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{319}{336}\right)\) | \(e\left(\frac{155}{336}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{61}{336}\right)\) |
| \(\chi_{7497}(1397,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{65}{336}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{47}{336}\right)\) | \(e\left(\frac{283}{336}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{29}{336}\right)\) |
| \(\chi_{7497}(1472,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{19}{336}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{205}{336}\right)\) | \(e\left(\frac{305}{336}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{55}{336}\right)\) |
| \(\chi_{7497}(1523,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{227}{336}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{221}{336}\right)\) | \(e\left(\frac{1}{336}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{215}{336}\right)\) |
| \(\chi_{7497}(1535,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{181}{336}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{43}{336}\right)\) | \(e\left(\frac{23}{336}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{241}{336}\right)\) |
| \(\chi_{7497}(1661,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{253}{336}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{307}{336}\right)\) | \(e\left(\frac{47}{336}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{25}{336}\right)\) |
| \(\chi_{7497}(1712,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{29}{336}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{83}{336}\right)\) | \(e\left(\frac{271}{336}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{137}{336}\right)\) |
| \(\chi_{7497}(1724,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{331}{336}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{229}{336}\right)\) | \(e\left(\frac{185}{336}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{127}{336}\right)\) |
| \(\chi_{7497}(1775,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{299}{336}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{149}{336}\right)\) | \(e\left(\frac{25}{336}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{335}{336}\right)\) |
| \(\chi_{7497}(1850,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{25}{336}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{199}{336}\right)\) | \(e\left(\frac{83}{336}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{37}{336}\right)\) |
| \(\chi_{7497}(1901,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{41}{336}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{71}{336}\right)\) | \(e\left(\frac{163}{336}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{101}{336}\right)\) |
| \(\chi_{7497}(2102,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{295}{336}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{265}{336}\right)\) | \(e\left(\frac{173}{336}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{235}{336}\right)\) |
| \(\chi_{7497}(2153,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{71}{336}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{41}{336}\right)\) | \(e\left(\frac{61}{336}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{11}{336}\right)\) |
| \(\chi_{7497}(2165,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{79}{336}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{145}{336}\right)\) | \(e\left(\frac{101}{336}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{211}{336}\right)\) |
| \(\chi_{7497}(2216,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{47}{336}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{65}{336}\right)\) | \(e\left(\frac{277}{336}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{83}{336}\right)\) |