Properties

Modulus 709
Structure \(C_{708}\)
Order 708

Learn more about

Show commands for: SageMath / Pari/GP

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(709)
pari: g = idealstar(,709,2)

Character group

sage: G.order()
pari: g.no
Order = 708
sage: H.invariants()
pari: g.cyc
Structure = \(C_{708}\)
sage: H.gens()
pari: g.gen
Generators = $\chi_{709}(2,\cdot)$

First 32 of 708 characters

Each row describes a character. When available, the columns show the orbit label, order of the character, whether the character is primitive, and several values of the character.

orbit label order primitive -1 1 2 3 4 5 6 7 8 9 10 11
\(\chi_{709}(1,\cdot)\) 709.a 1 No \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{709}(2,\cdot)\) 709.l 708 Yes \(-1\) \(1\) \(e\left(\frac{1}{708}\right)\) \(e\left(\frac{43}{177}\right)\) \(e\left(\frac{1}{354}\right)\) \(e\left(\frac{161}{354}\right)\) \(e\left(\frac{173}{708}\right)\) \(e\left(\frac{13}{177}\right)\) \(e\left(\frac{1}{236}\right)\) \(e\left(\frac{86}{177}\right)\) \(e\left(\frac{323}{708}\right)\) \(e\left(\frac{335}{354}\right)\)
\(\chi_{709}(3,\cdot)\) 709.i 177 Yes \(1\) \(1\) \(e\left(\frac{43}{177}\right)\) \(e\left(\frac{139}{177}\right)\) \(e\left(\frac{86}{177}\right)\) \(e\left(\frac{40}{177}\right)\) \(e\left(\frac{5}{177}\right)\) \(e\left(\frac{112}{177}\right)\) \(e\left(\frac{43}{59}\right)\) \(e\left(\frac{101}{177}\right)\) \(e\left(\frac{83}{177}\right)\) \(e\left(\frac{136}{177}\right)\)
\(\chi_{709}(4,\cdot)\) 709.k 354 Yes \(1\) \(1\) \(e\left(\frac{1}{354}\right)\) \(e\left(\frac{86}{177}\right)\) \(e\left(\frac{1}{177}\right)\) \(e\left(\frac{161}{177}\right)\) \(e\left(\frac{173}{354}\right)\) \(e\left(\frac{26}{177}\right)\) \(e\left(\frac{1}{118}\right)\) \(e\left(\frac{172}{177}\right)\) \(e\left(\frac{323}{354}\right)\) \(e\left(\frac{158}{177}\right)\)
\(\chi_{709}(5,\cdot)\) 709.k 354 Yes \(1\) \(1\) \(e\left(\frac{161}{354}\right)\) \(e\left(\frac{40}{177}\right)\) \(e\left(\frac{161}{177}\right)\) \(e\left(\frac{79}{177}\right)\) \(e\left(\frac{241}{354}\right)\) \(e\left(\frac{115}{177}\right)\) \(e\left(\frac{43}{118}\right)\) \(e\left(\frac{80}{177}\right)\) \(e\left(\frac{319}{354}\right)\) \(e\left(\frac{127}{177}\right)\)
\(\chi_{709}(6,\cdot)\) 709.l 708 Yes \(-1\) \(1\) \(e\left(\frac{173}{708}\right)\) \(e\left(\frac{5}{177}\right)\) \(e\left(\frac{173}{354}\right)\) \(e\left(\frac{241}{354}\right)\) \(e\left(\frac{193}{708}\right)\) \(e\left(\frac{125}{177}\right)\) \(e\left(\frac{173}{236}\right)\) \(e\left(\frac{10}{177}\right)\) \(e\left(\frac{655}{708}\right)\) \(e\left(\frac{253}{354}\right)\)
\(\chi_{709}(7,\cdot)\) 709.i 177 Yes \(1\) \(1\) \(e\left(\frac{13}{177}\right)\) \(e\left(\frac{112}{177}\right)\) \(e\left(\frac{26}{177}\right)\) \(e\left(\frac{115}{177}\right)\) \(e\left(\frac{125}{177}\right)\) \(e\left(\frac{145}{177}\right)\) \(e\left(\frac{13}{59}\right)\) \(e\left(\frac{47}{177}\right)\) \(e\left(\frac{128}{177}\right)\) \(e\left(\frac{37}{177}\right)\)
\(\chi_{709}(8,\cdot)\) 709.j 236 Yes \(-1\) \(1\) \(e\left(\frac{1}{236}\right)\) \(e\left(\frac{43}{59}\right)\) \(e\left(\frac{1}{118}\right)\) \(e\left(\frac{43}{118}\right)\) \(e\left(\frac{173}{236}\right)\) \(e\left(\frac{13}{59}\right)\) \(e\left(\frac{3}{236}\right)\) \(e\left(\frac{27}{59}\right)\) \(e\left(\frac{87}{236}\right)\) \(e\left(\frac{99}{118}\right)\)
\(\chi_{709}(9,\cdot)\) 709.i 177 Yes \(1\) \(1\) \(e\left(\frac{86}{177}\right)\) \(e\left(\frac{101}{177}\right)\) \(e\left(\frac{172}{177}\right)\) \(e\left(\frac{80}{177}\right)\) \(e\left(\frac{10}{177}\right)\) \(e\left(\frac{47}{177}\right)\) \(e\left(\frac{27}{59}\right)\) \(e\left(\frac{25}{177}\right)\) \(e\left(\frac{166}{177}\right)\) \(e\left(\frac{95}{177}\right)\)
\(\chi_{709}(10,\cdot)\) 709.l 708 Yes \(-1\) \(1\) \(e\left(\frac{323}{708}\right)\) \(e\left(\frac{83}{177}\right)\) \(e\left(\frac{323}{354}\right)\) \(e\left(\frac{319}{354}\right)\) \(e\left(\frac{655}{708}\right)\) \(e\left(\frac{128}{177}\right)\) \(e\left(\frac{87}{236}\right)\) \(e\left(\frac{166}{177}\right)\) \(e\left(\frac{253}{708}\right)\) \(e\left(\frac{235}{354}\right)\)
\(\chi_{709}(11,\cdot)\) 709.k 354 Yes \(1\) \(1\) \(e\left(\frac{335}{354}\right)\) \(e\left(\frac{136}{177}\right)\) \(e\left(\frac{158}{177}\right)\) \(e\left(\frac{127}{177}\right)\) \(e\left(\frac{253}{354}\right)\) \(e\left(\frac{37}{177}\right)\) \(e\left(\frac{99}{118}\right)\) \(e\left(\frac{95}{177}\right)\) \(e\left(\frac{235}{354}\right)\) \(e\left(\frac{7}{177}\right)\)
\(\chi_{709}(12,\cdot)\) 709.h 118 Yes \(1\) \(1\) \(e\left(\frac{29}{118}\right)\) \(e\left(\frac{16}{59}\right)\) \(e\left(\frac{29}{59}\right)\) \(e\left(\frac{8}{59}\right)\) \(e\left(\frac{61}{118}\right)\) \(e\left(\frac{46}{59}\right)\) \(e\left(\frac{87}{118}\right)\) \(e\left(\frac{32}{59}\right)\) \(e\left(\frac{45}{118}\right)\) \(e\left(\frac{39}{59}\right)\)
\(\chi_{709}(13,\cdot)\) 709.j 236 Yes \(-1\) \(1\) \(e\left(\frac{199}{236}\right)\) \(e\left(\frac{2}{59}\right)\) \(e\left(\frac{81}{118}\right)\) \(e\left(\frac{61}{118}\right)\) \(e\left(\frac{207}{236}\right)\) \(e\left(\frac{50}{59}\right)\) \(e\left(\frac{125}{236}\right)\) \(e\left(\frac{4}{59}\right)\) \(e\left(\frac{85}{236}\right)\) \(e\left(\frac{113}{118}\right)\)
\(\chi_{709}(14,\cdot)\) 709.l 708 Yes \(-1\) \(1\) \(e\left(\frac{53}{708}\right)\) \(e\left(\frac{155}{177}\right)\) \(e\left(\frac{53}{354}\right)\) \(e\left(\frac{37}{354}\right)\) \(e\left(\frac{673}{708}\right)\) \(e\left(\frac{158}{177}\right)\) \(e\left(\frac{53}{236}\right)\) \(e\left(\frac{133}{177}\right)\) \(e\left(\frac{127}{708}\right)\) \(e\left(\frac{55}{354}\right)\)
\(\chi_{709}(15,\cdot)\) 709.k 354 Yes \(1\) \(1\) \(e\left(\frac{247}{354}\right)\) \(e\left(\frac{2}{177}\right)\) \(e\left(\frac{70}{177}\right)\) \(e\left(\frac{119}{177}\right)\) \(e\left(\frac{251}{354}\right)\) \(e\left(\frac{50}{177}\right)\) \(e\left(\frac{11}{118}\right)\) \(e\left(\frac{4}{177}\right)\) \(e\left(\frac{131}{354}\right)\) \(e\left(\frac{86}{177}\right)\)
\(\chi_{709}(16,\cdot)\) 709.i 177 Yes \(1\) \(1\) \(e\left(\frac{1}{177}\right)\) \(e\left(\frac{172}{177}\right)\) \(e\left(\frac{2}{177}\right)\) \(e\left(\frac{145}{177}\right)\) \(e\left(\frac{173}{177}\right)\) \(e\left(\frac{52}{177}\right)\) \(e\left(\frac{1}{59}\right)\) \(e\left(\frac{167}{177}\right)\) \(e\left(\frac{146}{177}\right)\) \(e\left(\frac{139}{177}\right)\)
\(\chi_{709}(17,\cdot)\) 709.l 708 Yes \(-1\) \(1\) \(e\left(\frac{97}{708}\right)\) \(e\left(\frac{100}{177}\right)\) \(e\left(\frac{97}{354}\right)\) \(e\left(\frac{41}{354}\right)\) \(e\left(\frac{497}{708}\right)\) \(e\left(\frac{22}{177}\right)\) \(e\left(\frac{97}{236}\right)\) \(e\left(\frac{23}{177}\right)\) \(e\left(\frac{179}{708}\right)\) \(e\left(\frac{281}{354}\right)\)
\(\chi_{709}(18,\cdot)\) 709.j 236 Yes \(-1\) \(1\) \(e\left(\frac{115}{236}\right)\) \(e\left(\frac{48}{59}\right)\) \(e\left(\frac{115}{118}\right)\) \(e\left(\frac{107}{118}\right)\) \(e\left(\frac{71}{236}\right)\) \(e\left(\frac{20}{59}\right)\) \(e\left(\frac{109}{236}\right)\) \(e\left(\frac{37}{59}\right)\) \(e\left(\frac{93}{236}\right)\) \(e\left(\frac{57}{118}\right)\)
\(\chi_{709}(19,\cdot)\) 709.i 177 Yes \(1\) \(1\) \(e\left(\frac{163}{177}\right)\) \(e\left(\frac{70}{177}\right)\) \(e\left(\frac{149}{177}\right)\) \(e\left(\frac{94}{177}\right)\) \(e\left(\frac{56}{177}\right)\) \(e\left(\frac{157}{177}\right)\) \(e\left(\frac{45}{59}\right)\) \(e\left(\frac{140}{177}\right)\) \(e\left(\frac{80}{177}\right)\) \(e\left(\frac{1}{177}\right)\)
\(\chi_{709}(20,\cdot)\) 709.g 59 Yes \(1\) \(1\) \(e\left(\frac{27}{59}\right)\) \(e\left(\frac{42}{59}\right)\) \(e\left(\frac{54}{59}\right)\) \(e\left(\frac{21}{59}\right)\) \(e\left(\frac{10}{59}\right)\) \(e\left(\frac{47}{59}\right)\) \(e\left(\frac{22}{59}\right)\) \(e\left(\frac{25}{59}\right)\) \(e\left(\frac{48}{59}\right)\) \(e\left(\frac{36}{59}\right)\)
\(\chi_{709}(21,\cdot)\) 709.i 177 Yes \(1\) \(1\) \(e\left(\frac{56}{177}\right)\) \(e\left(\frac{74}{177}\right)\) \(e\left(\frac{112}{177}\right)\) \(e\left(\frac{155}{177}\right)\) \(e\left(\frac{130}{177}\right)\) \(e\left(\frac{80}{177}\right)\) \(e\left(\frac{56}{59}\right)\) \(e\left(\frac{148}{177}\right)\) \(e\left(\frac{34}{177}\right)\) \(e\left(\frac{173}{177}\right)\)
\(\chi_{709}(22,\cdot)\) 709.l 708 Yes \(-1\) \(1\) \(e\left(\frac{671}{708}\right)\) \(e\left(\frac{2}{177}\right)\) \(e\left(\frac{317}{354}\right)\) \(e\left(\frac{61}{354}\right)\) \(e\left(\frac{679}{708}\right)\) \(e\left(\frac{50}{177}\right)\) \(e\left(\frac{199}{236}\right)\) \(e\left(\frac{4}{177}\right)\) \(e\left(\frac{85}{708}\right)\) \(e\left(\frac{349}{354}\right)\)
\(\chi_{709}(23,\cdot)\) 709.l 708 Yes \(-1\) \(1\) \(e\left(\frac{511}{708}\right)\) \(e\left(\frac{25}{177}\right)\) \(e\left(\frac{157}{354}\right)\) \(e\left(\frac{143}{354}\right)\) \(e\left(\frac{611}{708}\right)\) \(e\left(\frac{94}{177}\right)\) \(e\left(\frac{39}{236}\right)\) \(e\left(\frac{50}{177}\right)\) \(e\left(\frac{89}{708}\right)\) \(e\left(\frac{203}{354}\right)\)
\(\chi_{709}(24,\cdot)\) 709.l 708 Yes \(-1\) \(1\) \(e\left(\frac{175}{708}\right)\) \(e\left(\frac{91}{177}\right)\) \(e\left(\frac{175}{354}\right)\) \(e\left(\frac{209}{354}\right)\) \(e\left(\frac{539}{708}\right)\) \(e\left(\frac{151}{177}\right)\) \(e\left(\frac{175}{236}\right)\) \(e\left(\frac{5}{177}\right)\) \(e\left(\frac{593}{708}\right)\) \(e\left(\frac{215}{354}\right)\)
\(\chi_{709}(25,\cdot)\) 709.i 177 Yes \(1\) \(1\) \(e\left(\frac{161}{177}\right)\) \(e\left(\frac{80}{177}\right)\) \(e\left(\frac{145}{177}\right)\) \(e\left(\frac{158}{177}\right)\) \(e\left(\frac{64}{177}\right)\) \(e\left(\frac{53}{177}\right)\) \(e\left(\frac{43}{59}\right)\) \(e\left(\frac{160}{177}\right)\) \(e\left(\frac{142}{177}\right)\) \(e\left(\frac{77}{177}\right)\)
\(\chi_{709}(26,\cdot)\) 709.k 354 Yes \(1\) \(1\) \(e\left(\frac{299}{354}\right)\) \(e\left(\frac{49}{177}\right)\) \(e\left(\frac{122}{177}\right)\) \(e\left(\frac{172}{177}\right)\) \(e\left(\frac{43}{354}\right)\) \(e\left(\frac{163}{177}\right)\) \(e\left(\frac{63}{118}\right)\) \(e\left(\frac{98}{177}\right)\) \(e\left(\frac{289}{354}\right)\) \(e\left(\frac{160}{177}\right)\)
\(\chi_{709}(27,\cdot)\) 709.g 59 Yes \(1\) \(1\) \(e\left(\frac{43}{59}\right)\) \(e\left(\frac{21}{59}\right)\) \(e\left(\frac{27}{59}\right)\) \(e\left(\frac{40}{59}\right)\) \(e\left(\frac{5}{59}\right)\) \(e\left(\frac{53}{59}\right)\) \(e\left(\frac{11}{59}\right)\) \(e\left(\frac{42}{59}\right)\) \(e\left(\frac{24}{59}\right)\) \(e\left(\frac{18}{59}\right)\)
\(\chi_{709}(28,\cdot)\) 709.h 118 Yes \(1\) \(1\) \(e\left(\frac{9}{118}\right)\) \(e\left(\frac{7}{59}\right)\) \(e\left(\frac{9}{59}\right)\) \(e\left(\frac{33}{59}\right)\) \(e\left(\frac{23}{118}\right)\) \(e\left(\frac{57}{59}\right)\) \(e\left(\frac{27}{118}\right)\) \(e\left(\frac{14}{59}\right)\) \(e\left(\frac{75}{118}\right)\) \(e\left(\frac{6}{59}\right)\)
\(\chi_{709}(29,\cdot)\) 709.i 177 Yes \(1\) \(1\) \(e\left(\frac{17}{177}\right)\) \(e\left(\frac{92}{177}\right)\) \(e\left(\frac{34}{177}\right)\) \(e\left(\frac{164}{177}\right)\) \(e\left(\frac{109}{177}\right)\) \(e\left(\frac{176}{177}\right)\) \(e\left(\frac{17}{59}\right)\) \(e\left(\frac{7}{177}\right)\) \(e\left(\frac{4}{177}\right)\) \(e\left(\frac{62}{177}\right)\)
\(\chi_{709}(30,\cdot)\) 709.j 236 Yes \(-1\) \(1\) \(e\left(\frac{165}{236}\right)\) \(e\left(\frac{15}{59}\right)\) \(e\left(\frac{47}{118}\right)\) \(e\left(\frac{15}{118}\right)\) \(e\left(\frac{225}{236}\right)\) \(e\left(\frac{21}{59}\right)\) \(e\left(\frac{23}{236}\right)\) \(e\left(\frac{30}{59}\right)\) \(e\left(\frac{195}{236}\right)\) \(e\left(\frac{51}{118}\right)\)
\(\chi_{709}(31,\cdot)\) 709.l 708 Yes \(-1\) \(1\) \(e\left(\frac{199}{708}\right)\) \(e\left(\frac{61}{177}\right)\) \(e\left(\frac{199}{354}\right)\) \(e\left(\frac{179}{354}\right)\) \(e\left(\frac{443}{708}\right)\) \(e\left(\frac{109}{177}\right)\) \(e\left(\frac{199}{236}\right)\) \(e\left(\frac{122}{177}\right)\) \(e\left(\frac{557}{708}\right)\) \(e\left(\frac{113}{354}\right)\)
\(\chi_{709}(32,\cdot)\) 709.l 708 Yes \(-1\) \(1\) \(e\left(\frac{5}{708}\right)\) \(e\left(\frac{38}{177}\right)\) \(e\left(\frac{5}{354}\right)\) \(e\left(\frac{97}{354}\right)\) \(e\left(\frac{157}{708}\right)\) \(e\left(\frac{65}{177}\right)\) \(e\left(\frac{5}{236}\right)\) \(e\left(\frac{76}{177}\right)\) \(e\left(\frac{199}{708}\right)\) \(e\left(\frac{259}{354}\right)\)