sage: H = DirichletGroup(2048)
pari: g = idealstar(,2048,2)
Character group
| sage: G.order()
pari: g.no
| ||
| Order | = | 1024 |
| sage: H.invariants()
pari: g.cyc
| ||
| Structure | = | \(C_{2}\times C_{512}\) |
| sage: H.gens()
pari: g.gen
| ||
| Generators | = | $\chi_{2048}(2047,\cdot)$, $\chi_{2048}(5,\cdot)$ |
First 32 of 1024 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.
| Character | Orbit | Order | Primitive | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2048}(1,\cdot)\) | 2048.a | 1 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
| \(\chi_{2048}(3,\cdot)\) | 2048.t | 512 | yes | \(-1\) | \(1\) | \(e\left(\frac{201}{512}\right)\) | \(e\left(\frac{163}{512}\right)\) | \(e\left(\frac{207}{256}\right)\) | \(e\left(\frac{201}{256}\right)\) | \(e\left(\frac{159}{512}\right)\) | \(e\left(\frac{301}{512}\right)\) | \(e\left(\frac{91}{128}\right)\) | \(e\left(\frac{21}{128}\right)\) | \(e\left(\frac{37}{512}\right)\) | \(e\left(\frac{103}{512}\right)\) |
| \(\chi_{2048}(5,\cdot)\) | 2048.s | 512 | yes | \(1\) | \(1\) | \(e\left(\frac{163}{512}\right)\) | \(e\left(\frac{1}{512}\right)\) | \(e\left(\frac{101}{256}\right)\) | \(e\left(\frac{163}{256}\right)\) | \(e\left(\frac{213}{512}\right)\) | \(e\left(\frac{495}{512}\right)\) | \(e\left(\frac{41}{128}\right)\) | \(e\left(\frac{103}{128}\right)\) | \(e\left(\frac{407}{512}\right)\) | \(e\left(\frac{365}{512}\right)\) |
| \(\chi_{2048}(7,\cdot)\) | 2048.r | 256 | no | \(-1\) | \(1\) | \(e\left(\frac{207}{256}\right)\) | \(e\left(\frac{101}{256}\right)\) | \(e\left(\frac{25}{128}\right)\) | \(e\left(\frac{79}{128}\right)\) | \(e\left(\frac{137}{256}\right)\) | \(e\left(\frac{75}{256}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{19}{256}\right)\) | \(e\left(\frac{1}{256}\right)\) |
| \(\chi_{2048}(9,\cdot)\) | 2048.q | 256 | no | \(1\) | \(1\) | \(e\left(\frac{201}{256}\right)\) | \(e\left(\frac{163}{256}\right)\) | \(e\left(\frac{79}{128}\right)\) | \(e\left(\frac{73}{128}\right)\) | \(e\left(\frac{159}{256}\right)\) | \(e\left(\frac{45}{256}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{37}{256}\right)\) | \(e\left(\frac{103}{256}\right)\) |
| \(\chi_{2048}(11,\cdot)\) | 2048.t | 512 | yes | \(-1\) | \(1\) | \(e\left(\frac{159}{512}\right)\) | \(e\left(\frac{213}{512}\right)\) | \(e\left(\frac{137}{256}\right)\) | \(e\left(\frac{159}{256}\right)\) | \(e\left(\frac{57}{512}\right)\) | \(e\left(\frac{475}{512}\right)\) | \(e\left(\frac{93}{128}\right)\) | \(e\left(\frac{51}{128}\right)\) | \(e\left(\frac{419}{512}\right)\) | \(e\left(\frac{433}{512}\right)\) |
| \(\chi_{2048}(13,\cdot)\) | 2048.s | 512 | yes | \(1\) | \(1\) | \(e\left(\frac{301}{512}\right)\) | \(e\left(\frac{495}{512}\right)\) | \(e\left(\frac{75}{256}\right)\) | \(e\left(\frac{45}{256}\right)\) | \(e\left(\frac{475}{512}\right)\) | \(e\left(\frac{289}{512}\right)\) | \(e\left(\frac{71}{128}\right)\) | \(e\left(\frac{41}{128}\right)\) | \(e\left(\frac{249}{512}\right)\) | \(e\left(\frac{451}{512}\right)\) |
| \(\chi_{2048}(15,\cdot)\) | 2048.p | 128 | no | \(-1\) | \(1\) | \(e\left(\frac{91}{128}\right)\) | \(e\left(\frac{41}{128}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{93}{128}\right)\) | \(e\left(\frac{71}{128}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{111}{128}\right)\) | \(e\left(\frac{117}{128}\right)\) |
| \(\chi_{2048}(17,\cdot)\) | 2048.o | 128 | no | \(1\) | \(1\) | \(e\left(\frac{21}{128}\right)\) | \(e\left(\frac{103}{128}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{51}{128}\right)\) | \(e\left(\frac{41}{128}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{65}{128}\right)\) | \(e\left(\frac{91}{128}\right)\) |
| \(\chi_{2048}(19,\cdot)\) | 2048.t | 512 | yes | \(-1\) | \(1\) | \(e\left(\frac{37}{512}\right)\) | \(e\left(\frac{407}{512}\right)\) | \(e\left(\frac{19}{256}\right)\) | \(e\left(\frac{37}{256}\right)\) | \(e\left(\frac{419}{512}\right)\) | \(e\left(\frac{249}{512}\right)\) | \(e\left(\frac{111}{128}\right)\) | \(e\left(\frac{65}{128}\right)\) | \(e\left(\frac{17}{512}\right)\) | \(e\left(\frac{75}{512}\right)\) |
| \(\chi_{2048}(21,\cdot)\) | 2048.s | 512 | yes | \(1\) | \(1\) | \(e\left(\frac{103}{512}\right)\) | \(e\left(\frac{365}{512}\right)\) | \(e\left(\frac{1}{256}\right)\) | \(e\left(\frac{103}{256}\right)\) | \(e\left(\frac{433}{512}\right)\) | \(e\left(\frac{451}{512}\right)\) | \(e\left(\frac{117}{128}\right)\) | \(e\left(\frac{91}{128}\right)\) | \(e\left(\frac{75}{512}\right)\) | \(e\left(\frac{105}{512}\right)\) |
| \(\chi_{2048}(23,\cdot)\) | 2048.r | 256 | no | \(-1\) | \(1\) | \(e\left(\frac{181}{256}\right)\) | \(e\left(\frac{71}{256}\right)\) | \(e\left(\frac{67}{128}\right)\) | \(e\left(\frac{53}{128}\right)\) | \(e\left(\frac{147}{256}\right)\) | \(e\left(\frac{73}{256}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{97}{256}\right)\) | \(e\left(\frac{59}{256}\right)\) |
| \(\chi_{2048}(25,\cdot)\) | 2048.q | 256 | no | \(1\) | \(1\) | \(e\left(\frac{163}{256}\right)\) | \(e\left(\frac{1}{256}\right)\) | \(e\left(\frac{101}{128}\right)\) | \(e\left(\frac{35}{128}\right)\) | \(e\left(\frac{213}{256}\right)\) | \(e\left(\frac{239}{256}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{151}{256}\right)\) | \(e\left(\frac{109}{256}\right)\) |
| \(\chi_{2048}(27,\cdot)\) | 2048.t | 512 | yes | \(-1\) | \(1\) | \(e\left(\frac{91}{512}\right)\) | \(e\left(\frac{489}{512}\right)\) | \(e\left(\frac{109}{256}\right)\) | \(e\left(\frac{91}{256}\right)\) | \(e\left(\frac{477}{512}\right)\) | \(e\left(\frac{391}{512}\right)\) | \(e\left(\frac{17}{128}\right)\) | \(e\left(\frac{63}{128}\right)\) | \(e\left(\frac{111}{512}\right)\) | \(e\left(\frac{309}{512}\right)\) |
| \(\chi_{2048}(29,\cdot)\) | 2048.s | 512 | yes | \(1\) | \(1\) | \(e\left(\frac{337}{512}\right)\) | \(e\left(\frac{379}{512}\right)\) | \(e\left(\frac{135}{256}\right)\) | \(e\left(\frac{81}{256}\right)\) | \(e\left(\frac{343}{512}\right)\) | \(e\left(\frac{213}{512}\right)\) | \(e\left(\frac{51}{128}\right)\) | \(e\left(\frac{125}{128}\right)\) | \(e\left(\frac{141}{512}\right)\) | \(e\left(\frac{95}{512}\right)\) |
| \(\chi_{2048}(31,\cdot)\) | 2048.n | 64 | no | \(-1\) | \(1\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{29}{64}\right)\) |
| \(\chi_{2048}(33,\cdot)\) | 2048.m | 64 | no | \(1\) | \(1\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{3}{64}\right)\) |
| \(\chi_{2048}(35,\cdot)\) | 2048.t | 512 | yes | \(-1\) | \(1\) | \(e\left(\frac{65}{512}\right)\) | \(e\left(\frac{203}{512}\right)\) | \(e\left(\frac{151}{256}\right)\) | \(e\left(\frac{65}{256}\right)\) | \(e\left(\frac{487}{512}\right)\) | \(e\left(\frac{133}{512}\right)\) | \(e\left(\frac{67}{128}\right)\) | \(e\left(\frac{45}{128}\right)\) | \(e\left(\frac{445}{512}\right)\) | \(e\left(\frac{367}{512}\right)\) |
| \(\chi_{2048}(37,\cdot)\) | 2048.s | 512 | yes | \(1\) | \(1\) | \(e\left(\frac{235}{512}\right)\) | \(e\left(\frac{281}{512}\right)\) | \(e\left(\frac{221}{256}\right)\) | \(e\left(\frac{235}{256}\right)\) | \(e\left(\frac{461}{512}\right)\) | \(e\left(\frac{343}{512}\right)\) | \(e\left(\frac{1}{128}\right)\) | \(e\left(\frac{15}{128}\right)\) | \(e\left(\frac{191}{512}\right)\) | \(e\left(\frac{165}{512}\right)\) |
| \(\chi_{2048}(39,\cdot)\) | 2048.r | 256 | no | \(-1\) | \(1\) | \(e\left(\frac{251}{256}\right)\) | \(e\left(\frac{73}{256}\right)\) | \(e\left(\frac{13}{128}\right)\) | \(e\left(\frac{123}{128}\right)\) | \(e\left(\frac{61}{256}\right)\) | \(e\left(\frac{39}{256}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{143}{256}\right)\) | \(e\left(\frac{21}{256}\right)\) |
| \(\chi_{2048}(41,\cdot)\) | 2048.q | 256 | no | \(1\) | \(1\) | \(e\left(\frac{221}{256}\right)\) | \(e\left(\frac{127}{256}\right)\) | \(e\left(\frac{27}{128}\right)\) | \(e\left(\frac{93}{128}\right)\) | \(e\left(\frac{171}{256}\right)\) | \(e\left(\frac{145}{256}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{233}{256}\right)\) | \(e\left(\frac{19}{256}\right)\) |
| \(\chi_{2048}(43,\cdot)\) | 2048.t | 512 | yes | \(-1\) | \(1\) | \(e\left(\frac{215}{512}\right)\) | \(e\left(\frac{317}{512}\right)\) | \(e\left(\frac{145}{256}\right)\) | \(e\left(\frac{215}{256}\right)\) | \(e\left(\frac{193}{512}\right)\) | \(e\left(\frac{243}{512}\right)\) | \(e\left(\frac{5}{128}\right)\) | \(e\left(\frac{11}{128}\right)\) | \(e\left(\frac{251}{512}\right)\) | \(e\left(\frac{505}{512}\right)\) |
| \(\chi_{2048}(45,\cdot)\) | 2048.s | 512 | yes | \(1\) | \(1\) | \(e\left(\frac{53}{512}\right)\) | \(e\left(\frac{327}{512}\right)\) | \(e\left(\frac{3}{256}\right)\) | \(e\left(\frac{53}{256}\right)\) | \(e\left(\frac{19}{512}\right)\) | \(e\left(\frac{73}{512}\right)\) | \(e\left(\frac{95}{128}\right)\) | \(e\left(\frac{17}{128}\right)\) | \(e\left(\frac{481}{512}\right)\) | \(e\left(\frac{59}{512}\right)\) |
| \(\chi_{2048}(47,\cdot)\) | 2048.p | 128 | no | \(-1\) | \(1\) | \(e\left(\frac{33}{128}\right)\) | \(e\left(\frac{43}{128}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{7}{128}\right)\) | \(e\left(\frac{37}{128}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{29}{128}\right)\) | \(e\left(\frac{79}{128}\right)\) |
| \(\chi_{2048}(49,\cdot)\) | 2048.o | 128 | no | \(1\) | \(1\) | \(e\left(\frac{79}{128}\right)\) | \(e\left(\frac{101}{128}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{9}{128}\right)\) | \(e\left(\frac{75}{128}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{19}{128}\right)\) | \(e\left(\frac{1}{128}\right)\) |
| \(\chi_{2048}(51,\cdot)\) | 2048.t | 512 | yes | \(-1\) | \(1\) | \(e\left(\frac{285}{512}\right)\) | \(e\left(\frac{63}{512}\right)\) | \(e\left(\frac{91}{256}\right)\) | \(e\left(\frac{29}{256}\right)\) | \(e\left(\frac{363}{512}\right)\) | \(e\left(\frac{465}{512}\right)\) | \(e\left(\frac{87}{128}\right)\) | \(e\left(\frac{89}{128}\right)\) | \(e\left(\frac{297}{512}\right)\) | \(e\left(\frac{467}{512}\right)\) |
| \(\chi_{2048}(53,\cdot)\) | 2048.s | 512 | yes | \(1\) | \(1\) | \(e\left(\frac{47}{512}\right)\) | \(e\left(\frac{261}{512}\right)\) | \(e\left(\frac{249}{256}\right)\) | \(e\left(\frac{47}{256}\right)\) | \(e\left(\frac{297}{512}\right)\) | \(e\left(\frac{171}{512}\right)\) | \(e\left(\frac{77}{128}\right)\) | \(e\left(\frac{3}{128}\right)\) | \(e\left(\frac{243}{512}\right)\) | \(e\left(\frac{33}{512}\right)\) |
| \(\chi_{2048}(55,\cdot)\) | 2048.r | 256 | no | \(-1\) | \(1\) | \(e\left(\frac{161}{256}\right)\) | \(e\left(\frac{107}{256}\right)\) | \(e\left(\frac{119}{128}\right)\) | \(e\left(\frac{33}{128}\right)\) | \(e\left(\frac{135}{256}\right)\) | \(e\left(\frac{229}{256}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{157}{256}\right)\) | \(e\left(\frac{143}{256}\right)\) |
| \(\chi_{2048}(57,\cdot)\) | 2048.q | 256 | no | \(1\) | \(1\) | \(e\left(\frac{119}{256}\right)\) | \(e\left(\frac{29}{256}\right)\) | \(e\left(\frac{113}{128}\right)\) | \(e\left(\frac{119}{128}\right)\) | \(e\left(\frac{33}{256}\right)\) | \(e\left(\frac{19}{256}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{27}{256}\right)\) | \(e\left(\frac{89}{256}\right)\) |
| \(\chi_{2048}(59,\cdot)\) | 2048.t | 512 | yes | \(-1\) | \(1\) | \(e\left(\frac{19}{512}\right)\) | \(e\left(\frac{209}{512}\right)\) | \(e\left(\frac{245}{256}\right)\) | \(e\left(\frac{19}{256}\right)\) | \(e\left(\frac{229}{512}\right)\) | \(e\left(\frac{31}{512}\right)\) | \(e\left(\frac{57}{128}\right)\) | \(e\left(\frac{23}{128}\right)\) | \(e\left(\frac{327}{512}\right)\) | \(e\left(\frac{509}{512}\right)\) |
| \(\chi_{2048}(61,\cdot)\) | 2048.s | 512 | yes | \(1\) | \(1\) | \(e\left(\frac{473}{512}\right)\) | \(e\left(\frac{339}{512}\right)\) | \(e\left(\frac{191}{256}\right)\) | \(e\left(\frac{217}{256}\right)\) | \(e\left(\frac{15}{512}\right)\) | \(e\left(\frac{381}{512}\right)\) | \(e\left(\frac{75}{128}\right)\) | \(e\left(\frac{101}{128}\right)\) | \(e\left(\frac{245}{512}\right)\) | \(e\left(\frac{343}{512}\right)\) |
| \(\chi_{2048}(63,\cdot)\) | 2048.l | 32 | no | \(-1\) | \(1\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{13}{32}\right)\) |