Properties

Modulus $3072$
Structure \(C_{256}\times C_{2}\times C_{2}\)
Order $1024$

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Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(3072)
 
pari: g = idealstar(,3072,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 1024
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{256}\times C_{2}\times C_{2}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{3072}(2047,\cdot)$, $\chi_{3072}(2053,\cdot)$, $\chi_{3072}(1025,\cdot)$

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 \(-1\) \(1\) \(5\) \(7\) \(11\) \(13\) \(17\) \(19\) \(23\) \(25\) \(29\) \(31\)
\(\chi_{3072}(1,\cdot)\) 3072.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{3072}(5,\cdot)\) 3072.bg 256 yes \(-1\) \(1\) \(e\left(\frac{129}{256}\right)\) \(e\left(\frac{101}{128}\right)\) \(e\left(\frac{85}{256}\right)\) \(e\left(\frac{239}{256}\right)\) \(e\left(\frac{7}{64}\right)\) \(e\left(\frac{151}{256}\right)\) \(e\left(\frac{7}{128}\right)\) \(e\left(\frac{1}{128}\right)\) \(e\left(\frac{251}{256}\right)\) \(e\left(\frac{17}{32}\right)\)
\(\chi_{3072}(7,\cdot)\) 3072.bc 128 no \(-1\) \(1\) \(e\left(\frac{101}{128}\right)\) \(e\left(\frac{57}{64}\right)\) \(e\left(\frac{73}{128}\right)\) \(e\left(\frac{75}{128}\right)\) \(e\left(\frac{3}{32}\right)\) \(e\left(\frac{83}{128}\right)\) \(e\left(\frac{35}{64}\right)\) \(e\left(\frac{37}{64}\right)\) \(e\left(\frac{7}{128}\right)\) \(e\left(\frac{13}{16}\right)\)
\(\chi_{3072}(11,\cdot)\) 3072.bi 256 yes \(1\) \(1\) \(e\left(\frac{85}{256}\right)\) \(e\left(\frac{73}{128}\right)\) \(e\left(\frac{57}{256}\right)\) \(e\left(\frac{219}{256}\right)\) \(e\left(\frac{19}{64}\right)\) \(e\left(\frac{35}{256}\right)\) \(e\left(\frac{19}{128}\right)\) \(e\left(\frac{85}{128}\right)\) \(e\left(\frac{215}{256}\right)\) \(e\left(\frac{21}{32}\right)\)
\(\chi_{3072}(13,\cdot)\) 3072.bh 256 no \(1\) \(1\) \(e\left(\frac{239}{256}\right)\) \(e\left(\frac{75}{128}\right)\) \(e\left(\frac{219}{256}\right)\) \(e\left(\frac{33}{256}\right)\) \(e\left(\frac{41}{64}\right)\) \(e\left(\frac{249}{256}\right)\) \(e\left(\frac{73}{128}\right)\) \(e\left(\frac{111}{128}\right)\) \(e\left(\frac{213}{256}\right)\) \(e\left(\frac{31}{32}\right)\)
\(\chi_{3072}(17,\cdot)\) 3072.y 64 no \(-1\) \(1\) \(e\left(\frac{7}{64}\right)\) \(e\left(\frac{3}{32}\right)\) \(e\left(\frac{19}{64}\right)\) \(e\left(\frac{41}{64}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{1}{64}\right)\) \(e\left(\frac{1}{32}\right)\) \(e\left(\frac{7}{32}\right)\) \(e\left(\frac{29}{64}\right)\) \(e\left(\frac{7}{8}\right)\)
\(\chi_{3072}(19,\cdot)\) 3072.bj 256 no \(-1\) \(1\) \(e\left(\frac{151}{256}\right)\) \(e\left(\frac{83}{128}\right)\) \(e\left(\frac{35}{256}\right)\) \(e\left(\frac{249}{256}\right)\) \(e\left(\frac{1}{64}\right)\) \(e\left(\frac{145}{256}\right)\) \(e\left(\frac{33}{128}\right)\) \(e\left(\frac{23}{128}\right)\) \(e\left(\frac{141}{256}\right)\) \(e\left(\frac{23}{32}\right)\)
\(\chi_{3072}(23,\cdot)\) 3072.be 128 no \(1\) \(1\) \(e\left(\frac{7}{128}\right)\) \(e\left(\frac{35}{64}\right)\) \(e\left(\frac{19}{128}\right)\) \(e\left(\frac{73}{128}\right)\) \(e\left(\frac{1}{32}\right)\) \(e\left(\frac{33}{128}\right)\) \(e\left(\frac{49}{64}\right)\) \(e\left(\frac{7}{64}\right)\) \(e\left(\frac{93}{128}\right)\) \(e\left(\frac{15}{16}\right)\)
\(\chi_{3072}(25,\cdot)\) 3072.bd 128 no \(1\) \(1\) \(e\left(\frac{1}{128}\right)\) \(e\left(\frac{37}{64}\right)\) \(e\left(\frac{85}{128}\right)\) \(e\left(\frac{111}{128}\right)\) \(e\left(\frac{7}{32}\right)\) \(e\left(\frac{23}{128}\right)\) \(e\left(\frac{7}{64}\right)\) \(e\left(\frac{1}{64}\right)\) \(e\left(\frac{123}{128}\right)\) \(e\left(\frac{1}{16}\right)\)
\(\chi_{3072}(29,\cdot)\) 3072.bg 256 yes \(-1\) \(1\) \(e\left(\frac{251}{256}\right)\) \(e\left(\frac{7}{128}\right)\) \(e\left(\frac{215}{256}\right)\) \(e\left(\frac{213}{256}\right)\) \(e\left(\frac{29}{64}\right)\) \(e\left(\frac{141}{256}\right)\) \(e\left(\frac{93}{128}\right)\) \(e\left(\frac{123}{128}\right)\) \(e\left(\frac{153}{256}\right)\) \(e\left(\frac{11}{32}\right)\)
\(\chi_{3072}(31,\cdot)\) 3072.u 32 no \(-1\) \(1\) \(e\left(\frac{17}{32}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{21}{32}\right)\) \(e\left(\frac{31}{32}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{23}{32}\right)\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{11}{32}\right)\) \(-i\)
\(\chi_{3072}(35,\cdot)\) 3072.bi 256 yes \(1\) \(1\) \(e\left(\frac{75}{256}\right)\) \(e\left(\frac{87}{128}\right)\) \(e\left(\frac{231}{256}\right)\) \(e\left(\frac{133}{256}\right)\) \(e\left(\frac{13}{64}\right)\) \(e\left(\frac{61}{256}\right)\) \(e\left(\frac{77}{128}\right)\) \(e\left(\frac{75}{128}\right)\) \(e\left(\frac{9}{256}\right)\) \(e\left(\frac{11}{32}\right)\)
\(\chi_{3072}(37,\cdot)\) 3072.bh 256 no \(1\) \(1\) \(e\left(\frac{25}{256}\right)\) \(e\left(\frac{93}{128}\right)\) \(e\left(\frac{205}{256}\right)\) \(e\left(\frac{87}{256}\right)\) \(e\left(\frac{15}{64}\right)\) \(e\left(\frac{191}{256}\right)\) \(e\left(\frac{111}{128}\right)\) \(e\left(\frac{25}{128}\right)\) \(e\left(\frac{3}{256}\right)\) \(e\left(\frac{9}{32}\right)\)
\(\chi_{3072}(41,\cdot)\) 3072.bf 128 no \(-1\) \(1\) \(e\left(\frac{63}{128}\right)\) \(e\left(\frac{27}{64}\right)\) \(e\left(\frac{107}{128}\right)\) \(e\left(\frac{17}{128}\right)\) \(e\left(\frac{9}{32}\right)\) \(e\left(\frac{105}{128}\right)\) \(e\left(\frac{25}{64}\right)\) \(e\left(\frac{63}{64}\right)\) \(e\left(\frac{69}{128}\right)\) \(e\left(\frac{15}{16}\right)\)
\(\chi_{3072}(43,\cdot)\) 3072.bj 256 no \(-1\) \(1\) \(e\left(\frac{61}{256}\right)\) \(e\left(\frac{81}{128}\right)\) \(e\left(\frac{65}{256}\right)\) \(e\left(\frac{243}{256}\right)\) \(e\left(\frac{11}{64}\right)\) \(e\left(\frac{123}{256}\right)\) \(e\left(\frac{43}{128}\right)\) \(e\left(\frac{61}{128}\right)\) \(e\left(\frac{79}{256}\right)\) \(e\left(\frac{29}{32}\right)\)
\(\chi_{3072}(47,\cdot)\) 3072.ba 64 no \(1\) \(1\) \(e\left(\frac{11}{64}\right)\) \(e\left(\frac{7}{32}\right)\) \(e\left(\frac{7}{64}\right)\) \(e\left(\frac{37}{64}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{61}{64}\right)\) \(e\left(\frac{13}{32}\right)\) \(e\left(\frac{11}{32}\right)\) \(e\left(\frac{9}{64}\right)\) \(e\left(\frac{7}{8}\right)\)
\(\chi_{3072}(49,\cdot)\) 3072.z 64 no \(1\) \(1\) \(e\left(\frac{37}{64}\right)\) \(e\left(\frac{25}{32}\right)\) \(e\left(\frac{9}{64}\right)\) \(e\left(\frac{11}{64}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{19}{64}\right)\) \(e\left(\frac{3}{32}\right)\) \(e\left(\frac{5}{32}\right)\) \(e\left(\frac{7}{64}\right)\) \(e\left(\frac{5}{8}\right)\)
\(\chi_{3072}(53,\cdot)\) 3072.bg 256 yes \(-1\) \(1\) \(e\left(\frac{133}{256}\right)\) \(e\left(\frac{121}{128}\right)\) \(e\left(\frac{169}{256}\right)\) \(e\left(\frac{171}{256}\right)\) \(e\left(\frac{35}{64}\right)\) \(e\left(\frac{243}{256}\right)\) \(e\left(\frac{35}{128}\right)\) \(e\left(\frac{5}{128}\right)\) \(e\left(\frac{231}{256}\right)\) \(e\left(\frac{21}{32}\right)\)
\(\chi_{3072}(55,\cdot)\) 3072.bc 128 no \(-1\) \(1\) \(e\left(\frac{107}{128}\right)\) \(e\left(\frac{23}{64}\right)\) \(e\left(\frac{71}{128}\right)\) \(e\left(\frac{101}{128}\right)\) \(e\left(\frac{13}{32}\right)\) \(e\left(\frac{93}{128}\right)\) \(e\left(\frac{13}{64}\right)\) \(e\left(\frac{43}{64}\right)\) \(e\left(\frac{105}{128}\right)\) \(e\left(\frac{3}{16}\right)\)
\(\chi_{3072}(59,\cdot)\) 3072.bi 256 yes \(1\) \(1\) \(e\left(\frac{81}{256}\right)\) \(e\left(\frac{53}{128}\right)\) \(e\left(\frac{229}{256}\right)\) \(e\left(\frac{31}{256}\right)\) \(e\left(\frac{55}{64}\right)\) \(e\left(\frac{199}{256}\right)\) \(e\left(\frac{119}{128}\right)\) \(e\left(\frac{81}{128}\right)\) \(e\left(\frac{235}{256}\right)\) \(e\left(\frac{17}{32}\right)\)
\(\chi_{3072}(61,\cdot)\) 3072.bh 256 no \(1\) \(1\) \(e\left(\frac{83}{256}\right)\) \(e\left(\frac{63}{128}\right)\) \(e\left(\frac{15}{256}\right)\) \(e\left(\frac{125}{256}\right)\) \(e\left(\frac{37}{64}\right)\) \(e\left(\frac{245}{256}\right)\) \(e\left(\frac{5}{128}\right)\) \(e\left(\frac{83}{128}\right)\) \(e\left(\frac{225}{256}\right)\) \(e\left(\frac{3}{32}\right)\)
\(\chi_{3072}(65,\cdot)\) 3072.q 16 no \(-1\) \(1\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(-i\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{13}{16}\right)\) \(-1\)
\(\chi_{3072}(67,\cdot)\) 3072.bj 256 no \(-1\) \(1\) \(e\left(\frac{243}{256}\right)\) \(e\left(\frac{31}{128}\right)\) \(e\left(\frac{175}{256}\right)\) \(e\left(\frac{221}{256}\right)\) \(e\left(\frac{5}{64}\right)\) \(e\left(\frac{213}{256}\right)\) \(e\left(\frac{37}{128}\right)\) \(e\left(\frac{115}{128}\right)\) \(e\left(\frac{193}{256}\right)\) \(e\left(\frac{19}{32}\right)\)
\(\chi_{3072}(71,\cdot)\) 3072.be 128 no \(1\) \(1\) \(e\left(\frac{109}{128}\right)\) \(e\left(\frac{33}{64}\right)\) \(e\left(\frac{113}{128}\right)\) \(e\left(\frac{3}{128}\right)\) \(e\left(\frac{11}{32}\right)\) \(e\left(\frac{75}{128}\right)\) \(e\left(\frac{59}{64}\right)\) \(e\left(\frac{45}{64}\right)\) \(e\left(\frac{95}{128}\right)\) \(e\left(\frac{5}{16}\right)\)
\(\chi_{3072}(73,\cdot)\) 3072.bd 128 no \(1\) \(1\) \(e\left(\frac{91}{128}\right)\) \(e\left(\frac{39}{64}\right)\) \(e\left(\frac{55}{128}\right)\) \(e\left(\frac{117}{128}\right)\) \(e\left(\frac{29}{32}\right)\) \(e\left(\frac{45}{128}\right)\) \(e\left(\frac{61}{64}\right)\) \(e\left(\frac{27}{64}\right)\) \(e\left(\frac{57}{128}\right)\) \(e\left(\frac{11}{16}\right)\)
\(\chi_{3072}(77,\cdot)\) 3072.bg 256 yes \(-1\) \(1\) \(e\left(\frac{31}{256}\right)\) \(e\left(\frac{59}{128}\right)\) \(e\left(\frac{203}{256}\right)\) \(e\left(\frac{113}{256}\right)\) \(e\left(\frac{25}{64}\right)\) \(e\left(\frac{201}{256}\right)\) \(e\left(\frac{89}{128}\right)\) \(e\left(\frac{31}{128}\right)\) \(e\left(\frac{229}{256}\right)\) \(e\left(\frac{15}{32}\right)\)
\(\chi_{3072}(79,\cdot)\) 3072.bb 64 no \(-1\) \(1\) \(e\left(\frac{45}{64}\right)\) \(e\left(\frac{17}{32}\right)\) \(e\left(\frac{17}{64}\right)\) \(e\left(\frac{3}{64}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{43}{64}\right)\) \(e\left(\frac{11}{32}\right)\) \(e\left(\frac{13}{32}\right)\) \(e\left(\frac{31}{64}\right)\) \(e\left(\frac{1}{8}\right)\)
\(\chi_{3072}(83,\cdot)\) 3072.bi 256 yes \(1\) \(1\) \(e\left(\frac{103}{256}\right)\) \(e\left(\frac{99}{128}\right)\) \(e\left(\frac{51}{256}\right)\) \(e\left(\frac{169}{256}\right)\) \(e\left(\frac{17}{64}\right)\) \(e\left(\frac{193}{256}\right)\) \(e\left(\frac{17}{128}\right)\) \(e\left(\frac{103}{128}\right)\) \(e\left(\frac{125}{256}\right)\) \(e\left(\frac{7}{32}\right)\)
\(\chi_{3072}(85,\cdot)\) 3072.bh 256 no \(1\) \(1\) \(e\left(\frac{157}{256}\right)\) \(e\left(\frac{113}{128}\right)\) \(e\left(\frac{161}{256}\right)\) \(e\left(\frac{147}{256}\right)\) \(e\left(\frac{43}{64}\right)\) \(e\left(\frac{155}{256}\right)\) \(e\left(\frac{11}{128}\right)\) \(e\left(\frac{29}{128}\right)\) \(e\left(\frac{111}{256}\right)\) \(e\left(\frac{13}{32}\right)\)
\(\chi_{3072}(89,\cdot)\) 3072.bf 128 no \(-1\) \(1\) \(e\left(\frac{121}{128}\right)\) \(e\left(\frac{61}{64}\right)\) \(e\left(\frac{45}{128}\right)\) \(e\left(\frac{55}{128}\right)\) \(e\left(\frac{31}{32}\right)\) \(e\left(\frac{31}{128}\right)\) \(e\left(\frac{47}{64}\right)\) \(e\left(\frac{57}{64}\right)\) \(e\left(\frac{35}{128}\right)\) \(e\left(\frac{9}{16}\right)\)
\(\chi_{3072}(91,\cdot)\) 3072.bj 256 no \(-1\) \(1\) \(e\left(\frac{185}{256}\right)\) \(e\left(\frac{61}{128}\right)\) \(e\left(\frac{109}{256}\right)\) \(e\left(\frac{183}{256}\right)\) \(e\left(\frac{47}{64}\right)\) \(e\left(\frac{159}{256}\right)\) \(e\left(\frac{15}{128}\right)\) \(e\left(\frac{57}{128}\right)\) \(e\left(\frac{227}{256}\right)\) \(e\left(\frac{25}{32}\right)\)
\(\chi_{3072}(95,\cdot)\) 3072.w 32 no \(1\) \(1\) \(e\left(\frac{3}{32}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{15}{32}\right)\) \(e\left(\frac{29}{32}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{5}{32}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{17}{32}\right)\) \(i\)