Properties

Modulus $448$
Structure \(C_{48}\times C_{2}\times C_{2}\)
Order $192$

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Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(448)
 
pari: g = idealstar(,448,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 192
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{48}\times C_{2}\times C_{2}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{448}(127,\cdot)$, $\chi_{448}(197,\cdot)$, $\chi_{448}(129,\cdot)$

First 32 of 192 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 \(-1\) \(1\) \(3\) \(5\) \(9\) \(11\) \(13\) \(15\) \(17\) \(19\) \(23\) \(25\)
\(\chi_{448}(1,\cdot)\) 448.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{448}(3,\cdot)\) 448.bm 48 yes \(1\) \(1\) \(e\left(\frac{11}{48}\right)\) \(e\left(\frac{1}{48}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{5}{48}\right)\) \(e\left(\frac{5}{16}\right)\) \(i\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{31}{48}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{1}{24}\right)\)
\(\chi_{448}(5,\cdot)\) 448.bk 48 yes \(-1\) \(1\) \(e\left(\frac{1}{48}\right)\) \(e\left(\frac{11}{48}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{31}{48}\right)\) \(e\left(\frac{7}{16}\right)\) \(i\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{29}{48}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{11}{24}\right)\)
\(\chi_{448}(9,\cdot)\) 448.bh 24 no \(1\) \(1\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{5}{8}\right)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{12}\right)\)
\(\chi_{448}(11,\cdot)\) 448.bl 48 yes \(-1\) \(1\) \(e\left(\frac{5}{48}\right)\) \(e\left(\frac{31}{48}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{35}{48}\right)\) \(e\left(\frac{11}{16}\right)\) \(-i\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{48}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{7}{24}\right)\)
\(\chi_{448}(13,\cdot)\) 448.bf 16 yes \(-1\) \(1\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{9}{16}\right)\) \(-i\) \(-i\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{8}\right)\)
\(\chi_{448}(15,\cdot)\) 448.k 4 no \(-1\) \(1\) \(i\) \(i\) \(-1\) \(-i\) \(-i\) \(-1\) \(1\) \(i\) \(1\) \(-1\)
\(\chi_{448}(17,\cdot)\) 448.bb 12 no \(-1\) \(1\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{12}\right)\) \(-i\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\)
\(\chi_{448}(19,\cdot)\) 448.bm 48 yes \(1\) \(1\) \(e\left(\frac{31}{48}\right)\) \(e\left(\frac{29}{48}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{1}{48}\right)\) \(e\left(\frac{1}{16}\right)\) \(i\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{35}{48}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{5}{24}\right)\)
\(\chi_{448}(23,\cdot)\) 448.bj 24 no \(-1\) \(1\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{1}{8}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{12}\right)\)
\(\chi_{448}(25,\cdot)\) 448.bh 24 no \(1\) \(1\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{7}{8}\right)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{11}{12}\right)\)
\(\chi_{448}(27,\cdot)\) 448.bd 16 yes \(1\) \(1\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{15}{16}\right)\) \(-i\) \(i\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{1}{8}\right)\)
\(\chi_{448}(29,\cdot)\) 448.bc 16 no \(1\) \(1\) \(e\left(\frac{1}{16}\right)\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{16}\right)\) \(e\left(\frac{5}{16}\right)\) \(-i\) \(i\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{8}\right)\)
\(\chi_{448}(31,\cdot)\) 448.q 6 no \(1\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{448}(33,\cdot)\) 448.n 6 no \(-1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{448}(37,\cdot)\) 448.bn 48 yes \(1\) \(1\) \(e\left(\frac{1}{48}\right)\) \(e\left(\frac{11}{48}\right)\) \(e\left(\frac{1}{24}\right)\) \(e\left(\frac{7}{48}\right)\) \(e\left(\frac{7}{16}\right)\) \(i\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{29}{48}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{11}{24}\right)\)
\(\chi_{448}(39,\cdot)\) 448.bj 24 no \(-1\) \(1\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{7}{8}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{11}{12}\right)\)
\(\chi_{448}(41,\cdot)\) 448.v 8 no \(-1\) \(1\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(i\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(-1\) \(1\) \(e\left(\frac{5}{8}\right)\) \(i\) \(-i\)
\(\chi_{448}(43,\cdot)\) 448.be 16 no \(-1\) \(1\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{13}{16}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{3}{16}\right)\) \(-i\) \(-i\) \(e\left(\frac{3}{16}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{5}{8}\right)\)
\(\chi_{448}(45,\cdot)\) 448.bk 48 yes \(-1\) \(1\) \(e\left(\frac{23}{48}\right)\) \(e\left(\frac{13}{48}\right)\) \(e\left(\frac{23}{24}\right)\) \(e\left(\frac{41}{48}\right)\) \(e\left(\frac{1}{16}\right)\) \(-i\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{43}{48}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{13}{24}\right)\)
\(\chi_{448}(47,\cdot)\) 448.z 12 no \(1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{7}{12}\right)\) \(-i\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\)
\(\chi_{448}(51,\cdot)\) 448.bl 48 yes \(-1\) \(1\) \(e\left(\frac{31}{48}\right)\) \(e\left(\frac{29}{48}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{25}{48}\right)\) \(e\left(\frac{1}{16}\right)\) \(i\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{35}{48}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{5}{24}\right)\)
\(\chi_{448}(53,\cdot)\) 448.bn 48 yes \(1\) \(1\) \(e\left(\frac{29}{48}\right)\) \(e\left(\frac{31}{48}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{11}{48}\right)\) \(e\left(\frac{11}{16}\right)\) \(i\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{25}{48}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{7}{24}\right)\)
\(\chi_{448}(55,\cdot)\) 448.x 8 no \(1\) \(1\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(i\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(1\) \(1\) \(e\left(\frac{5}{8}\right)\) \(-i\) \(-i\)
\(\chi_{448}(57,\cdot)\) 448.u 8 no \(1\) \(1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(-i\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(-1\) \(-1\) \(e\left(\frac{3}{8}\right)\) \(-i\) \(i\)
\(\chi_{448}(59,\cdot)\) 448.bm 48 yes \(1\) \(1\) \(e\left(\frac{41}{48}\right)\) \(e\left(\frac{43}{48}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{23}{48}\right)\) \(e\left(\frac{7}{16}\right)\) \(-i\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{37}{48}\right)\) \(e\left(\frac{17}{24}\right)\) \(e\left(\frac{19}{24}\right)\)
\(\chi_{448}(61,\cdot)\) 448.bk 48 yes \(-1\) \(1\) \(e\left(\frac{19}{48}\right)\) \(e\left(\frac{17}{48}\right)\) \(e\left(\frac{19}{24}\right)\) \(e\left(\frac{13}{48}\right)\) \(e\left(\frac{5}{16}\right)\) \(-i\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{23}{48}\right)\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{17}{24}\right)\)
\(\chi_{448}(65,\cdot)\) 448.i 3 no \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{3}\right)\)
\(\chi_{448}(67,\cdot)\) 448.bl 48 yes \(-1\) \(1\) \(e\left(\frac{35}{48}\right)\) \(e\left(\frac{25}{48}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{5}{48}\right)\) \(e\left(\frac{13}{16}\right)\) \(i\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{7}{48}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{1}{24}\right)\)
\(\chi_{448}(69,\cdot)\) 448.bf 16 yes \(-1\) \(1\) \(e\left(\frac{11}{16}\right)\) \(e\left(\frac{9}{16}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{16}\right)\) \(e\left(\frac{7}{16}\right)\) \(i\) \(i\) \(e\left(\frac{15}{16}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{1}{8}\right)\)
\(\chi_{448}(71,\cdot)\) 448.w 8 no \(-1\) \(1\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(-i\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(1\) \(-1\) \(e\left(\frac{7}{8}\right)\) \(i\) \(i\)
\(\chi_{448}(73,\cdot)\) 448.bg 24 no \(-1\) \(1\) \(e\left(\frac{7}{24}\right)\) \(e\left(\frac{5}{24}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{13}{24}\right)\) \(e\left(\frac{1}{8}\right)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{24}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{12}\right)\)