Fungrim home page

Fungrim entry: 338b5c

Table of GqprimitiveG_{q}^{\text{primitive}} and GqGqprimitiveG_{q} \setminus G_{q}^{\text{primitive}} for 1q301 \le q \le 30
qq #Gq\# G_{q} #Gqprimitive\# G_{q}^{\text{primitive}} \ell such that χq.\chi_{q \, . \, \ell} is primitive \ell such that χq.\chi_{q \, . \, \ell} is non-primitive
111{1}{}
210{}{1}
321{2}{1}
421{3}{1}
543{2, 3, 4}{1}
620{}{1, 5}
765{2, 3, 4, 5, 6}{1}
842{3, 5}{1, 7}
964{2, 4, 5, 7}{1, 8}
1040{}{1, 3, 7, 9}
11109{2, 3, 4, 5, 6, 7, 8, 9, 10}{1}
1241{11}{1, 5, 7}
131211{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}{1}
1460{}{1, 3, 5, 9, 11, 13}
1583{2, 8, 14}{1, 4, 7, 11, 13}
1684{3, 5, 11, 13}{1, 7, 9, 15}
171615{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}{1}
1860{}{1, 5, 7, 11, 13, 17}
191817{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}{1}
2083{3, 7, 19}{1, 9, 11, 13, 17}
21125{2, 5, 11, 17, 20}{1, 4, 8, 10, 13, 16, 19}
22100{}{1, 3, 5, 7, 9, 13, 15, 17, 19, 21}
232221{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22}{1}
2482{5, 11}{1, 7, 13, 17, 19, 23}
252016{2, 3, 4, 6, 8, 9, 11, 12, 13, 14, 16, 17, 19, 21, 22, 23}{1, 7, 18, 24}
26120{}{1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25}
271812{2, 4, 5, 7, 11, 13, 14, 16, 20, 22, 23, 25}{1, 8, 10, 17, 19, 26}
28125{3, 11, 19, 23, 27}{1, 5, 9, 13, 15, 17, 25}
292827{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28}{1}
3080{}{1, 7, 11, 13, 17, 19, 23, 29}
Table data: (q,#Gq,#Gqprimitive,P,N)\left(q, \# G_{q}, \# G_{q}^{\text{primitive}}, P, N\right) such that Gqprimitive={χq.:P}  and  GqGqprimitive={χq.:N}G_{q}^{\text{primitive}} = \left\{ \chi_{q \, . \, \ell} : \ell \in P \right\} \;\mathbin{\operatorname{and}}\; G_{q} \setminus G_{q}^{\text{primitive}} = \left\{ \chi_{q \, . \, \ell} : \ell \in N \right\}
Definitions:
Fungrim symbol Notation Short description
PrimitiveDirichletCharactersGqprimitiveG_{q}^{\text{primitive}} Primitive Dirichlet characters with given modulus
DirichletGroupGqG_{q} Dirichlet characters with given modulus
Cardinality#S\# S Set cardinality
DirichletCharacterχq.\chi_{q \, . \, \ell} Dirichlet character
Source code for this entry:
Entry(ID("338b5c"),
    Description("Table of", PrimitiveDirichletCharacters(q), "and", SetMinus(DirichletGroup(q), PrimitiveDirichletCharacters(q)), "for", LessEqual(1, q, 30)),
    Table(TableRelation(Tuple(q, Cardinality(DirichletGroup(q)), Cardinality(PrimitiveDirichletCharacters(q)), P, N), And(Equal(PrimitiveDirichletCharacters(q), Set(DirichletCharacter(q, ell), For(ell), Element(ell, P))), Equal(SetMinus(DirichletGroup(q), PrimitiveDirichletCharacters(q)), Set(DirichletCharacter(q, ell), For(ell), Element(ell, N))))), TableHeadings(q, Cardinality(DirichletGroup(q)), Cardinality(PrimitiveDirichletCharacters(q)), Description(ell, "such that", DirichletCharacter(q, ell), "is primitive"), Description(ell, "such that", DirichletCharacter(q, ell), "is non-primitive")), List(Tuple(1, 1, 1, Set(1), Set()), Tuple(2, 1, 0, Set(), Set(1)), Tuple(3, 2, 1, Set(2), Set(1)), Tuple(4, 2, 1, Set(3), Set(1)), Tuple(5, 4, 3, Set(2, 3, 4), Set(1)), Tuple(6, 2, 0, Set(), Set(1, 5)), Tuple(7, 6, 5, Set(2, 3, 4, 5, 6), Set(1)), Tuple(8, 4, 2, Set(3, 5), Set(1, 7)), Tuple(9, 6, 4, Set(2, 4, 5, 7), Set(1, 8)), Tuple(10, 4, 0, Set(), Set(1, 3, 7, 9)), Tuple(11, 10, 9, Set(2, 3, 4, 5, 6, 7, 8, 9, 10), Set(1)), Tuple(12, 4, 1, Set(11), Set(1, 5, 7)), Tuple(13, 12, 11, Set(2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12), Set(1)), Tuple(14, 6, 0, Set(), Set(1, 3, 5, 9, 11, 13)), Tuple(15, 8, 3, Set(2, 8, 14), Set(1, 4, 7, 11, 13)), Tuple(16, 8, 4, Set(3, 5, 11, 13), Set(1, 7, 9, 15)), Tuple(17, 16, 15, Set(2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16), Set(1)), Tuple(18, 6, 0, Set(), Set(1, 5, 7, 11, 13, 17)), Tuple(19, 18, 17, Set(2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18), Set(1)), Tuple(20, 8, 3, Set(3, 7, 19), Set(1, 9, 11, 13, 17)), Tuple(21, 12, 5, Set(2, 5, 11, 17, 20), Set(1, 4, 8, 10, 13, 16, 19)), Tuple(22, 10, 0, Set(), Set(1, 3, 5, 7, 9, 13, 15, 17, 19, 21)), Tuple(23, 22, 21, Set(2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22), Set(1)), Tuple(24, 8, 2, Set(5, 11), Set(1, 7, 13, 17, 19, 23)), Tuple(25, 20, 16, Set(2, 3, 4, 6, 8, 9, 11, 12, 13, 14, 16, 17, 19, 21, 22, 23), Set(1, 7, 18, 24)), Tuple(26, 12, 0, Set(), Set(1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25)), Tuple(27, 18, 12, Set(2, 4, 5, 7, 11, 13, 14, 16, 20, 22, 23, 25), Set(1, 8, 10, 17, 19, 26)), Tuple(28, 12, 5, Set(3, 11, 19, 23, 27), Set(1, 5, 9, 13, 15, 17, 25)), Tuple(29, 28, 27, Set(2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28), Set(1)), Tuple(30, 8, 0, Set(), Set(1, 7, 11, 13, 17, 19, 23, 29)))))

Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-08-27 09:56:25.682319 UTC