Fungrim entry: 338b5c

Table of

G^{\text{Primitive}}_{q}

and

G_{q} \setminus G^{\text{Primitive}}_{q}

for

1 \le q \le 30

$q$	$\# G_{q}$	$\# G^{\text{Primitive}}_{q}$	$\ell$ such that $\chi_{q \, . \, \ell}$ is primitive	$\ell$ such that $\chi_{q \, . \, \ell}$ is non-primitive
1	1	1	{1}	{}
2	1	0	{}	{1}
3	2	1	{2}	{1}
4	2	1	{3}	{1}
5	4	3	{2, 3, 4}	{1}
6	2	0	{}	{1, 5}
7	6	5	{2, 3, 4, 5, 6}	{1}
8	4	2	{3, 5}	{1, 7}
9	6	4	{2, 4, 5, 7}	{1, 8}
10	4	0	{}	{1, 3, 7, 9}
11	10	9	{2, 3, 4, 5, 6, 7, 8, 9, 10}	{1}
12	4	1	{11}	{1, 5, 7}
13	12	11	{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}	{1}
14	6	0	{}	{1, 3, 5, 9, 11, 13}
15	8	3	{2, 8, 14}	{1, 4, 7, 11, 13}
16	8	4	{3, 5, 11, 13}	{1, 7, 9, 15}
17	16	15	{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}	{1}
18	6	0	{}	{1, 5, 7, 11, 13, 17}
19	18	17	{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}	{1}
20	8	3	{3, 7, 19}	{1, 9, 11, 13, 17}
21	12	5	{2, 5, 11, 17, 20}	{1, 4, 8, 10, 13, 16, 19}
22	10	0	{}	{1, 3, 5, 7, 9, 13, 15, 17, 19, 21}
23	22	21	{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22}	{1}
24	8	2	{5, 11}	{1, 7, 13, 17, 19, 23}
25	20	16	{2, 3, 4, 6, 8, 9, 11, 12, 13, 14, 16, 17, 19, 21, 22, 23}	{1, 7, 18, 24}
26	12	0	{}	{1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25}
27	18	12	{2, 4, 5, 7, 11, 13, 14, 16, 20, 22, 23, 25}	{1, 8, 10, 17, 19, 26}
28	12	5	{3, 11, 19, 23, 27}	{1, 5, 9, 13, 15, 17, 25}
29	28	27	{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}
30	8	0	{}	{1, 7, 11, 13, 17, 19, 23, 29}

Table data:

\left(q, \# G_{q}, \# G^{\text{Primitive}}_{q}, P, N\right)

such that

G^{\text{Primitive}}_{q} = \left\{ \chi_{q \, . \, \ell} : \ell \in P \right\} \;\mathbin{\operatorname{and}}\; G_{q} \setminus G^{\text{Primitive}}_{q} = \left\{ \chi_{q \, . \, \ell} : \ell \in N \right\}

Definitions:

Fungrim symbol	Notation	Short description
`PrimitiveDirichletCharacters`	$G^{\text{Primitive}}_{q}$	Primitive Dirichlet characters with given modulus
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`Cardinality`	$\# S$	Set cardinality
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	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

Dirichlet characters