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Fact 1: (1979, Conway and Norton)$^{1}$

"There are $194-22-9=\color{blue}{163\,}$ $\mathbb{Z}$-independent McKay-Thompson series for the Monster."

Note: There are 194 (linear) irreducible representations of $\mathbb{M}$, hence 194 conjugacy classes. Of these, there are 22 that are complex quadratic valued. Of the remaining, there are 9 linear dependencies.$^{2}$

Fact 2: (2004, C. Cummins)$^{3}$

"The genus 0 moonshine groups have $132 + 1 + 4 + 5 + 13 + 1 + 7=\color{blue}{163}$ equivalence classes with period 1."

Note: There are exactly $6486$ genus $0$ moonshine groups, but only $371$ equivalence classes. Of these, $310\,$ have a rational representative. (Curiously, $\color{red}{310} =163+67+43+19+11+7$.)

\begin{array}{|c|r|c|c|c|c|c|c|c|c|c|c|} \hline \text{Period}& & 15^* & 11^*& 7^*& 3^*& 2^*& i& i\,2& 2 & 5 &\color{brown}{\text{Total}}\\ \hline 1&132& 0& 1& 4& 5& 0& 13& 1& 0& 7 & \color{blue}{163}\\ 2&120& 0& 0& 2& 1& 7& 4& 2& 2& 2& 140\\ 3& 26& 1& 0& 1& 4& 0& 2& 0& 0& 0& 34\\ 4& 16& 0& 0& 0& 0& 2& 0& 0& 0& 0& 18\\ 6& 10& 0& 0& 0& 0& 0& 0& 0& 0& 0& 10\\ 8& 3& 0& 0& 0& 0& 0& 0& 0& 0& 0& 3\\ 12& 2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2\\ 24& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\\ \hline \color{brown}{\text{Total}}&\color{red}{310}& 1& 1& 7& 10& 9& 19& 3& 2& 9& \color{brown}{371}\\ \hline \end{array}

Question: While the value $163$ was calculated differently in Facts 1 and 2, are they just different ways of saying the same thing, and that one implies the other?

References:

1. Monstrous Moonshine, by J.H. Conway & S.P. Norton.
2. Sporadic and Exceptional, by Yang-Hui He & John McKay.
3. Congruence Subgroups of Groups Commensurable with PSL (2, Z) of Genus 0 and 1, by C. J. Cummins.
4. Tables by C. J. Cummins.

Fact 1: (1979, Conway and Norton)$^{1}$

"There are $194-22-9=\color{blue}{163\,}$ $\mathbb{Z}$-independent McKay-Thompson series for the Monster."

Note: There are 194 (linear) irreducible representations of $\mathbb{M}$, hence 194 conjugacy classes. Of these, there are 22 that are complex quadratic valued. Of the remaining, there are 9 linear dependencies.$^{2}$

Fact 2: (2004, C. Cummins)$^{3}$

"The genus 0 moonshine groups have $\color{blue}{132 + 0 + 1 + 4 + 5 + 0 + 13 + 1 + 0 + 7=163}$ equivalence classes with period 1."

Note: There are exactly $6486$ genus $0$ moonshine groups, but only $\color{brown}{371}$ equivalence classes. Of these, $310\,$ have a rational representative. (Curiously, $\color{red}{310} =163+67+43+19+11+7$.)

\begin{array}{|c|r|c|c|c|c|c|c|c|c|c|c|} \hline \text{Period}& & 15^* & 11^*& 7^*& 3^*& 2^*& i& i\,2& 2 & 5 &\color{brown}{\text{Total}}\\ \hline 1&\color{blue}{132}& \color{blue}0& \color{blue}1& \color{blue}4& \color{blue}5& \color{blue}0& \color{blue}{13}& \color{blue}1& \color{blue}0& \color{blue}7 & \color{blue}{163}\\ 2&120& 0& 0& 2& 1& 7& 4& 2& 2& 2& 140\\ 3& 26& 1& 0& 1& 4& 0& 2& 0& 0& 0& 34\\ 4& 16& 0& 0& 0& 0& 2& 0& 0& 0& 0& 18\\ 6& 10& 0& 0& 0& 0& 0& 0& 0& 0& 0& 10\\ 8& 3& 0& 0& 0& 0& 0& 0& 0& 0& 0& 3\\ 12& 2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2\\ 24& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\\ \hline \color{brown}{\text{Total}}&\color{red}{310}& 1& 1& 7& 10& 9& 19& 3& 2& 9& \color{brown}{371}\\ \hline \end{array}

Question: While the value $163$ was calculated differently in Facts 1 and 2, are they just different ways of saying the same thing, and that one implies the other?

References:

1. Monstrous Moonshine, by J.H. Conway & S.P. Norton.
2. Sporadic and Exceptional, by Yang-Hui He & John McKay.
3. Congruence Subgroups of Groups Commensurable with PSL (2, Z) of Genus 0 and 1, by C. J. Cummins.
4. Tables by C. J. Cummins.

Fact 1: There are $194-22-9=\color{blue}{163\,}$ $\mathbb{Z}$-independent McKay-Thompson series for the Monster.

Note: There are 194 (linear) irreducible representations of $\mathbb{M}$, hence 194 conjugacy classes. Of these, there are 22 that are complex quadratic valued. Of the remaining, there are 9 linear dependencies.

Fact 2: There are $132 + 1 + 4 + 5 + 13 + 1 + 7=\color{blue}{163}$ equivalence classes with period $1$.

Note: There are exactly $6486$ genus $0$ moonshine groups, but only $371$ equivalence classes. Of these, $\color{red}{310} =163+67+43+19+11+7$ have a rational representative.

\begin{array}{|c|r|c|c|c|c|c|c|c|c|c|c|} \hline \text{Period}& & 15^* & 11^*& 7^*& 3^*& 2^*& i& i\,2& 2 & 5 &\color{brown}{\text{Total}}\\ \hline 1&132& 0& 1& 4& 5& 0& 13& 1& 0& 7 & \color{blue}{163}\\ 2&120& 0& 0& 2& 1& 7& 4& 2& 2& 2& 140\\ 3& 26& 1& 0& 1& 4& 0& 2& 0& 0& 0& 34\\ 4& 16& 0& 0& 0& 0& 2& 0& 0& 0& 0& 18\\ 6& 10& 0& 0& 0& 0& 0& 0& 0& 0& 0& 10\\ 8& 3& 0& 0& 0& 0& 0& 0& 0& 0& 0& 3\\ 12& 2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2\\ 24& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\\ \hline \color{brown}{\text{Total}}&\color{red}{310}& 1& 1& 7& 10& 9& 19& 3& 2& 9& \color{brown}{371}\\ \hline \end{array}

Q: While the value $163$ was calculated differently in Facts 1 and 2, are they just different ways of saying the same thing, and that one implies the other?

References:

1. Sporadic and Exceptional, by Yang-Hui He & John McKay.
2. Congruence Subgroups of Groups Commensurable with PSL (2, Z) of Genus 0 and 1, by C. J. Cummins
3. Tables by C. J. Cummins.

Fact 1: (1979, Conway and Norton)$^{1}$

"There are $194-22-9=\color{blue}{163\,}$ $\mathbb{Z}$-independent McKay-Thompson series for the Monster."

Note: There are 194 (linear) irreducible representations of $\mathbb{M}$, hence 194 conjugacy classes. Of these, there are 22 that are complex quadratic valued. Of the remaining, there are 9 linear dependencies.$^{2}$

Fact 2: (2004, C. Cummins)$^{3}$

"The genus 0 moonshine groups have $132 + 1 + 4 + 5 + 13 + 1 + 7=\color{blue}{163}$ equivalence classes with period 1."

Note: There are exactly $6486$ genus $0$ moonshine groups, but only $371$ equivalence classes. Of these, $310\,$ have a rational representative. (Curiously, $\color{red}{310} =163+67+43+19+11+7$.)

\begin{array}{|c|r|c|c|c|c|c|c|c|c|c|c|} \hline \text{Period}& & 15^* & 11^*& 7^*& 3^*& 2^*& i& i\,2& 2 & 5 &\color{brown}{\text{Total}}\\ \hline 1&132& 0& 1& 4& 5& 0& 13& 1& 0& 7 & \color{blue}{163}\\ 2&120& 0& 0& 2& 1& 7& 4& 2& 2& 2& 140\\ 3& 26& 1& 0& 1& 4& 0& 2& 0& 0& 0& 34\\ 4& 16& 0& 0& 0& 0& 2& 0& 0& 0& 0& 18\\ 6& 10& 0& 0& 0& 0& 0& 0& 0& 0& 0& 10\\ 8& 3& 0& 0& 0& 0& 0& 0& 0& 0& 0& 3\\ 12& 2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2\\ 24& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\\ \hline \color{brown}{\text{Total}}&\color{red}{310}& 1& 1& 7& 10& 9& 19& 3& 2& 9& \color{brown}{371}\\ \hline \end{array}

Question: While the value $163$ was calculated differently in Facts 1 and 2, are they just different ways of saying the same thing, and that one implies the other?

References:

1. Monstrous Moonshine, by J.H. Conway & S.P. Norton.
2. Sporadic and Exceptional, by Yang-Hui He & John McKay.
3. Congruence Subgroups of Groups Commensurable with PSL (2, Z) of Genus 0 and 1, by C. J. Cummins.
4. Tables by C. J. Cummins.

Fact 1: There are $194-22-9=\color{blue}{163\,}$ $\mathbb{Z}$-independent McKay-Thompson series for the Monster.

(Note: There are 194 (linear) irreducible representations of $\mathbb{M}$, hence 194 conjugacy classes. Of these, there are 22 that are complex quadratic valued. Of the remaining, there are 9 linear dependencies.)

Fact 2: There are $132 + 1 + 4 + 5 + 13 + 1 + 7=\color{blue}{163}$ equivalence classes with period $1$.

(Note: There are exactly $6486$ genus $0$ moonshine groups, but only $371$ equivalence classes. Of these, $\color{red}{310} =163+67+43+19+11+7$ have a rational representative.)

\begin{array}{|c|r|c|c|c|c|c|c|c|c|c|c|} \hline \text{Period}& & 15^* & 11^*& 7^*& 3^*& 2^*& i& i\,2& 2 & 5 &\color{brown}{\text{Total}}\\ \hline 1&132& 0& 1& 4& 5& 0& 13& 1& 0& 7 & \color{blue}{163}\\ 2&120& 0& 0& 2& 1& 7& 4& 2& 2& 2& 140\\ 3& 26& 1& 0& 1& 4& 0& 2& 0& 0& 0& 34\\ 4& 16& 0& 0& 0& 0& 2& 0& 0& 0& 0& 18\\ 6& 10& 0& 0& 0& 0& 0& 0& 0& 0& 0& 10\\ 8& 3& 0& 0& 0& 0& 0& 0& 0& 0& 0& 3\\ 12& 2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2\\ 24& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\\ \hline \color{brown}{\text{Total}}&\color{red}{310}& 1& 1& 7& 10& 9& 19& 3& 2& 9& \color{brown}{371}\\ \hline \end{array}

Q: While the value $163$ was calculated differently in Facts 1 and 2, are they just different ways of saying the same thing, and that one implies the other?

References:

1. Sporadic and Exceptional, by Yang-Hui He & John McKay.
2. Congruence Subgroups of Groups Commensurable with PSL (2, Z) of Genus 0 and 1, by C. J. Cummins
3. Tables by C. J. Cummins.

Fact 1: There are $194-22-9=\color{blue}{163\,}$ $\mathbb{Z}$-independent McKay-Thompson series for the Monster.

Note: There are 194 (linear) irreducible representations of $\mathbb{M}$, hence 194 conjugacy classes. Of these, there are 22 that are complex quadratic valued. Of the remaining, there are 9 linear dependencies.

Fact 2: There are $132 + 1 + 4 + 5 + 13 + 1 + 7=\color{blue}{163}$ equivalence classes with period $1$.

Note: There are exactly $6486$ genus $0$ moonshine groups, but only $371$ equivalence classes. Of these, $\color{red}{310} =163+67+43+19+11+7$ have a rational representative.

\begin{array}{|c|r|c|c|c|c|c|c|c|c|c|c|} \hline \text{Period}& & 15^* & 11^*& 7^*& 3^*& 2^*& i& i\,2& 2 & 5 &\color{brown}{\text{Total}}\\ \hline 1&132& 0& 1& 4& 5& 0& 13& 1& 0& 7 & \color{blue}{163}\\ 2&120& 0& 0& 2& 1& 7& 4& 2& 2& 2& 140\\ 3& 26& 1& 0& 1& 4& 0& 2& 0& 0& 0& 34\\ 4& 16& 0& 0& 0& 0& 2& 0& 0& 0& 0& 18\\ 6& 10& 0& 0& 0& 0& 0& 0& 0& 0& 0& 10\\ 8& 3& 0& 0& 0& 0& 0& 0& 0& 0& 0& 3\\ 12& 2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 2\\ 24& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1\\ \hline \color{brown}{\text{Total}}&\color{red}{310}& 1& 1& 7& 10& 9& 19& 3& 2& 9& \color{brown}{371}\\ \hline \end{array}

Q: While the value $163$ was calculated differently in Facts 1 and 2, are they just different ways of saying the same thing, and that one implies the other?

References:

1. Sporadic and Exceptional, by Yang-Hui He & John McKay.
2. Congruence Subgroups of Groups Commensurable with PSL (2, Z) of Genus 0 and 1, by C. J. Cummins
3. Tables by C. J. Cummins.