Paper IV: Cyclic Analytic Functions, Reciprocal Families, and Spectral Zero Data

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1 \begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\sffamily\small,line cap=round,line join=round]
2 \node[font=\sffamily\bfseries\footnotesize] at (2.00000,3.90000) {COEFFICIENT LANES};
3 \node[font=\sffamily\bfseries\footnotesize] at (6.50000,3.90000) {RECIPROCAL COEFFICIENTS};
4 \node[font=\sffamily\bfseries\footnotesize] at (11.00000,3.90000) {ZERO ORBITS};
5 \draw[fill=black!10,draw=black!40,line width=.35pt] (0.25000,2.60000) rectangle ++(0.90000,0.65000);
6 \node[] at (0.70000,2.92500) {$z^{0}$};
7 \draw[fill=black!10,draw=black!40,line width=.35pt] (1.15000,2.60000) rectangle ++(0.90000,0.65000);
8 \node[] at (1.60000,2.92500) {$z^{3}$};
9 \draw[fill=black!10,draw=black!40,line width=.35pt] (2.05000,2.60000) rectangle ++(0.90000,0.65000);
10 \node[] at (2.50000,2.92500) {$z^{6}$};
11 \draw[fill=black!10,draw=black!40,line width=.35pt] (2.95000,2.60000) rectangle ++(0.90000,0.65000);
12 \node[] at (3.40000,2.92500) {$z^{9}$};
13 \draw[fill=black!0,draw=black!40,line width=.35pt] (0.25000,1.85000) rectangle ++(0.90000,0.65000);
14 \node[] at (0.70000,2.17500) {$z^{1}$};
15 \draw[fill=black!0,draw=black!40,line width=.35pt] (1.15000,1.85000) rectangle ++(0.90000,0.65000);
16 \node[] at (1.60000,2.17500) {$z^{4}$};
17 \draw[fill=black!0,draw=black!40,line width=.35pt] (2.05000,1.85000) rectangle ++(0.90000,0.65000);
18 \node[] at (2.50000,2.17500) {$z^{7}$};
19 \draw[fill=black!0,draw=black!40,line width=.35pt] (2.95000,1.85000) rectangle ++(0.90000,0.65000);
20 \node[] at (3.40000,2.17500) {$z^{10}$};
21 \draw[fill=black!0,draw=black!40,line width=.35pt] (0.25000,1.10000) rectangle ++(0.90000,0.65000);
22 \node[] at (0.70000,1.42500) {$z^{2}$};
23 \draw[fill=black!0,draw=black!40,line width=.35pt] (1.15000,1.10000) rectangle ++(0.90000,0.65000);
24 \node[] at (1.60000,1.42500) {$z^{5}$};
25 \draw[fill=black!0,draw=black!40,line width=.35pt] (2.05000,1.10000) rectangle ++(0.90000,0.65000);
26 \node[] at (2.50000,1.42500) {$z^{8}$};
27 \draw[fill=black!0,draw=black!40,line width=.35pt] (2.95000,1.10000) rectangle ++(0.90000,0.65000);
28 \node[] at (3.40000,1.42500) {$z^{11}$};
29 \node[] at (2.00000,-0.10000) {$m\equiv j\pmod 3$};
30 \draw[fill=black!12,draw=black!40,line width=.35pt] (5.10000,2.35000) rectangle ++(0.85000,0.85000);
31 \node[] at (5.52500,2.77500) {$1$};
32 \draw[fill=black!0,draw=black!40,line width=.35pt] (5.95000,2.35000) rectangle ++(0.85000,0.85000);
33 \node[] at (6.37500,2.77500) {$0$};
34 \draw[fill=black!0,draw=black!40,line width=.35pt] (6.80000,2.35000) rectangle ++(0.85000,0.85000);
35 \node[] at (7.22500,2.77500) {$0$};
36 \draw[fill=black!12,draw=black!40,line width=.35pt] (5.10000,1.50000) rectangle ++(0.85000,0.85000);
37 \node[] at (5.52500,1.92500) {$- \frac{1}{2}$};
38 \draw[fill=black!12,draw=black!40,line width=.35pt] (5.95000,1.50000) rectangle ++(0.85000,0.85000);
39 \node[] at (6.37500,1.92500) {$1$};
40 \draw[fill=black!0,draw=black!40,line width=.35pt] (6.80000,1.50000) rectangle ++(0.85000,0.85000);
41 \node[] at (7.22500,1.92500) {$0$};
42 \draw[fill=black!12,draw=black!40,line width=.35pt] (5.10000,0.65000) rectangle ++(0.85000,0.85000);
43 \node[] at (5.52500,1.07500) {$\frac{1}{24}$};
44 \draw[fill=black!12,draw=black!40,line width=.35pt] (5.95000,0.65000) rectangle ++(0.85000,0.85000);
45 \node[] at (6.37500,1.07500) {$- \frac{1}{2}$};
46 \draw[fill=black!12,draw=black!40,line width=.35pt] (6.80000,0.65000) rectangle ++(0.85000,0.85000);
47 \node[] at (7.22500,1.07500) {$1$};
48 \node[] at (6.50000,-0.10000) {$1,\quad\frac12,\quad\frac5{24},\ldots$};
49 \draw[black!50] (11.00000,1.60000) -- (12.45000,1.60000);
50 \draw[fill=black] (11.70000,1.60000) circle (0.055);
51 \draw[fill=black] (12.35000,1.60000) circle (0.055);
52 \draw[black!50] (11.00000,1.60000) -- (11.00000,3.05000);
53 \draw[fill=black] (11.00000,2.30000) circle (0.055);
54 \draw[fill=black] (11.00000,2.95000) circle (0.055);
55 \draw[black!50] (11.00000,1.60000) -- (9.55000,1.60000);
56 \draw[fill=black] (10.30000,1.60000) circle (0.055);
57 \draw[fill=black] (9.65000,1.60000) circle (0.055);
58 \draw[black!50] (11.00000,1.60000) -- (11.00000,0.15000);
59 \draw[fill=black] (11.00000,0.90000) circle (0.055);
60 \draw[fill=black] (11.00000,0.25000) circle (0.055);
61 \node[] at (11.00000,-0.10000) {$\{\rho,i\rho,-\rho,-i\rho\}$};
62 \draw[black!25] (4.25000,-0.40000) -- (4.25000,4.10000);
63 \draw[black!25] (8.70000,-0.40000) -- (8.70000,4.10000);
64 \end{tikzpicture}

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