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% dischecke.tex
\documentclass[10pt]{article}
\textwidth=1.2\textwidth
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\title{Discriminants of Hecke Algebras}
\include{macros}
\begin{document}
\maketitle
\begin{abstract}
For $N<250$ we compute the discriminants of the Hecke algebras associated to
weight $2$ cusp forms and newforms of level $N$ and determine
the primes $p$ so that the cusp $\infty$ is a Weierstrass point
on $X_0(N)/\Fp$. 
\end{abstract}
\section{Introduction}
Let $S_2(N)$ be the space of
weight $2$ cusp forms for $\Gamma_0(N)$. The
Hecke algebra $\T\subset\End(S_2(N))$ is a finite commutative
$\Z$-algebra. Its discriminant $\Delta=\disc(\T)$ is important in
studying congruences between modular forms.  Let $\Delta^{\new}$ be
the discriminant of the new Hecke algebra
$\T^{\new}\subset\End(S_2^{\new}(N))$.
 By ``multiplicity one'' $\T^{\new}$ is a subring
of a product of fields and $\tilde{\T}^{\new}$ is the product
of the rings of integers of these fields.

Let $W$ be the sub $\Z$-module of $\T$ generated by the Hecke
operator $T_1,T_2,\ldots T_g$ where $g$ is the genus of $X_0(N)$. 
When $N$ is prime $W$ has finite index 
in $\T$, but when $\infty$ is a Weierstrass point on $X_0(N)$ the index
will not be finite.  If $p\nmid N$, the cusp $\infty$ is a Weierstrass point on 
$X_0(N)/\Fp$ iff $p|[\T:W]$. 
%\section{Algorithms}
\section{Tables}
The notation is as above.
\newpage
\begin{center}
\begin{tabular}{|l||c|c|c|c|}\hline
$N$ & $\Delta^{\new}$ & $\Delta/\Delta^{\new}$ 
    & $[\tilde{\T}^{\new}:\T^{\new}]$ & $[\T:W]$\\ \hline\hline

11 & $1$ & $1$ & $1$ & $1$\\ \hline

14 & $1$ & $1$ & $1$ & $1$\\ \hline

15 & $1$ & $1$ & $1$ & $1$\\ \hline

17 & $1$ & $1$ & $1$ & $1$\\ \hline

19 & $1$ & $1$ & $1$ & $1$\\ \hline

20 & $1$ & $1$ & $1$ & $1$\\ \hline

21 & $1$ & $1$ & $1$ & $1$\\ \hline

23 & $5$ & $1$ & $1$ & $1$\\ \hline

24 & $1$ & $1$ & $1$ & $1$\\ \hline

26 & $2^2$ & $1$ & $2$ & $1$\\ \hline

27 & $1$ & $1$ & $1$ & $1$\\ \hline

29 & $2^3$ & $1$ & $1$ & $1$\\ \hline

30 & $1$ & $-2^2.7$ & $1$ & $1$\\ \hline

31 & $5$ & $1$ & $1$ & $1$\\ \hline

32 & $1$ & $1$ & $1$ & $1$\\ \hline

33 & $1$ & $-3^2.11$ & $1$ & $1$\\ \hline

34 & $1$ & $-2^2.7$ & $1$ & $1$\\ \hline

35 & $2^2.17$ & $1$ & $2$ & $1$\\ \hline

36 & $1$ & $1$ & $1$ & $1$\\ \hline

37 & $2^2$ & $1$ & $2$ & $1$\\ \hline

38 & $2^2$ & $-2^3.3^2$ & $2$ & $1$\\ \hline

39 & $2^5$ & $1$ & $2$ & $1$\\ \hline

40 & $1$ & $0$ & $1$ & $?$\\ \hline

41 & $2^2.37$ & $1$ & $1$ & $1$\\ \hline

42 & $1$ & $2^9.7$ & $1$ & $1$\\ \hline

43 & $2^5$ & $1$ & $2$ & $2$\\ \hline

44 & $1$ & $-2^6$ & $1$ & $1$\\ \hline

45 & $1$ & $2^2$ & $1$ & $1$\\ \hline

46 & $1$ & $2^2.5^4.41$ & $1$ & $1$\\ \hline

47 & $19.103$ & $1$ & $1$ & $1$\\ \hline

48 & $1$ & $0$ & $1$ & $?$\\ \hline

49 & $1$ & $1$ & $1$ & $1$\\ \hline

50 & $2^2$ & $1$ & $2$ & $1$\\ \hline

51 & $2^2.17$ & $-2^6.3$ & $2$ & $3$\\ \hline

52 & $1$ & $2^4.3^2$ & $1$ & $3$\\ \hline

53 & $2^4.37$ & $1$ & $2$ & $3$\\ \hline

54 & $2^2$ & $-2^3.3^4$ & $2$ & $\infty$\\ \hline

55 & $2^5$ & $-7^2.19$ & $2$ & $1$\\ \hline

56 & $2^2$ & $0$ & $2$ & $?$\\ \hline

57 & $2^2.3^2$ & $-2^7$ & $2.3$ & $1$\\ \hline

58 & $2^2$ & $2^{10}.17$ & $2$ & $2$\\ \hline

59 & $2^3.31.557$ & $1$ & $1$ & $1$\\ \hline

61 & $2^4.37$ & $1$ & $2$ & $2$\\ \hline

62 & $2^4.3$ & $2^2.5^2.11^2.41$ & $2$ & $3$\\ \hline

63 & $2^4.3$ & $2^4$ & $2$ & $2$\\ \hline

64 & $1$ & $0$ & $1$ & $\infty$\\ \hline

65 & $2^{11}.3$ & $1$ & $2^3$ & $3$\\ \hline

66 & $2^4$ & $-2^8.3^4.5^2.7.11^2$ & $2^2$ & $2$\\ \hline

67 & $2^4.5^4$ & $1$ & $2^2.5$ & $2^3$\\ \hline

68 & $2^2.3$ & $-2^8.3^2.7$ & $1$ & $2^2$\\ \hline

69 & $2^4.5$ & $5^2.7^2.11^2$ & $2^2$ & $2$\\ \hline

70 & $1$ & $2^{18}.3^2.5.17^2$ & $1$ & $3$\\ \hline


71 & $3^4.257^2$ & $1$ & $3^2$ & $1$\\ \hline

\end{tabular}

\begin{tabular}{|l||c|c|c|c|}\hline
$N$ & $\Delta^{\new}$ & $\Delta/\Delta^{\new}$ 
    & $[\tilde{\T}^{\new}:\T^{\new}]$ & $[\T:W]$\\ \hline\hline

72 & $1$ & $0$ & $1$ & $\infty$\\ \hline

73 & $2^4.3^2.5.13$ & $1$ & $2^2.3$ & $2$\\ \hline

74 & $2^4.5.13$ & $2^9.3^2.5^2$ & $2^2$ & $11$\\ \hline

75 & $2^2.3^2$ & $2^2.3^2$ & $2.3$ & $1$\\ \hline

76 & $1$ & $-2^{11}.3^6$ & $1$ & $1$\\ \hline

77 & $2^6.5^3$ & $-2^7.3^3$ & $2^3.5$ & $5$\\ \hline

78 & $1$ & $-2^{22}.3.5^2.7^3.11.17$ & $1$ & $2^2$\\ \hline

79 & $2^2.83.983$ & $1$ & $2$ & $2^3$\\ \hline

80 & $2^2$ & $0$ & $2$ & $?$\\ \hline

81 & $2^2.3$ & $0$ & $1$ & $?$\\ \hline

82 & $2^5$ & $-2^{10}.23.37^2$ & $2$ & $2$\\ \hline

83 & $2^4.197.11497$ & $1$ & $2$ & $2^2$\\ \hline

84 & $2^2$ & $-2^{22}.3^4.7$ & $2$ & $3$\\ \hline

85 & $2^{11}.3$ & $-2^{10}$ & $2^3$ & $7$\\ \hline

86 & $2^4.3.5.7$ & $-2^{14}.3^2.5^2.7^2$ & $2^2$ & $41$\\ \hline

87 & $2^4.5.229$ & $2^6.23^2.73$ & $2^2$ & $2^3$\\ \hline

88 & $2^6.17$ & $0$ & $2^3$ & $?$\\ \hline

89 & $2^6.5^3.6689$ & $1$ & $2.5$ & $3^2$\\ \hline

90 & $2^4$ & $-2^{22}.7^3$ & $2^2$ & $1$\\ \hline

91 & $2^{15}.79$ & $1$ & $2^5$ & $2$\\ \hline

92 & $2^2$ & $2^{12}.3^2.5^7.41$ & $2$ & $1$\\ \hline

93 & $2^4.5.229$ & $2^{12}.5^2$ & $2^2$ & $2^3$\\ \hline

94 & $2^5$ & $2^4.19^2.47^2.103^2.457$ & $2$ & $7$\\ \hline

95 & $2^{12}.37.709$ & $-3^4.11$ & $2^3$ & $11$\\ \hline

96 & $2^2$ & $0$ & $2$ & $\infty$\\ \hline

97 & $2^6.7^2.2777$ & $1$ & $2^3$ & $2.3$\\ \hline

98 & $2^5$ & $-2^{12}.7$ & $2$ & $\infty$\\ \hline

99 & $2^4.3^4$ & $-2^6.3^8.11$ & $2^2.3^2$ & $2$\\ \hline

100 & $1$ & $2^8.3^4$ & $1$ & $\infty$\\ \hline

101 & $2^8.17568767$ & $1$ & $2$ & $3^3$\\ \hline

102 & $2^4$ & $2^{41}.3^6.7^2.17^2$ & $2^2$ & $3.5^2$\\ \hline

103 & $2^4.5.17.411721$ & $1$ & $2^2$ & $2^2.7$\\ \hline

104 & $2^2.17$ & $0$ & $2$ & $?$\\ \hline

105 & $2^4.5$ & $-2^{30}.5^2.7.11.13.17^2$ & $2^2$ & $2.3$\\ \hline

106 & $2^4.3^2$ & $2^{18}.5^2.7.37^2.151$ & $2^2.3$ & $2.13$\\ \hline

107 & $2^6.5.7.1667.19079$ & $1$ & $2^2$ & $2^2.5$\\ \hline

108 & $1$ & $0$ & $1$ & $\infty$\\ \hline

109 & $2^{10}.7^2.7537$ & $1$ & $2^5$ & $2.5$\\ \hline

110 & $2^8.3.11$ & $-2^{22}.5^2.7^7.17.19^2$ & $2^4$ & $2.3$\\ \hline

111 & $2^{12}.37.389$ & $2^4.3.5^2.7^2.11$ & $2^3$ & $2^2.3$\\ \hline

112 & $2^6$ & $0$ & $2^3$ & $?$\\ \hline

113 & $2^{10}.3^4.7^2.11^2.107$ & $1$ & $2^4.3.11$ & $2^2.17$\\ \hline

114 & $2^4$ & $-2^{36}.3^{12}.5^6.7.11^2$ & $2^2$ & $3^5$\\ \hline

115 & $2^6.5.17^2.53$ & $2^{12}.5^4.11$ & $2^3$ & $2^2$\\ \hline

116 & $2^4.5^2$ & $2^{29}.3^4.17$ & $2^2.5$ & $2^2.7$\\ \hline

117 & $2^{11}.3$ & $2^{24}$ & $2^3$ & $2^4$\\ \hline

118 & $2^4.3^2$ & $-2^{18}.7.19^2.31^2.557^2$ & $2^2.3$ & $223$\\ \hline

119 & $2^8.71.131.311.1459$ & $-2^6.3^3$ & $2^4$ & $2^4$\\ \hline

120 & $2^2$ & $0$ & $2$ & $\infty$\\ \hline

121 & $2^4.3^2$ & $2^6.3^2$ & $2^2.3$ & $\infty$\\ \hline

122 & $2^6.13.229$ & $2^{18}.7.13^2.37^2.151$ & $2^3$ & $2.83$\\ \hline

123 & $2^{15}.79$ & $-2^8.5^2.23^2.37^2.191$ & $2^5$ & $13$\\ \hline

124 & $2^2$ & $2^{20}.3^6.5^3.11^4.41$ & $2$ & $3^2$\\ \hline
\end{tabular}

\begin{tabular}{|l||c|c|c|c|}\hline
$N$ & $\Delta^{\new}$ & $\Delta/\Delta^{\new}$ 
    & $[\tilde{\T}^{\new}:\T^{\new}]$ & $[\T:W]$\\ \hline\hline

125 & $2^8.5^{12}.11$ & $0$ & $2^2.5^4$ & $\infty$\\ \hline

126 & $2^2$ & $2^{49}.3^8.5^2.7^3$ & $2$ & $2$\\ \hline

127 & $2^6.3^4.7.86235899$ & $1$ & $2^3$ & $2^2.29$\\ \hline

128 & $2^{16}$ & $0$ & $2^8$ & $\infty$\\ \hline

129 & $2^{12}.5^2.71$ & $-2^{23}.3^2.5^2.7^2$ & $2^3.5$ & $17$\\ \hline

130 & $2^4$ & $-2^{38}.3^6.5^4.7^3.11.17.19$ & $2^2$ & $5.7$\\ \hline

131 & $2^9.5.46141.75619573$ & $1$ & $2$ & $2.5^2$\\ \hline

132 & $2^2$ & $-2^{32}.3^{12}.5^6.7.11^4$ & $2$ & $2^3$\\ \hline

133 & $2^{16}.5^2.13.229$ & $-3^5.7^2$ & $2^8$ & $2.43$\\ \hline

134 & $2^6.3^4.11.43$ & $-2^{10}.3^2.5^{12}.19^2.41$ & $2^3$ & $29.31$\\ \hline

135 & $2^6.3^4.13^2$ & $0$ & $2^3.3^2$ & $\infty$\\ \hline

136 & $2^8.5$ & $0$ & $2^4$ & $?$\\ \hline

137 & $2^{10}.5^2.29.401.895241$ & $1$ & $2^4$ & $2^5.5$\\ \hline

138 & $2^8.5$ & $2^{32}.3^3.5^{10}.7^5.11^6.41^2$ & $2^4$ & $2^4.7^2$\\ \hline

139 & $2^8.3^2.7^2.997.2151701$ & $1$ & $2^4.3$ & $2^3.41$\\ \hline

140 & $2^2$ & $2^{43}.3^{11}.5^4.17^3$ & $2$ & $2^2.3$\\ \hline

141 & $2^{14}.3^2.17$ & $3^2.7^2.19^2.43^2.103^2.3529$ & $2^7.3$ & $2^2.5.13$\\ \hline

142 & $2^8.3^2$ & $2^6.3^{22}.7.103.257^4$ & $2^4.3$ & $3.67$\\ \hline

143 & $2^{10}.5.7.19.103.5560463$ & $-2^6.3^6$ & $2^5$ & $2.3^2$\\ \hline

144 & $2^2$ & $0$ & $2$ & $\infty$\\ \hline

145 & $2^{21}.37^2$ & $2^6.5^4.7^2.19^2$ & $2^7$ & $2.7$\\ \hline

146 & $2^{12}.101.389$ & $-2^{12}.3^{10}.5^2.7.13^2.17.19^2$ & $2^3$ & $97$\\ \hline

147 & $2^{14}.3^2.7^4$ & $-2^{14}.3^3$ & $2^4.3.7^2$ & $\infty$\\ \hline

148 & $2^2.17$ & $2^{27}.3^{10}.5^6.13^2$ & $2$ & $3^2.11^2$\\ \hline

149 & $2^{12}.7^2.234893.1252037$ & $1$ & $2^3$ & $107$\\ \hline

150 & $2^4$ & $-2^{34}.3^{12}.5^8.7^3.11^2$ & $2^2$ & $\infty$\\ \hline

151 & $2^6.7^2.11.67^2.257.439867$ & $1$ & $2^3.67$ & $2.11^2$\\ \hline

152 & $2^8.31^2$ & $0$ & $2^4$ & $?$\\ \hline

153 & $2^{12}.3^2.17$ & $-2^{30}.3^5.17^2$ & $2^6.3$ & $3^3$\\ \hline

154 & $2^8.5$ & $2^{56}.3^{14}.5^8.7.11$ & $2^4$ & $2.23$\\ \hline

155 & $2^{24}.29.73.5077$ & $5^4.7^4.19^2$ & $2^{10}$ & $2.73$\\ \hline

156 & $2^2$ & $2^{57}.3^{11}.5^4.7^5.11^2.17$ & $2$ & $2^2.3$\\ \hline

157 & $2^{13}.61.397.48795779$ & $1$ & $2^5$ & $2.13$\\ \hline

158 & $2^{13}.3^3.5^2$ & $2^{22}.7.17.53^2.83^2.271.983^2$ & $2^5.3.5$ & $2^2.541$\\ \hline

159 & $2^8.19.103.1054013$ & $2^8.3.7^2.37^2.107^2.227$ & $2^4$ & $2^4.3.5$\\ \hline

160 & $2^{13}$ & $0$ & $2^5$ & $\infty$\\ \hline

161 & $2^{16}.5^3.37.536777$ & $2^{12}.5^2.19^2.29$ & $2^6.5$ & $2.97$\\ \hline

162 & $2^4.3^4$ & $0$ & $2^2.3^2$ & $\infty$\\ \hline

163 & $2^{15}.3^2.65657.82536739$ & $1$ & $2^6.3$ & $2^2.347$\\ \hline

164 & $2^4.1613$ & $-2^{40}.3^6.23.37^3$ & $1$ & $2^3.5$\\ \hline

165 & $2^{17}.3.37$ & $-2^{50}.3^5.5^2.7^5.11^2.19^2$ & $2^5$ & $2^2.3$\\ \hline

166 & $2^6.5.229$ & $-2^{26}.7.71.131^2.197^2.11497^2$ & $2^3$ & $2.5.479$\\ \hline

167 & $2^4.5.8269.5103536431379173$ & $1$ & $2^2$ & $2.3^2$\\ \hline

168 & $2^2$ & $0$ & $2$ & $\infty$\\ \hline

169 & $2^8.3.7^4.13^4$ & $1$ & $2^3.13^2$ & $\infty$\\ \hline

170 & $2^{12}.5^2.17$ & $2^{68}.3^8.5^3.7^5.17$ & $2^6.5$ & $839$\\ \hline

171 & $2^{16}.3^7.11^2$ & $-2^{33}.3^8$ & $2^7.3^2$ & $2.3^3$\\ \hline

172 & $2^5$ & $-2^{39}.3^{10}.5^6.7^6$ & $2$ & $41^2$\\ \hline

173 & $2^{14}.5^2.7.29.5608385124289$ & $1$ & $2^4$ & $79$\\ \hline

174 & $2^8.5^2$ & $-2^{38}.3^3.5^4.7^4.11^3.13^2.17^2\ldots$ & $2^4.5$ & $2^6.5.11^2$\\
    &  & $\ldots 23^4.41.47.73^2.229^2$ & &\\ \hline

175 & $2^{12}.3^4.5^6.17$ & $2^{18}.3^4.17^2$ & $2^6.3^2.5^2$ & $2^2$\\ \hline

176 & $2^{14}.17$ & $0$ & $2^7$ & $?$\\ \hline
\end{tabular}

\begin{tabular}{|l||c|c|c|c|}\hline
$N$ & $\Delta^{\new}$ & $\Delta/\Delta^{\new}$ 
    & $[\tilde{\T}^{\new}:\T^{\new}]$ & $[\T:W]$\\ \hline\hline

177 & $2^{16}.5^3.229$ & $-2^6.31^4.229^2.557^2.20627$ & $2^8$ & $2^4.149$\\ \hline

178 & $2^{16}.71$ & $-2^{30}.5^6.7^4.199.6689^2$ & $2^5$ & $2^2.487$\\ \hline

179 & $2^{12}.3^4.7^2.313.137707.536747147$ & $1$ & $2^3.3^2$ & $2.59$\\ \hline

180 & $1$ & $-2^{73}.3^{16}.5.7^3$ & $1$ & $\infty$\\ \hline

181 & $2^{16}.5^2.7.61.397.595051637$ & $1$ & $2^5$ & $2^4.3^3$\\ \hline

182 & $2^8.3^2$ & $2^{51}.3^{20}.5^4.7^6.11^2.23.79^2$ & $2^4.3$ & $2^2.3^2.5.37$\\ \hline

183 & $2^{22}.37.127.5623$ & $2^{27}.3^2.19^2.23.37^2$ & $2^5$ & $2^2.5^4$\\ \hline

184 & $2^{14}.3^2.17$ & $0$ & $2^7.3$ & $?$\\ \hline

185 & $2^{26}.3^2.23029.60869$ & $2^{24}.3^2.5$ & $2^9.3$ & $193$\\ \hline

186 & $2^8.17$ & $2^{62}.3^5.5^8.7^2.11^6.13.19^2\ldots $ & $2^4$ & $2^2.5^5$\\ 
    &           & $\ldots 41^2.127.229^2$ && \\ \hline

187 & $2^{28}.3.5^2.17.37.8461$ & $2^{24}.3^4.11$ & $2^{11}.5$ & $2.3^2.11$\\ \hline

188 & $2^4.5.13$ & $2^{34}.3^6.19^3.47^4.103^3.457$ & $2^2$ & $7^2$\\ \hline

189 & $2^{16}.3^9.7$ & $0$ & $2^6.3^4$ & $\infty$\\ \hline

190 & $2^8.17$ & $-2^{56}.3^{16}.5^3.11^4.13^2.37^2\ldots $ & $2^4$ & $11.53$\\ 
    &           & $\ldots 113.151.709^2 $ && \\ \hline

191 & $2^4.3^3.5.382146223.319500117632677$ & $1$ & $2^2$ & $2^2.223$\\ \hline

192 & $2^{12}$ & $0$ & $2^6$ & $\infty$\\ \hline

193 & $2^{14}.5.11^2.17.103.401.4153.680059$ & $1$ & $2^7.11$ & $2^7$\\ \hline

194 & $2^{20}.7^2.137.223$ & $-2^{14}.7^5.67^2.71^2.433.2777^2$ & $2^4.7$ & $3.22283$\\ \hline

195 & $2^{14}.3^2.79$ & $-2^{77}.3^9.5^2.7^6.11^2.13$ & $2^6.3$ & $2.3^2.5.7$\\ \hline

196 & $2^5.7^4$ & $-2^{38}.3^8.7$ & $2.7^2$ & $\infty$\\ \hline

197 & $2^{18}.5^2.61.397.35217676193989$ & $1$ & $2^6.5$ & $7.383$\\ \hline

198 & $2^{10}$ & $-2^{78}.3^{24}.5^6.7^5.11^2$ & $2^5$ & $2.7^2$\\ \hline

199 & $2^8.3.5^3.29.31.71^2.347.947.37316093$ & $1$ & $2^4.71$ & $2^4.241$\\ \hline

200 & $2^8.3^2.5^4$ & $0$ & $2^4.3.5^2$ & $\infty$\\ \hline

201 & $2^{18}.3^2.37.269.953$ & $-2^{15}.3^2.5^{10}.19^2.29^2.61.109$ & $2^7.3$ & $3.239$\\ \hline

202 & $2^8.3^4.10273$ & $2^{33}.3^4.17^4.17568767^2$ & $2^4$ & $2^3.2381$\\ \hline

203 & $2^{35}.3^7.17.29.37.7547$ & $2^{22}.5^2.7^2$ & $2^{14}.3^3$ & $2^3.193$\\ \hline

204 & $2^2$ & $2^{102}.3^{23}.7^2.11^2.13.17^3$ & $2$ & $2^2.3.5^4$\\ \hline

205 & $2^{30}.3^2.5.13.229^2$ & $-2^{14}.13^2.31^2.37^2.239$ & $2^{15}.3$ & $7^2.23$\\ \hline

206 & $2^{16}.3^2.5^2.13.29.359$ & $2^{12}.3^2.5^2.17^4.19^2.67^2\ldots $ & $2^6.3.5$ & $17.2843$\\ 
    &           & $\ldots 1801.411721^2$ && \\ \hline

207 & $2^{20}.3^4.5^2.11^2$ & $2^{28}.3^8.5^5.7^2.11^4$ & $2^7.3^2.11$ & $2^6$\\ \hline

208 & $2^{14}.3^2.17$ & $0$ & $2^7.3$ & $?$\\ \hline

209 & $2^{32}.3^4.15427.2002061$ & $2^{10}.3^4.5^5.7.19$ & $2^9$ & $3.11^2$\\ \hline

210 & $2^8$ & $2^{176}.3^{14}.5^8.7^7.11^2.13^2.17^4$ & $2^4$ & $2^3.3$\\ \hline

211 & $2^{14}.3.5.7^4.41^2.43.229.52184516509$ & $1$ & $2^6.7.41$ & $2^5.5.23$\\ \hline

212 & $2^4.3^3.7^3$ & $2^{56}.3^{12}.5^4.7.37^3.151$ & $2.7$ & $2^2.13^3$\\ \hline

213 & $2^{20}.3^2.5^4.13.89$ & $3^{14}.5^2.19^2.37.61^2.229.257^4$ & $2^{10}.3$ & $2^3.3.17.53$\\ \hline

214 & $2^{16}.3^6$ & $-2^{24}.5^4.7^4.11^2.41.109^2.521\ldots$  & $2^6.3^2$ & $3.67.4903$\\
    &           & $\ldots 1667^2.19079^2$ && \\ \hline

215 & $2^{14}.3.7^4.101.107.321821.1933097$ & $-2^{22}.5^2.31^2.41$ & $2^7.7^2$ & $2^2.3^6$\\ \hline

216 & $2^8.3^4$ & $0$ & $2^4.3^2$ & $\infty$\\ \hline

217 & $2^{31}.3^8.11.31.557.619$ & $5^2.19^2.31^2.281$ & $2^{14}$ & $2^4.5.43$\\ \hline

218 & $2^{19}.3^4.5.11^2.23$ & $2^{34}.3^4.7^7.41^2.167.601.7537^2$ & $2^7.11$ & $2.7.21227$\\ \hline

219 & $2^{24}.3^2.29.73.1189637$ & $-2^{16}.3^8.5^4.13^2.17^2.23.29^2.61$ & $2^{10}.3$ & $3^2.41.71$\\ \hline

220 & $2^2$ & $2^{77}.3^{16}.5^4.7^{11}.11^4.17.19^3$ & $2$ & $2.3.5$\\ \hline

221 & $2^{40}.3^3.5^6.7.37.109^2.229$ & $-2^{12}.3^3$ & $2^{17}.3.5^2$ & $5.107$\\ \hline

222 & $2^8.3^2$ & $-2^{60}.3^{14}.5^{10}.7^4.11^4.13^5.19^2\ldots$ & $2^4.3$ & $2^4.3^5.11^2$\\ 
    &           & $\ldots23^2.31.37^2.109.389^2.409$ && \\ \hline

223 & $2^{18}.7^2.19.103.3995922697473293141$ & $1$ & $2^6.7$ & $2.3^2.5.641$\\ \hline

224 & $2^{18}.5^2$ & $0$ & $2^9$ & $\infty$\\ \hline
\end{tabular}

\begin{tabular}{|l||c|c|c|c|}\hline
$N$ & $\Delta^{\new}$ & $\Delta/\Delta^{\new}$ 
    & $[\tilde{\T}^{\new}:\T^{\new}]$ & $[\T:W]$\\ \hline\hline

225 & $2^{12}.3^2.5^5$ & $2^{38}.3^{10}$ & $2^6.3.5^2$ & $\infty$\\ \hline

226 & $2^{23}.3.5^3$ & $-2^{34}.3^{12}.7^9.11^4.23.41^2\ldots$  & $2^7$ & $2^2.190129$\\ 
    & & $\ldots 107^2.167$ & & \\\hline

227 & $2^{19}.5^3.7^4.13^2.29.31^2\ldots $ & $1$ & $2^7.5.7.13.31$ & $2^2.29^2$\\ 
    & $\ldots 13591.57139.273349$ & & & \\\hline

228 & $2^4.3.11$ & $-2^{88}.3^{36}.5^{12}.7.11^4$ & $2^2$ & $2^2.3^9$\\ \hline

229 & $2^{24}.107.17467.39555937\times\ldots$ & $1$ & $2^9$ & $2^2.11.193$\\ 
    & $\ldots 53625889$ & & & \\ \hline

230 & $2^{16}.3^2.5.7.13.367$ & $2^{52}.3^2.5^{18}.11^4.17^4.19^2\ldots$ & $2^8$ & $2^2.5^2.53$\\ 
   & & $\ldots41^2.43^2.47.53^2$ & & \\\hline

231 & $2^{20}.3^4.5^3.7.31.229$ & $-2^{59}.3^{16}.5^{12}.7^3.11^3$ & $2^{10}.5$ & $2^2.29$\\ \hline

232 & $2^{20}.71$ & $0$ & $2^7$ & $?$\\ \hline

233 & $2^{22}.3^7.53.139.653.4127\times \ldots$ & $1$ & $2^7.3^3$ & $2^2.3.11.197$\\
    & $\ldots24989.8388019$ & & & \\\hline

234 & $2^6.3^2.5^2$ & $-2^{118}.3^{13}.5^6.7^9.11.17^3$ & $2^3.3.5$ & $2^5.3^2$\\ \hline

235 & $2^{18}.3^2.73.1453.3851.1916279$ & $2^{24}.3^8.19^3.103^2$ & $2^9.3$ & $2^2.3.17.31$\\ \hline

236 & $2^2.3.7^2.107$ & $-2^{55}.3^{12}.7.19^4.31^3.557^3$ & $2.7$ & $223^2$\\ \hline

237 & $2^{18}.19.103.1705391537$ & $2^{28}.3^2.5^2.11^2.31^2.83^2.983^2$ & $2^6$ & $2^6.5.97$\\ \hline

238 & $2^{16}.5$ & $-2^{73}.3^{11}.5^2.7^4.61^2.71^2.131^2\ldots $ & $2^8$ & $2^2.13.331$\\ 
    & &    $\ldots 311^2.337.479.1459^2$ & & \\\hline

239 & $2^6.7^2.2833.51817.97423\times \ldots$ & $1$ & $2^3$ & $2^2.3.11^2$\\ 
    & $\ldots 1174779433.8920940047$ & & & \\\hline

240 & $2^{10}$ & $0$ & $2^5$ & $\infty$\\ \hline

241 & $2^{23}.97.1489.20857\times \ldots$ & $1$ & $2^7$ & $2^7.151$\\
    &      $\ldots  651474368435017$  & & &\\\hline

242 & $2^{18}.3^2.5^2.11^8$ & $2^{41}.3^{12}.5^4.7^2$ & $2^7.11^4$ & $\infty$\\ \hline

243 & $2^{13}.3^{40}$ & $0$ & $2^4.3^{15}$ & $\infty$\\ \hline

244 & $2^4.5077$ & $2^{60}.3^{10}.7.13^6.37^3.151.229^2$ & $2$ & $2^2.3.5.83^2$\\ \hline

245 & $2^{36}.3^4.7^8.17$ & $-2^{28}.3^8.5.17^2$ & $2^{12}.3^2.7^4$ & $2^2$\\ \hline

246 & $2^{12}.3^2.5^2$ & $2^{96}.3^8.5^{10}.7^6.11^2.23^7.37^4\ldots$ & $2^6.3.5$ & $7.73^2$\\ 
   & & $\ldots 79^2.191^2$ & & \\\hline

247 & $2^{30}.3^4.5^2.11.31^2.619\times\ldots$ & $-2^6.3^4.5^2$ & $2^{15}.31$ & $2^2.29.241$\\ 
   & $\ldots 57713.2655049$ & & & \\\hline

248 & $2^{22}.3.11.79$ & $0$ & $2^{10}$ & $?$\\ \hline

249 & $2^{33}.389.23029$ & $-2^8.3^2.5^2.11.31^2.197^2.2711^2\ldots$ & $2^{11}$ & $2.13.41.89$\\
    & & $\ldots 11497^2.295201$ & & \\ \hline

250 & $2^8.5^{12}$ & $0$ & $2^4.5^4$ & $\infty$\\ \hline

251 & $2^{14}.5^2.29.373\times\ldots $ & $1$ & $2^4$ & $2^2.3.797$\\ 
    & $8768135668531.2006012696666681$ & & & \\ \hline

252 & $2^2$ & $2^{136}.3^{30}.5^2.7^3$ & $2$ & $\infty$\\ \hline

253 & $2^{28}.3^4.13^2.2711.3187.170701$ & $-2^{12}.3^2.5^7.7.13.19^2.23^2$ & $2^{14}$ & $2.4507$\\ \hline

254 & $2^{28}.3^4.17.569$ &  $2^{16}.3^{18}.7^2.17^2.41.71\times\ldots$ & $2^{11}$ & $7.189713$\\
   & & $\ldots 383^2.1231.86235899^2$ & & \\ \hline

255 & $2^{23}.5.13.229.1721$ & $-2^{99}.3^{14}.5^2.11^3.13^3.17^4$ & $2^{10}$ & $2^3.3^3.379$\\ \hline

256 & $2^{31}$ & $0$ & $2^{14}$ & $\infty$\\ \hline

257 & $2^{29}.29.479.71711.409177\times\ldots$ & $1$ & $2^{11}$ & $2.29.3251$\\ 
    & $\ldots 654233.32354821$ & & & \\ \hline


258 & $2^{12}.3^2.5^2$ & $-2^{104}.3^{14}.5^{16}.7^{17}\times\ldots$  & $2^6.3.5$ & $7^3.199^2$\\ 
   & & $\ldots 11^2.17.19^2.37.71^2.109$ & & \\\hline

259 & $2^{43}.3^9.5^3.7^2.17^3.29.37.167$ & $2^4.3^{12}.7^4$ & $2^{20}.3.17$ & $2^2.3.29.193$\\ \hline

260 & $2^6.3.47$ & $-2^{117}.3^{26}.5^6.7^5.11^2.17.19^2$ & $2^2$ & $\infty$\\ \hline

261 & $2^{19}.3^4.5^3.23^2.229$ & $2^{47}.3^8.5^2.23^4.73.229^2$ & $2^8.3^2.23$ & $2^{10}.5$\\ \hline

\end{tabular}


\begin{tabular}{|l||c|c|c|c|}\hline
$N$ & $\Delta^{\new}$ & $\Delta/\Delta^{\new}$ 
    & $[\tilde{\T}^{\new}:\T^{\new}]$ & $[\T:W]$\\ \hline\hline

262 & $2^{19}.3^3.5.11^2.13$ & $-2^{43}.3^4.5^2.11^2.71\times\ldots $ & $2^7.3.11$ & $29.482689$\\ 
    & & $\ldots 313^2.46141^2.75619573^2$ & & \\\hline

263 & $2^{10}.11.61.397.15631853\times\ldots$& $1$ & $2^5$ & $2^3.19.223$\\
    & $\ldots 34867513.97092067.252746489$ & & & \\\hline


264 & $2^6.3^2$ & $0$ & $2^3.3$ & $\infty$\\ \hline

265 & $2^{37}.3^2.5^4.7^3.11^2.13.17^2.113$ & $2^{26}.3^4.5^3.31^2.37^2.239$ & $2^{13}.5.7.11.17$ & $2.3^3.103$\\ \hline

266 & $2^{16}.5.7.13.29.67$ & $2^{51}.3^{31}.5^6.7^6.11^6.13^4.17.19^3.41.47.229^2$ & $2^8$ & $3.216679$\\ \hline

267 & $2^{26}.3^4.5^2.7^4.13^2.97.241$ & $-2^{17}.3^4.5^8.11.13^2.17^2.37.113^2.911.6689^2$ & $2^{13}.5.7$ & $3^2.17.41.271$\\ \hline

268 & $2^4.3^3.5.7$ & $-2^{46}.3^{20}.5^{20}.11^2.19^4.41.43^2$ & $2^2.3$ & $29^2.31^2$\\ \hline

269 & $2^{22}.3^2.43.151.27767.65657.5550873754172978311$ & $1$ & $2^6.3$ & $11^2.19^2$\\ \hline

270 & $2^8$ & $0$ & $2^4$ & $\infty$\\ \hline

271 & $2^{12}.3^2.1367.6091.592661.1132673.14171513.172450541$ & $1$ & $2^6.3$ & $2.3.17239$\\ \hline

272 & $2^{28}.3^3.5$ & $0$ & $2^{13}.3$ & $?$\\ \hline

273 & $2^{25}.79.4357$ & $-2^{111}.3^9.5^4.7^6.11.79^2$ & $2^9$ & $2^2.3.313$\\ \hline

274 & $2^{20}.3^2.19.37.1321$ & $-2^{26}.5^4.7^2.11^2.29^2.109^2.149^2.401^2.769.47143.895241^2$ & $2^7.3$ & $2.11.304937$\\ \hline

275 & $2^{31}.3^8.5^{12}.7^2.11^2.13^2$ & $-2^{28}.3^6.5^4.7^4.19$ & $2^{12}.3^3.5^5.7$ & $2^2$\\ \hline

276 & $2^{10}.5$ & $-2^{108}.3^{19}.5^{21}.7^9.11^{11}.41^2$ & $2^2$ & $2^4.7^4$\\ \hline

277 & $2^{22}.5^2.19.29.37.137^2.92767.1530091.25531570859$ & $1$ & $2^{10}.5.137$ & $2.3^2.8311$\\ \hline

278 & $2^{22}.3^4.7.17^2.103.107$ & $-2^{38}.3^4.7^7.41^2.167.271^3.997^2.2151701^2$ & $2^8.17$ & $6084451$\\ \hline

279 & $2^{28}.3^5.5^2.229.1373^2$ & $2^{60}.3^8.5^5.229^2$ & $2^{11}.3^2$ & $2^9.5^3$\\ \hline

280 & $2^{18}.3.11.17$ & $0$ & $2^9$ & $\infty$\\ \hline

281 & $2^{22}.3.5.181.857.8388019.2647382149.1778899342669$ & $1$ & $2^7$ & $2^2.79309$\\ \hline

282 & $2^{23}.3^2.37$ & $2^{96}.3^{13}.5^4.7^6.17^2.19^4.23^2.43^4.47^4.103^4.457^2.3529^2$ & $2^8$ & $2^7.5^6$\\ \hline

283 & $2^{40}.349.1297.413713.73199099.5832488839$ & $1$ & $2^9$ & $2^7.2687$\\ \hline

284 & $2^6.3^5.107$ & $2^{52}.3^{52}.7.103.257^6$ & $2^3$ & $3^2.67^2$\\ \hline

285 & $2^{34}.3^3.7$ & $-2^{80}.3^{23}.5^5.7^2.11^3.19^3.23.37^2.47^2.73.709^2$ & $2^{12}.3$ & $2.3.10771$\\ \hline

286 & $2^{16}.3^2.31^2$ & $-2^{70}.3^{22}.5^9.7^4.13^2.19^4.103^2.139^2.257.5560463^2$ & $2^8.3$ & $120671$\\ \hline

287 & $2^{28}.3.5^2.7^2.61^2.97^2.103.257.211039.1798619$ & $-2^8.13^2.19^2.37^4.1847$ & $2^{14}.61.97$ & $2^2.3.823$\\ \hline

288 & $2^{20}.3^2$ & $0$ & $2^{10}.3$ & $\infty$\\ \hline

289 & $2^{23}.3^{16}.13^2.17^8$ & $2^6.3^4$ & $2^6.3^4.17^4$ & $\infty$\\ \hline

290 & $2^{20}.3^3.7.13^2.23.67$ & $-2^{84}.3^6.5^{10}.7^{11}.11.13^2.17^3.19^5.23^2.31.37^4.151$ & $2^{10}$ & $2^3.5.97.421$\\ \hline

291 & $2^{32}.3^4.5^4.11^2.13.19.457.16657$ & $-2^{22}.7^4.11^2.13^2.23^2.139^2.491.1933.2777^2$ & $2^{15}.3^2.5.11$ & $73.167$\\ \hline

292 & $2^7.5.41^2$ & $-2^{64}.3^{28}.5^3.7.13^3.17.19^4.101^2.389^2$ & $2^2$ & $2.3^2.97^2$\\ \hline

293 & $2^{26}.3^2.29.233.2351^2.69763.42711913589792108923$ & $1$ & $2^{10}$ & $67759$\\ \hline

294 & $2^{12}.3^2.7^4$ & $2^{147}.3^{16}.5^6.7^{21}.17^2$ & $2^6.3.7^2$ & $\infty$\\ \hline

295 & $2^{36}.3^4.7^4.43.37199.8055869$ & $-2^6.7^2.31^2.107^2.557^2.947^2.271499$ & $2^{15}$ & $2.223.241$\\ \hline

296 & $2^{18}.11.53.83.229$ & $0$ & $2^9$ & $?$\\ \hline

297 & $2^{30}.3^{24}.47^2$ & $0$ & $2^{11}.3^{10}$ & $\infty$\\ \hline

298 & $2^{21}.3.5^2.7.13^2.103.107$ & $2^{36}.3^6.7^4.13^2.23^2.29^2.41.239.1847.234893^2.1252037^2$ & $2^8.5$ & $2^3.2972969$\\ \hline

299 & $2^{50}.3.5^9.7.17^3.197.5936311524617$ & $5^2.11^2.17^4.43^2$ & $2^{19}.5^3.17$ & $2^3.3769$\\ \hline

300 & $2^4.5^4$ & $2^{113}.3^{42}.5^{16}.7^3.11^4$ & $2^2.5^2$ & $\infty$\\ \hline

301 & $2^{38}.19.31.37.103.9739.81509.32366197$ & $-2^{30}.7^3.17^2.113$ & $2^{19}$ & $2^2.3.13.17.61$\\ \hline

302 & $2^{33}.3^4.389.1613$ & $2^{18}.3^8.5^4.7^7.11^2.41^2.67^4.167.257^2.431^2.16937.439867^2$ & $2^{11}.3^2$ & $17.877.4457$\\ \hline

303 & $2^{35}.89.3083683.3678833$ & $2^{39}.3^2.7^2.59.1103^2.4021.17568767^2$ & $2^{15}$ & $2^4.61.199$\\ \hline

304 & $2^{26}.3^4.31^2$ & $0$ & $2^{13}.3^2$ & $?$\\ \hline

305 & $2^{52}.3^4.31.43.977.2777.82219057$ & $2^8.3^6.11.37^4.139^2.2371$ & $2^{19}$ & $2^2.3^4.223$\\ \hline

306 & $2^{24}.3^4$ & $2^{183}.3^{30}.5^4.7^4.17^6$ & $2^9.3$ & $3^5.5^4$\\ \hline

307 & $2^{22}.3^2.5^5.11^2.13^3.107^2.457.3697.21577.974513.568380457$ & $1$ & $2^{11}.3.5^2.11.13.107$ & $2^3.3^2.16529$\\ \hline

308 & $2^{12}.3.127$ & $-2^{140}.3^{38}.5^{17}.7.11^2$ & $2^3$ & $2^9.23$\\ \hline

309 & $2^{24}.5.37.81509.109363884517$ & $2^{32}.3^4.5^8.11^2.17^2.89^2.397.411721^2$ & $2^{10}$ & $2^6.3.11^2.421$\\ \hline

310 & $2^{23}.3^2.37$ & $2^{119}.3^{10}.5^{12}.7^9.11^4.17.19^4.23^2.29^2.41^2.73^2.5077^2$ & $2^8$ & $2.5^2.7.73.991$\\ \hline
\end{tabular}

\begin{tabular}{|l||c|c|c|c|}\hline
$N$ & $\Delta^{\new}$ & $\Delta/\Delta^{\new}$ 
    & $[\tilde{\T}^{\new}:\T^{\new}]$ & $[\T:W]$\\ \hline\hline

311 & $2^8.5^2.29.3013091897.2106873009119126062143259000543887593$ & $1$ & $2^4$ & $2.610843$\\ \hline

312 & $2^{18}$ & $0$ & $2^9$ & $\infty$\\ \hline

313 & $2^{24}.5.41^2.8619587.9614923.130838023.2164322751511$ & $1$ & $2^{11}.41$ & $2^5.19.241$\\ \hline

314 & $2^{14}.3.47.53.337.907.176130259$ & $2^{30}.5^2.7.19^2.61^2.113^2.397^2.709^2.743.1489.48795779^2$ & $2^7$ & $2.13^2.127.461$\\ \hline

315 & $2^{24}.3^4.5^3.17$ & $-2^{158}.3^{16}.5^6.7^3.11.13.17^3$ & $2^9.3^2.5$ & $2^3.3$\\ \hline

316 & $2^6.3^4.13^2$ & $2^{92}.3^{18}.5^4.7.17.53^4.83^3.271.983^3$ & $2^3.3^2$ & $2^4.541^2$\\ \hline

317 & $2^{26}.7.367.3217.660603043.14989400036918065702697531$ & $1$ & $2^{11}$ & $5.167.14627$\\ \hline

318 & $2^{16}.5^2.17.41$ & $-2^{84}.3^{16}.5^8.7^7.11^4.13^2.17^2.19^2.37^4.43^2.71.103^2.107^4.151^2.227^2.263.1054013^2$ & $2^8.5$ & $2^4.3^2.5^4$\\ \hline

319 & $2^{34}.3^4.23^2.127.1063.1433.2459.2777.6709.65551$ & $-2^{12}.3^4.5^2.7.17^2.29.167^2.239$ & $2^{17}.23$ & $2.13.59.197$\\ \hline

320 & $2^{39}$ & $0$ & $2^{18}$ & $\infty$\\ \hline

321 & $2^{32}.5^2.21803.24443.826957$ & $-2^{12}.5^4.7^2.29^2.41^2.61.1667^2.14051^2.19079^2.1229279$ & $2^{12}$ & $2^7.11.73.109$\\ \hline

322 & $2^{28}.3^3.5.79$ & $-2^{100}.3^5.5^{18}.7^3.11^4.19^4.23^2.29^2.37^2.41^3.151.536777^2$ & $2^{12}.3$ & $2^3.13.41.53.67$\\ \hline

323 & $2^{48}.7^2.17^3.19.73.103.353.1453.1627.1697.3851.17299$ & $2^{12}.3^5.5^3.7^2.59$ & $2^{23}.7.17$ & $2^2.767603$\\ \hline

324 & $2^4.3^8$ & $0$ & $2^2.3^4$ & $\infty$\\ \hline

325 & $2^{49}.3^9.5^{12}.7^4.37^2$ & $2^{50}.3^{10}$ & $2^{17}.3^4.5^6.7^2$ & $2^2.3.17$\\ \hline

326 & $2^{18}.17^2.617.28921.482689$ & $-2^{35}.3^{16}.5^2.13^2.47^2.61^2.263.14831.65657^2.82536739^2$ & $2^9.17$ & $311.102305897$\\ \hline

327 & $2^{37}.5.37.139.1023203.1033895651$ & $2^{40}.3.7^4.13^2.107^2.139^2.491.577.7537^2$ & $2^{11}$ & $2^6.3.281.463$\\ \hline

328 & $2^{28}.3^3.37.197$ & $0$ & $2^{11}.3$ & $?$\\ \hline

329 & $2^{46}.5^3.7^4.11.13^2.17.37^3.1447.1609.51607$ & $3^6.19^4.31^2.83^2.103^2.18617$ & $2^{20}.5.13.37$ & $2^4.11.1801$\\ \hline

330 & $2^8$ & $2^{264}.3^{26}.5^{21}.7^{22}.11^{10}.17^3.19^4.23.37^2$ & $2^4$ & $2.3.5.1153$\\ \hline

331 & $2^{32}.3^2.53^2.229.1399.21911.205493.6363601.584461573862449$ & $1$ & $2^{12}.3.53$ & $2^2.1120529$\\ \hline

332 & $2^9.7^3.29^2$ & $-2^{82}.3^{12}.5^2.7.71.131^4.197^3.229^2.11497^3$ & $2^2.29$ & $2^2.5^2.479^2$\\ \hline

333 & $2^{42}.3^7.5^2.7^2.37.389$ & $2^{72}.3^9.5^4.7^4.11.37^2.389^2$ & $2^{13}.3^2.5.7$ & $2^{10}.3^5$\\ \hline

334 & $2^{27}.5^2.7^3.67.733$ & $2^{18}.3^{10}.5^4.7^4.11^4.41.113^2.8269^2.1951993.5103536431379173^2$ & $2^{12}.7$ & $113.3715823$\\ \hline

335 & $2^{31}.3.5.29.71.83.179^2.887.26393.6262079.23057641$ & $-2^{22}.3^{12}.5^8.11^2.41.59^2$ & $2^{13}.179$ & $2^2.3.331.431$\\ \hline

336 & $2^{16}.3^2$ & $0$ & $2^8.3$ & $\infty$\\ \hline

337 & $2^{28}.113.593.2791.2963615537.747945736667.4122851467451$ & $1$ & $2^{13}$ & $2^4.3^3.27299$\\ \hline

338 & $2^{16}.3^2.7^6.13^8$ & $2^{36}.3^8.5^2.7^{14}.13^8.41^4.167^2$ & $2^8.3.7.13^4$ & $\infty$\\ \hline

339 & $2^{41}.7^2.13^2.17.71^2.8297.470621$ & $-2^{29}.3^{18}.7^6.11^8.13^5.107^2.167.647$ & $2^{14}.7.13.71$ & $2^2.5^2.127.11801$\\ \hline

340 & $2^6.101$ & $-2^{198}.3^{37}.5^8.7^7.17^3$ & $2^2$ & $\infty$\\ \hline

341 & $2^{44}.3.5^3.89.151.1121599.344460847.14444130109$ & $-2^{14}.3^3.5^{10}.7^4.13^2$ & $2^{16}$ & $31.9043$\\ \hline

342 & $2^{12}.3^4.5^2$ & $-2^{184}.3^{58}.5^{16}.7^3.11^6$ & $2^6.3^2.5$ & $2.3^9.5^2$\\ \hline

343 & $2^{27}.7^{48}.13^2.29^3.41^2$ & $0$ & $3.33521863.4906673923$ & $\infty$\\ \hline

344 & $2^{26}.3.229.1999567$ & $0$ & $2^{11}$ & $?$\\ \hline

345 & $2^{46}.3^2.5^4.79$ & $-2^{100}.3^{13}.5^{19}.7^4.11^{13}.13^2.17^4.23.53^2$ & $2^{18}.5^2$ & $2^2.7^2.15991$\\ \hline

346 & $2^{18}.7^2.229.2777.2075621$ & $2^{40}.5^4.7^6.29^2.31^2.67^2.79^2.311.1279.1289.5608385124289^2$ & $2^9.7$ & $2^3.18947.256957$\\ \hline

347 & $2^{27}.5.7^2.19^2.331.349.479.617.1797330450291217.918291275915301361$ & $1$ & $2^{10}.7.19$ & $2^6.3.7.19709$\\ \hline

348 & $2^8$ & $2^{129}.3^{27}.5^{17}.7^{10}.11^7.13^4.17^2.23^6.41.47.73^3.229^3$ & $2^4$ & $2^8.5.11^4$\\ \hline

349 & $2^{28}.13.103.1118857.72318613.6771977049413.1313981654817031$ & $1$ & $2^{11}$ & $2^3.3^2.239.1531$\\ \hline

350 & $2^{24}.3^4.5^6$ & $2^{138}.3^{34}.5^{29}.11^4.17^6.41^2$ & $2^9.3.5^3$ & $2^3.3$\\ \hline
\end{tabular}

\begin{tabular}{|l||c|c|c|c|}\hline
$N$ & $\Delta^{\new}$ & $\Delta/\Delta^{\new}$ 
    & $[\tilde{\T}^{\new}:\T^{\new}]$ & $[\T:W]$\\ \hline\hline

351 & $2^{32}.3^{17}.5^4.13^2.19.37^2$ & $0$ & $2^{12}.3^8$ & $\infty$\\ \hline

352 & $2^{42}.17^2$ & $0$ & $2^{21}$ & $\infty$\\ \hline

353 & $2^{34}.3^2.5.127^2.229.114641.551801.12611821.7779730837.24314514437$ & $1$ & $2^{15}.3.127$ & $2^2.3.47.26927$\\ \hline

354 & $2^{28}.5^2.11.79$ & $-2^{98}.3^{14}.5^{12}.7^4.11^6.19^6.29^2.31^8.41^2.127.229^6.557^4.20627^2$ & $2^{12}.5$ & $2^{15}.3^4.5$\\ \hline

355 & $2^{51}.5^2.19.29.103.62581037.933591331$ & $3^{10}.5.7^2.19.61^2.103^2.257^4.359^2.3671$ & $2^{23}$ & $2^6.3^2.5^2.13.37$\\ \hline

356 & $2^9.3.4079.31547$ & $-2^{114}.3^{14}.5^9.7^6.71^2.199.6689^3$ & $2$ & $2^5.11.487^2$\\ \hline

357 & $2^{49}.3.79.113$ & $-2^{99}.3^{16}.7^4.17^4.47^2.71^2.131^2.223^2.311^2.397.1459^2.1753$ & $2^{18}$ & $2^2.17.86341$\\ \hline

358 & $2^{22}.3^3.5^3.7.11^4.17^3$ & $-2^{46}.3^8.5^2.7^4.13^2.17^4.29^2.79^2.239.313^2.3257.137707^2.536747147^2$ & $2^{11}.3.11^2$ & $251.4099597$\\ \hline

359 & $2^{24}.3^4.2777.16512254293.64542630435970307.2171776478013633068927$ & $1$ & $2^{11}$ & $2^6.5.123059$\\ \hline

360 & $2^{12}.3^2$ & $0$ & $2^6.3$ & $\infty$\\ \hline

361 & $2^{28}.3^{10}.5^{11}.11^4.19^{12}$ & $2^6.3^2.5^2$ & $2^{12}.3.5^2.11^2.19^6$ & $\infty$\\ \hline

362 & $2^{26}.5^4.17^3.6359.1134769$ & $2^{40}.3^2.5^4.7^4.17^2.23^2.29^2.47.61^2.71.97.397^2.797^2.5297.595051637^2$ & $2^{10}.5.17$ & $2^2.2609601041$\\ \hline

363 & $2^{36}.3^9.5^7.11^{18}$ & $2^{38}.3^{22}.5^8.11^4$ & $2^{15}.3^3.5.11^8$ & $\infty$\\ \hline

364 & $2^{15}.3^2$ & $2^{125}.3^{63}.5^{10}.7^{12}.11^4.23.79^3$ & $2^5$ & $2^7.3^2.5.17.37$\\ \hline

365 & $2^{51}.3^3.5^4.7^3.47.313^2.353783.9377894173$ & $-2^{20}.3^{12}.5^4.11^2.13^4.19^2.29.269$ & $2^{19}.3.5^2$ & $3^2.37.71473$\\ \hline

366 & $2^{16}.3^2.13^2.17$ & $-2^{162}.3^{11}.5^4.7^6.13^{10}.17^3.19^6.23^3.31^2.37^6.127^2.137.151^3.229^2.5623^2$ & $2^8.3.13$ & $2^4.5^7.1277^2$\\ \hline

367 & $2^{22}.7.81421.251387.418175501.15354151381.13144405392643360366681$ & $1$ & $2^{11}$ & $2^5.3197809$\\ \hline

368 & $2^{40}.3^4.5^3.17$ & $0$ & $2^{20}.3^2.5$ & $?$\\ \hline

369 & $2^{41}.3^6.5^2.23^2.37^3.79$ & $-2^{86}.3^{12}.5^4.23^4.37^3.79^2.191$ & $2^{15}.3^3.5.23$ & $7.13^2$\\ \hline

370 & $2^{28}.3^2.11.223$ & $-2^{148}.3^{28}.5^{10}.7.11^2.13^4.19^2.23.31^2.61.193.269.23029^2.60869^2$ & $2^{12}$ & $2^4.72709379$\\ \hline

371 & $2^{62}.5^3.7.157.229.64627370911.76723322773093$ & $2^{30}.3^3.5^2.37^2.191^2.643$ & $2^{25}.5$ & $2^5.3.5.76103$\\ \hline

372 & $2^{10}.5^2.17$ & $-2^{181}.3^{31}.5^{13}.7^4.11^{12}.13^2.17^2.19^4.41^2.127.229^3$ & $2^5.5$ & $2^2.3^5.5^8$\\ \hline

373 & $2^{32}.7.11^3.23.199.673.2143.1542194372227.72819251148518000363297$ & $1$ & $2^{13}.11$ & $2^4.3.1520821$\\ \hline

374 & $2^{28}.5^2.157.257.11117.17417$ & $2^{142}.3^{20}.5^{18}.7^2.11^2.13^2.17^2.23.37^4.8461^2$ & $2^{14}$ & $11.3259.109111$\\ \hline

375 & $2^{24}.5^{24}.101^2$ & $0$ & $2^{12}.5^8$ & $\infty$\\ \hline

376 & $2^{33}.5^5.41^2.61$ & $0$ & $2^{15}$ & $?$\\ \hline

377 & $2^{70}.3^3.5^2.7^2.829.991.36497.202817.400031.54126547$ & $2^6.7^2.23^3.31^2.79$ & $2^{30}.3.7$ & $181.118057$\\ \hline

378 & $2^{16}.3^8$ & $0$ & $2^8.3^4$ & $\infty$\\ \hline

379 & $2^{28}.59.317.421.278329.5698591.2117788336277.2851210737989187265253$ & $1$ & $2^{13}$ & $2^5.3^3.7.229.809$\\ \hline

380 & $2^{15}.3$ & $2^{158}.3^{47}.5^9.7^2.11^7.13^4.17^2.19.37^3.113.151.709^3$ & $2^5$ & $2^2.3^4.11.53$\\ \hline

381 & $2^{32}.11^2.15427.81509.98361184089911$ & $2^{36}.3^{10}.5^2.7^2.19^2.71.109^2.647^2.2003.86235899^2$ & $2^{12}.11$ & $2^6.3.7^3.13.43.113$\\ \hline

382 & $2^{28}.7^2.11.13^2.619.44171$ & $2^{20}.3^6.5^4.11^4.41.967^2.1511.6277^2.63703.382146223^2.319500117632677^2$ & $2^{13}$ & $5.59.9181.50459$\\ \hline

383 & $2^{16}.5.11^2.13.72893.3151861.16141144314299.178236551484825400362837637090811$ & $1$ & $2^8.11$ & $2^7.2480227$\\ \hline

384 & $2^{40}$ & $0$ & $2^{20}$ & $\infty$\\ \hline

385 & $2^{61}.3.37^3.2837$ & $-2^{114}.3^{18}.5^{17}.7^8.11.17^2.19^5.389$ & $2^{24}$ & $2^4.3.5.29.71$\\ \hline

386 & $2^{32}.5^2.14653.22961.9659021$ & $-2^{30}.3^2.5^4.7.11^4.17^2.19^2.29^2.31^2.103^2.229^2.271^2.337.401^2.4153^2.244553.680059^2$ & $2^{12}$ & $3^2.13^2.74891419$\\ \hline

387 & $2^{51}.3^9.5^8.7^2.13^2.71$ & $-2^{106}.3^{16}.5^6.7^4.71^2$ & $2^{20}.3^4.5^3.7$ & $2.7^3.17^2$\\ \hline

388 & $2^8.7^2.179.1297$ & $-2^{90}.3^{16}.7^{11}.67^4.71^4.137^2.223^2.433.2777^3$ & $2^3$ & $2^3.3^3.22283^2$\\ \hline

389 & $2^{53}.3^4.5^6.31^2.37.389.3881.215517113148241.477439237737571441$ & $1$ & $2^{17}.3^2.5.31$ & $7^2.67.173.863$\\ \hline



\end{tabular}
\end{center}

%\begin{thebibliography}{HHHHHHH}         
%\bibitem[S]{stein} W. Stein, {\em Generating the Hecke algebra
%as a $\Z$-module}.  preprint, 1998. 
%\end{thebibliography} \normalsize\vspace*{1 cm}

\end{document}



\\ table.gp -- make the discriminant table.

\\discnew=discnormal=discsk=disct1tg=vector(1000,x,0);

\rdiscnew
\rdiscnormal
\rdiscsk
\rdisct1tg

{pfac(n,
  i, f)=
  if(n<0, print1("-"); n=-n);
  if(n==0 || n==1,print1(n); return;);
  f=factor(n);
  for(i=1,matsize(f)[1],
    if(i>1,print1("."));
    print1(f[i,1]);
    if(f[i,2]>1 && f[i,2]<10, print1("^",f[i,2]));
    if(f[i,2]>=10, print1("^{",f[i,2],"}"));
  );
}

{pfac2(n,
  i, f)=
  if(n<0, print1("-"); n=-n);
  if(n==0 || n==1,print1(n); return;);
  f=factor(n);
  for(i=1,matsize(f)[1],
    if(i>1,print1("*"));
    print1(f[i,1]);
    if(f[i,2]>1, print1("^",f[i,2]));
  );
}

{dsk(N)=
  if(type(discsk[N])=="t_VEC",return);
  print1("\ndiscsk[",N,"] = ");
  pfac2(discsk[N]);
}

{dsknew(N)=
  if(type(discnew[N])=="t_VEC",return);
  print1("\ndiscnew[",N,"] = ");
  pfac2(discnew[N]);
}

{entry(N)=
  if(type(discnew[N])=="t_VEC",return);
  print1("\n",N," & $");
  pfac(discnew[N]);
  print1("$ & $");
  pfac(discsk[N]/discnew[N]);
  print1("$ & $"); 
  pfac(floor(sqrt(discnew[N]/discnorm[N])));
  if(!issquare(discnew[N] % discnorm[N]),
     print("ERROR ERROR ERROR!!! in [Ttildenew:Tnew] at level ",N);return;);
  print1("$ & $");
  if(type(disct1tg[N])=="t_VEC",
     print1("\\infty"),
  \\ else
  if(disct1tg[N]==0,
     print1("?"),
  \\ else
     pfac(floor(sqrt(disct1tg[N]/discsk[N])));
     if(!issquare(disct1tg[N] % discsk[N]),
       print("ERROR ERROR ERROR!!! in [T:W] at level ",N);return;)
  ));
  print1("$\\\\ \\hline\n");
}

{entry2(N)=
  if(type(discnew[N])=="t_VEC",return);
  if(type(disct1tg[N])=="t_VEC",
    return);
  \\ else
  if(disct1tg[N]!=discsk[N],return);
  if(disct1tg[N]==0,
     print(N,"?"),
  \\ else
     if(disct1tg[N]==discsk[N] && !isprime(N),
       print(N));
  );
}