JOURNAL OF RESEARCH of the Nati onal Bureau of Standards- B. Mathematics and Mathemati cal Physics Vol. 67B, N o. 1, Ja nuary- March 1963 Tables of Genera of Groups of Linear Fractional Transformations * Harriet Fell, Morris Newman, and Edward Ordman ( October 29, 1962) The genera of the groups r o(n), r *(n) together with certain associated Ilumber-theoretic fun ct ions are given for 1 n 1000. The 2X2 modul ar group r and its subgroups are of fundamental importance in the theory of auto- morphi c functions (in particular the elliptic modular functions) and in the theorY of Riemann surfaces. Leti G be such a subgroup. Then the substitutions , aT +b T =-- · cr + d (1) of G map the upp er T haH-plane onto itself, and the points of this half-plane are partition ed into equival- ence classes modulo G. A set of points consisting of one point from each equivalence class is termed a fundamental set, and if G is of finite index in r there is a simple way of selecting a standard fundamental set R, which is fully described in Ford 's book [1]1 or in Gunning's book [3]. The set R is commonly called a fundamental region. After appropriate identifications of sides and verti ces ar e mad e R b e- comes a surface whose genus g plays a central role in th e study of G. Among the subgroups of finite index in r the congruence subgroups hav e been studied most ext ensi vely and amon g the congl'llence subgroups the (nonnormal) subgroups r o (n) arc of primary importan ce. These are defined for every natural numb er n as the t ot ality of substitutions (1) where a, b, c, d are rational int egers, ad - bc = I, and c= O (mod n). If the substitution , 1 T = - - nT is adjoined to ro Cn) the larger group so obtained (in general not a subgroup of r ) is dono ted by r "(n). Formulas for the genera go (n), g* (n ) of r o(n) , r *(n) , r es pectively, have been given by F. Kl ein [6], E. Heclm [4 , 5], and R. Fricke [2]. Denote the numb er of solutions of the congruence x 2 -x+ I = 0 ( mod n), O::;:x::;:n- I (2) by ))I(n), and the numb er of solutions of the con- gruence *'1"11 0 compu tation of the ta bles was car ri ed out by th e fi rst ancl third a uth ors at the suggestion of the second a uth or, as a summer project at NBS. Th e work was support ed by the Office of Naval Resea rch. 1 Figures in brackets indicate th e li t.c r at ure refe rences at the end of this paper. 61 x Z+ I=O (modn ), O::;:x::;:n - I b y))z(n) . Set }l (n )= n II (1+!) ' qln q (3 ) wher e denot es the greatest common divisor of d and and cp is the Eul er function. Th en ))I(n) is the number of inequivalent elliptic verti ces of period 3 of th e fundam ental region R" of ro(n), ))2( n) the numb er of inequivalent elliptic vertices of period 2 of R", }l en) the index of ro (n) in r , and tJo(n) the num- ber of inequivalent parabolic vertices of R n. Th e genus go(n) of ro (n) is then Furthermor e let h( - 4n) denote the number or classes of primitive positive binary quadr atic forms of discriminant - 4n , a nd set r 2n = 7(mod 8) 8),n>3 \.. 1 otherwise. Then th e genus g* ( n) or r* (n) is given by In this article we give the act ual numerical va lu es or go(n) , g* (n) Jor 1 ::;:n'5: 1000 together with the values of the associated fun ct ion s ))l(n), ))z(n) , }l (n ) , tJ o(n) , h( - 4n ). These were comput ed on the IBU 7090 of t he NBS in a negli gible nmount of time. The r es ulting tables are of considerabl e inter est however and should prove useful in many numb er-