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O, B(2) as a little shrinks to zero. so we want k ~I d log r<~> r - -n ~. · = (-1) n 1, and ~I+(~], dim ml(2,k,-l) ~ 1 + [~]. as before. (s) =~ 2 E' Cn 2 + m ) n,m£l: which has signature (2,1,1). -5 , ~(1), MODULAR FORMS AND DIRICIILET SERIES I-38 2 00 LEMMA J. ~(2,~,1), belongs to ! ::1 "(~) at e mn -r 2n=-"" l r and Its only zero in -r = "(-r), "(-r + 2) ls 0-condition. -r l/2 = "(-r). Poisson sumroatton formula: B(2) = -1. is clearly holomorphic tn "(-r) and satisfies the show " ( T) - The theta-function Im -r > 0, holomorphtc at ""• We want now to For this we apply the if t f(x + n) n=-co converges absolutely, uniformly on compact subsets, to a continuously differentiable function where F(x), x is a real variable, then F(x) = ~ 9 2Tr1nx o.

2) and hence that is a fundamental domain for B(2) one checks we conclude 3 in has index 2, = 6. (1'• 1(2)) (G(2)r(2) :r(2)) = 2, Now of rc2> ~ r ! sLC2,~1~>. 1s onto and so 0(2) r by and B(2) G(2), two~ (points where it meets the boundary of the upper hnlf plane), and -1; +1 is equivalent to and so 1s not counted. 0 at each cusp. =-1> T + i c t 2 at + 2 ,-lJ 0) as follows: at points not equivalent to T "'• -1 2) T ~ T has an angle of into a Riemann surface (of genus by assigning local parameters 1) under i' 1 em• at t ,.

The reason for these choices ls as follows, at the three corners i, "'• -1, Except a neighborhood of DIRICHLET WITH FUNCTIONAL EQUATION SERI&~ 1-33 contains no equivalent point, whence 1). ll. ""• whence 2), 3). ~. 1nl, or T + 1 1 ls the appropriate thus local variable Rt -1. ) We now investigate the meaning or the 0-con- dftlon: LEMMA 1. J i then tnh(t) ls "quasit _ -2rrl/T+l - e ' MODULAR FORMS AND DIRICHLET SERIES I-34 where and h(t) n 2 0; tn = e- 2 n1n/T+l. tn general, and ts called the f(T) at ~· T ..