Preprint (July 12, 1999), Version 0.1

The first few nonmaximal orders attached to weight two newforms on Γ_{0}(N)

W.A. Stein

Department of Mathematics, University of California, Berkeley, CA 94720, USA

Introduction

Let f ∈ S_{2}(Γ_{0}(N)) be a weight two newform. Attach to f the ring

O_{f} = Z[… a_{n} …]

generated by the Fourier coefficients of f. This ring is an order in the totally real number field K_{f} = Q[… a_{n} …]. David Carleton asked me for examples in which O_{f} is not equal to the maximal order in K_{f}, equivalently, for which Spec(O_{f}) is not normal. Armand Brumer pointed out that Birch had long ago laboriously compiled tables of f so that O_{f} appeared to be non-maximal. Unfortunately, we do not have access to Birch’s tables and it was perhaps possible that in Birch’s examples, given the limited computing resources of the day, that he had not computed enough of the a_{n}’s to generate the full ring O_{f}. When N = 1 and the weight grows, Jochnowitz showed that O_{f} is far from normal.

Data gathering

Using HECKE [3], the author computed the index of O_{f} in the maximal order for every newform f of level N ≤ 450. This was done as follows:

1. Compute enough a_{n} to generate O_{f} as a Z-module using the Sturm bound [1, 2, 4].

2. Compute the discriminant of O_{f} by embedding a matrix representation of O_{f} into a space of column vectors, finding an integral basis, and computing the determinant of the trace pairing.

3. Compute the discriminant of K_{f} using LiDIA.

There are 1775 newforms of level N ≤ 450. Of these, 93 have the property that O_{f} is not maximal. The distribution is as follows, where index is the index of O_{f} in the maximal order.

Table 1. Distribution of indexes

index  1  2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
number 1682 77 1 12 1 0 0 1 0 0 0 0 0 0 0 1

Table 2. Each newform of level ≤ 450 with O_{f} having given index:

index f
2  69B, 77D, 105B, 136C, 138D, 154D, 160C, 165C, 171E, 195E, 207D, 213E,

217D, 221D, 221G, 224C, 224D, 225F, 226D, 238F, 255D, 256E, 260B, 272E, 273E, 282E, 299B, 299G, 301D, 305C, 305D, 310E, 315F, 320G, 322E, 322G, 323E, 329F, 337B, 340B, 355E, 357H, 359D, 363G, 368I, 376D, 377F, 385H, 390H, 392G, 399F, 399G, 406G, 410E, 410G, 410I, 414F, 415D, 416D, 417F, 426G, 429H, 433C, 434I, 435E, 435J, 437F, 438H, 438I, 442F, 442H, 442I, 445A, 445F, 445G, 448I, 448J

3  271B
4  219E, 291H, 293B, 303E, 371E, 387J, 389E, 395H, 413F, 416F, 431F, 437H
5  401B
8  371F
16 257B

Table 3. The first example exhibiting each index.

index f  [K_{f} : Q]  disc(K_{f})
2 69B  2  5
3 271B  16  1367 · 6091 · 1132673 · 14171513 · 172450541
4 219E  6  2^{2} · 1189637
5 401B  21  2^{8} · 19 · 163 · 71742740351 · 388881803749 · 34393898968391
8 371F  11  2^{8} · 157 · 76723322773093
16 257B  14  2^{7} · 29 · 479 · 71711 · 409177 · 654233

References

[1] A. Agashe and W.A. Stein, Generating the Hecke algebra, Preprint (1999).

[2] W.A. Stein, Congruences between modular forms, Notes (1999).

[3] , HECKE: The modular forms calculator, Software (available online) (1999).

[4] J. Sturm, On the congruence of modular forms, Number theory (New York, 1984–1985), Springer, Berlin, 1987, pp. 275–280.