Richard Hoekstra
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A census of singular 3-adic Hecke factors at prime level

Paper VI in the singular Hecke node series
Richard Hoekstra · 2026 · PDF

Setup

Let N be a prime. Write VN = S2(Γ0(N)) ⊗ Q for the rational cuspidal modular-symbols space, and MN = VN ⊗ F3 for its mod-3 reduction. Define

QN := ker(T22 : MN → MN),   qN = dimF3 QN,   fN = minimal polynomial of T2|QN over F3.

When qN > 0, the syndrome algebra is

AN := spanF3{ Tℓ|QN ∣ ℓ ∈ {3,5,7,11,13,17,19,23}, ℓ ≠ N } ⊂ EndF3(QN).

Classification rule

Definition (Category rule) Each prime level N is assigned exactly one of four categories:

Smooth — qN = 0. The mod-3 Hecke action has no local singularity.

Split node — qN > 0, and AN contains a nontrivial idempotent that lifts to Z3 via Hensel's lemma.

Cusp — qN > 0, fN = x, and no split idempotent lift exists.

Non-split node — qN > 0, fN = x2, and no split idempotent lift exists.

The split/non-split distinction is implemented by searching for a nontrivial mod-3 idempotent in AN and attempting to lift it to Z3. The computation proceeds by: (1) forming MN from ModularSymbols(N,2).cuspidal_subspace(), (2) computing QN, (3) building AN from a small prime Hecke set, and (4) running the Hensel lifting test.

Census: N ≤ 500

The scan covers 91 primes in the range 11 ≤ N ≤ 500.

Summary (91 primes)
Smooth56Cusp19
Split node16Non-split node0

All 16 split-node primes (N ≤ 500)

Leveldim S2qNalg dimT2 rankfN
716221x2
12710331x2
16313331x2
16714220x
19916441x2
26922221x2
27122221x2
30725220x
33727330x
35930221x2
42134221x2
43936442x2
44337221x2
44937551x2
48740772x2
49941662x2

Level 163 is the largest Heegner prime among the split nodes. The q-dimensions range from 2 to 7, with both minimal polynomial types (x and x2) represented. Notably, this initial scan finds zero non-split nodes below 500.

Sample cusp levels (N ≤ 500)

Of the 19 cusp primes in this range, all have qN = 1 except N = 167 (qN = 2). All cusp levels have fN = x and T2 rank 0. The first ten:

Leveldim S2qNr3r9r27r81
19112222
37212222
73512111
101812222
109812111
113912211
1311112222
1491212111
1511212111
1811412221

Cusp levels are characterized by a one-dimensional kernel with stable 3-adic valuations. The remaining cusp primes below 500 are: 223, 293, 379, 383, 397, 431, 433, 463, 467.

Census: N ≤ 1000

Extending the scan to all 164 primes in the range 11 ≤ N ≤ 1000:

Summary (164 primes)
Smooth95Cusp39
Split node17Non-split node13

The 17 split-node primes up to 1000 are:

127, 163, 199, 307, 337, 449, 487, 499, 541, 577, 631, 773, 811, 829, 857, 883, 919.

The 13 non-split-node primes up to 1000 are:

71, 269, 271, 359, 421, 439, 443, 509, 523, 571, 601, 947, 997.
The non-split phenomenon. The initial N ≤ 500 census finds 16 split nodes and 0 non-split nodes. The extended N ≤ 1000 scan, using a refined Hensel lifting test, reclassifies several of those levels and discovers 13 non-split nodes. Seven of these (71, 269, 271, 359, 421, 439, 443) fall below 500 but were originally classified as split by the earlier algorithm. The remaining six (509, 523, 571, 601, 947, 997) appear only in the 500–1000 range. All 13 non-split nodes have fN = x2 and qN ∈ {2, 4}, with the idempotent search failing to produce a lift.

Asymptotics

The split-node density among primes scanned:

RangePrimesSplitDensity
N ≤ 500918 (refined)0.088
N ≤ 1000164170.104

The q-dimensions occurring in each category across N ≤ 1000:

CategoryqN values
Smooth0
Cusp1, 2
Split node2, 3, 4, 5, 6, 7
Non-split node2, 4
Conjecture (positive density) Both ratios

#{ split node N ≤ X } / #{ primes p ≤ X, p ≠ 3 }   and   #{ non-split node N ≤ X } / #{ primes p ≤ X, p ≠ 3 }

have nonzero limits as X → ∞.

Data and paper

Full data table (N ≤ 1000) View PDF Full paper (HTML)