3-adic Discriminant vs. Singular Quotient Scan

This scan compares two possible selectors for singular mod-3 behavior at prime level N <= 199:

• the 3-adic discriminant valuation v_3(disc(T_N)) of the full Hecke algebra;
• the quotient dimension dim ker(T_2^2) on the canonical mod-3 Hecke module.

level	dim S_2	v3(disc)	dim ker(T_2^2)
19	1	0	1
37	2	0	1
71	6	4	2
73	5	2	1
101	8	0	1
109	8	0	1
113	9	4	1
127	10	4	3
131	11	0	1
139	11	2	0
149	12	0	1
151	12	0	1
163	13	2	3
167	14	0	2
179	15	4	0
181	14	0	1
191	16	3	0
199	16	1	4

Read

The naive rule

number of generations = v_3(disc) + 1

fails immediately in the scanned range.

Counterexamples in both directions:

• 19: v_3(disc)=0 but dim ker(T_2^2)=1.
• 167: v_3(disc)=0 but dim ker(T_2^2)=2.
• 179: v_3(disc)=4 but dim ker(T_2^2)=0.
• 191: v_3(disc)=3 but dim ker(T_2^2)=0.
• 199: v_3(disc)=1 but dim ker(T_2^2)=4.

So the existence or dimension of the canonical mod-3 singular quotient is not determined by the 3-adic discriminant valuation alone.

At level 163 both invariants are nontrivial (v_3(disc)=2, dim ker(T_2^2)=3), but that coincidence does not extend to a general law in the first scan up to 199.