Weight and level

Weight controls the transformation rule; level chooses which transformations are required.

Start with a concrete example

Think of a two-by-two table as instructions for moving a point. Here the table is [1, 0; c, 1]. For one actual example of a level-3 restriction, the lower-left entry c must be divisible by 3. Change c. Weight has a different role: it is an exponent in the transformation rule. Compare 2¹ = 2 with 2² = 4 to see what changing an exponent does; this arithmetic alone is not the full transformation rule.

Move the number or condition

c = 3; remainder after division by 3 is 0.

How this example differs from the definition

The diagram illustrates the Γ₀(3) matrix condition with determinant 1. Weight is a separate exponent in the transformation rule; it is not the number of notes or a physical weight.

A small check: try the idea

Compare the numbers in the question with the diagram you are moving. You can try again.

In our level-3 matrix example, does c = 4 satisfy the lower-left entry condition?

Read the answer and reasoning

No; 4 is not divisible by 3. Set c to 4: 4 = 3×1+1. Size is not the test; divisibility is. Weight is a separate part of the transformation rule.

Build the idea in three steps

Weight k is an integer parameter in the transformation law of a form. It is not the mass of an instrument or a difficulty rating.
For a matrix with rows (a,b) and (c,d), its determinant is ad − bc. Level N selects integer matrices with determinant one and c divisible by N; the selected transformations are called Γ₀(N).
CuspForm (Gamma0 N) k fixes both parameters. Changing either can change which forms exist.

A small example

CuspForm (Gamma0 2) 2 means cusp forms at level two and weight two, not “second-level beginners”.

Lean definition · original excerpt

Mathlib · db584cd · CongruenceSubgroups.lean

79def Gamma0 : Subgroup SL(2, ℤ) where80  carrier := { g | (g 1 0 : ZMod N) = 0 }

How to read this Lean

The membership condition tests A 1 0, the bottom-left matrix entry. N ∣ that entry is the level condition. Weight is the separate k parameter in ModularForm and CuspForm.

Original source ↗ · L79–80 Open the complete saved source

Where this idea is used

8. The secret duet 9. Two doors to the concert 10. A three-note audition 11. Borrow a better passport 12. The workshops agree 13. Pip’s impossible quilt 14. The shrinking reservation 15. The concert of absolute silence 16. The room with no room

Weight and level

Start with a concrete example

A small check: try the idea

Build the idea in three steps

A small example

Lean definition · original excerpt

How to read this Lean

Where this idea is used

Mika’s help desk

One thing at a time

In simpler words

A small clue