{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "4CDFWST967"
    },
    {
      "field": "uses",
      "target": "CS1NVJBZMH"
    },
    {
      "field": "uses",
      "target": "RWXXNZNQYJ"
    },
    {
      "field": "defines",
      "target": "X9PBTW2J8D"
    }
  ],
  "body": "\n## Summary\n\nThe Hecke operators $T_n$, for $n \\ge 1$, are linear maps on the space $M_k$ of [[modular forms|GWKN5VG78A]] of weight $k$.\nThey average a modular form over the sublattices of index $n$ of a lattice.\nThey commute with each other and preserve the [[cusp forms|A50Y012RBP]].\nA common eigenform of all $T_n$, suitably normalized, has multiplicative Fourier coefficients and an L-function with an Euler product.\n\n## Explanations\n\n### L1 human\n\nA modular form can be seen as a function of lattices in the plane.\nThe operator $T_n$ replaces the value at a lattice by the sum of the values at all its sublattices of index $n$.\nThis produces a new modular form of the same weight.\nSince the space of forms is small, a form is often sent to a multiple of itself.\nFor $\\Delta$ of weight $12$, $T_n \\Delta = \\tau(n) \\Delta$, which explains why the [[Ramanujan tau function|6DNZAGCSQS]] is multiplicative.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "11F25",
      "11F11"
    ],
    "name": "Hecke operator",
    "same_as": {
      "wikidata": "Q1592959",
      "wikipedia": "Hecke operator"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hecke_operator"
      }
    ],
    "tag": "S4SV3QZ8B1"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/S4SV3QZ8B1/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/S4SV3QZ8B1.md",
  "msc": [
    "11F25",
    "11F11"
  ],
  "name": "Hecke operator",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/S4S/S4SV3QZ8B1.md",
  "status": null,
  "tag": "S4SV3QZ8B1",
  "uri": "m2://S4SV3QZ8B1",
  "url": "https://math2.tamnd.com/wiki/hecke-operator/",
  "writing": {
    "tag": "S4SV3QZ8B1",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}