History of Hecke polynomials of level one cusp forms

     values $a_p/p^{(k-1)/2}$. With this convention, the weight $12$ rows are $x-\tau(p)$     for Ramanujan's tau function.-  comment-maeda: Maeda's conjecture, stated in CITE{HidaMaeda} and summarized in CITE{GhitzaMcAndrew},-    says that for each $m>1$ the characteristic polynomial of $T_m$ on $S_k$ is irreducible-    over $\mathbb{Q}$ and has Galois group $\mathfrak{S}_{\dim S_k}$. For the rows-    in this table, every polynomial is irreducible over $\mathbb{Q}$; the Galois groups-    were not checked.-  comment-zero-space: There are no cusp forms of level one below weight $12$, and-    none in weight $14$, by CITE{formula-degree}.+  comment-maeda: The version of Maeda's conjecture stated in CITE{HidaMaeda} and summarized+    in CITE{GhitzaMcAndrew} says that for each $m>1$ the characteristic polynomial+    of $T_m$ on $S_k$ is irreducible over $\mathbb{Q}$ and has Galois group $\mathfrak{S}_{\dim+    S_k}$. For the rows in this table, every polynomial is irreducible over $\mathbb{Q}$;+    the Galois groups were not checked.+  comment-zero-space: Formula CITE{formula-degree} shows that there are no cusp forms+    of level one below weight $12$, and none in weight $14$. Formulas:   formula-q-action: If $f(q)=\sum_{n\geq1}a_nq^n$, then $(T_pf)(q)=\sum_{n\geq1}(a_{pn}+p^{k-1}a_{n/p})q^n$,

     $j$ for $\mathrm{SL}_2(\mathbb{Z})$. Then $S_k(\mathrm{SL}_2(\mathbb{Z}))=\Delta\,M_{k-12}$.     The graded ring $M_*=\bigoplus_j M_j$ is $\mathbb{C}[E_4,E_6]$, generated by HREF{Q-expansion_of_the_Eisenstein_series_E4}[$E_4$]-    and HREF{Q-expansion_of_the_Eisenstein_series_E6}[$E_6$]. Thus the products $\Delta-    E_4^aE_6^b$ with $4a+6b=k-12$ form a basis of $S_k$.+    and HREF{Q-expansion_of_the_Eisenstein_series_E6}[$E_6$] together. Thus the products+    $\Delta E_4^aE_6^b$ with $4a+6b=k-12$ form a basis of $S_k$.   formula-tau: For $k=12$, $\chi_{12,p}(x)=x-\tau(p)$, where $\tau(n)$ is the coefficient     of $q^n$ in the modular discriminant $\Delta$.

     and HREF{Q-expansion_of_the_Eisenstein_series_E6}[$E_6$] from their divisor-sum     $q$-expansions, forms HREF{Q-expansion_of_the_modular_discriminant}[$\Delta$]-    as $(E_4^3-E_6^2)/1728$, builds the basis $\Delta E_4^aE_6^b$ in CITE{formula-level-one-basis},-    and applies CITE{formula-q-action} coefficient by coefficient. It expresses the-    result in that exact basis over $\mathbb{Q}$ and takes $\det(xI-T_p)$ by a permutation-    expansion, requiring every coefficient to be an integer before it returns the-    polynomial. Every row is required to equal PARI/GP's `charpoly(mfheckemat(mfinit([1,k],0),p))`-    CITE{PARIHecke}. The weight $12$ rows were compared with the stored Ramanujan-    tau values in the table of the modular discriminant.+    as $(E_4^3-E_6^2)/1728$, builds the basis $\Delta E_4^aE_6^b$ described in CITE{formula-level-one-basis}+    and applies the Hecke action from CITE{formula-q-action} coefficient by coefficient.+    It expresses the result in that exact basis over $\mathbb{Q}$ and takes $\det(xI-T_p)$+    by a permutation expansion, requiring every coefficient to be an integer before+    it returns the polynomial. Every row is required to equal PARI/GP's `charpoly(mfheckemat(mfinit([1,k],0),p))`+    documented by CITE{PARIHecke} in the PARI/GP manual. The weight $12$ rows were+    compared with the stored Ramanujan tau values in the table of the modular discriminant. Display properties:   number-header: $\chi_{k,p}(x)$