20 problems

For $N\in\mathbb{N}$, let $S_2^{\rm new}(\Gamma_0(N))$ denote the space of weight-two new cusp forms on $\Gamma_0(N)$, and let $\dim S_2^{\rm new}(\Gamma_0(N))$ be its dimension. M…

For a prime number $r$, let $O(1),\dotsc,O(s)$ be the newform orbits of $S_2\big(\Gamma_0(r)\big)$. For a subset $W\subset\{1,\dotsc,s\}$, write … for the sum of the sizes of the s…

Fix a weight $k\geq 6$, and let $f$ range over newforms. Roberts's conjecture. Only finitely many newforms $f$ have rational coefficient field, equivalently … This is a weaker gene…

Let $p$ be a prime, and consider the weight-$2$ newforms for $\Gamma_0(p)$ in each of the two Atkin–Lehner eigenspaces. Maeda-type conjecture. In each eigenspace, the newforms shou…

For each prime $p$, let $\operatorname{perdim}(p)$ be the number of weight-$2$ newforms $f$ for $\Gamma_0(p)$ such that $w(f)=-f$ and $L(f,1)=0$. Positive-even-rank dimension conje…

Let $m\geq 2$ be a fixed non-square integer, and let $T_m^{\operatorname{new}}(N,k)$ denote the restriction of the $m$-th Hecke operator to the new subspace…

Let $G_n$ be the unramified unitary group under consideration, let $\pi$ be an irreducible generic representation of $G_n$ with conductor exponent $a_\pi$, and let…

Let $G_n={\rm SO}_{2n+1}$ be the split odd special orthogonal group of rank $n\geq 1$ over $F$, let $U_n\subset G_n$ be its upper triangular maximal unipotent subgroup, and let…

Let $a_3(n)$ and $a_4(n)$ be the Fourier coefficients of newforms … whose coefficient fields are $\mathbb{Q}\left(\sqrt{2}\right)$ and $\mathbb{Q}\left(\sqrt{3}\right)$, respective…

Let $\eta_i$ be weight-two newforms, and let $\check r_i,\check t_i\in\mathbb{N}$. Suppose … where, for fixed $i$, the sequence $a_{i,n}$ is increasing, consists of positive intege…

Let $N=MN'$ be square-free and let $k\geq 4$. Let $g\in M_k(M)$ be a newform of level $M$, with Hecke eigenvalues $\lambda_p$ for $p\mid N'$. Let $\varepsilon:D(N')\to\{\pm1\}$ be…

Let $p$) be prime, let $S_k(p)$ denote the space of cusp forms of weight $k$ and level $p$, and let a Galois orbit mean an orbit of newforms under the action of the Galois group on…

For each even weight $k$ and level $N$, let $Q_k(N)$ be the set of quadratic-twist classes of classical newforms with rational coefficients and no complex multiplication, represent…

Let $Q_k(N)$ denote the set of quadratic-twist classes of classical newforms of weight $k$ and minimal level $N$ with rational coefficients and no complex multiplication. Finitenes…

Roberts' conjecture. There are no non-CM newforms with rational coefficients and weight $k'\geq 52$. This conjecture is used to support the expected finiteness of the search: combi…

For $N=2,3$, let $\mathcal S_k^{\mathrm{new}}(\Gamma_0(N))^{\pm}$ denote the subspaces of newforms of weight $k$ and level $\Gamma_0(N)$ on which the Atkin–Lehner involution acts b…

Eichler–Shimura–Manin correspondence. There are isomorphisms defined over $\mathbb C$

Let $F$ be a number field, let $G=\operatorname{GL}_2$, and let $\pi$ be an automorphic representation with finite conductor $N$ and local conductor exponents $n_v$ and central-cha…

Let $f$ be a newform on $\operatorname{GL}_2(\mathbb{A}_{\mathbb{Q}})$ with level $M$ and conductor $N$, and let $N_1$ be the smallest integer such that $N\mid N_1^2$. Assume that…

Let $n_0(N,k)$ be the smallest nonnegative integer such that, for any newforms $f,g\in S_k(\Gamma_0(N))$, equality $a_n(f)=a_n(g)$ for all $n\leq n_0(N,k)$ implies $f=g$. Let…