Lfunctions
The highest priority for Lfunctions is to reorganize the code so that all the information is stored in a database. This will make it much easier to implement the other planned changes. This work has been started for the Lfunctions of genus 2 curves.
Navigation pages
 Reorganize the degree 3 and 4 navigation pages to allow browsing in terms of functional equation parameters in addition to browsing by underlying object.
 Add searching by first zero, coefficients, functional equation parameters, and underlying object.
 Show the sign of the Dirichlet Lfunctions as a color code on the browsing graph.
Home pages
 Reorganize the home pages so that they look more like other home pages. In particular, have an "Invariants" section near the top.
 Add section for Special Values.
 Allow the user to choose arithmetic or analytic normalization (for arithmetic Lfunctions).
 Smoothly handle the zeros and Zfunction plot in the cases where not enough coefficients are available. (Will be automatic for Lfunctions coming from the database.)
 Display more coefficients for Lfunctions with very sparse Dirichlet series, and condense the display for floating point coefficients.
 Recognize nonprimitive Lfunctions and provide links to the primitive components. (sym^2 of CM curves, Dedekind zetafunction in the nonabelian case, etc)
 Links to more friends: dual Lfunction (if not selfdual), twists, symmetric powers, other powers.
 Next/previous navigation for more Lfunctions (e.g. GL(2) Maass forms).
Available data
 Add the Lfunctions of: Maass forms (some currently limited by the number of available coefficients), $sym^2$ of CM elliptic curves, $sym^2$ of other GL(2) objects, Siegel modular forms, Hilbert modular forms (Asai Lfunction), Artin representations.
Elliptic modular forms
Navigation pages
 Add a paragraph above each browsable table (Gamma_0(N) and Gamma_1(N)) describing how much data is currently available
Home pages
 Give an option to show the dimensions of the whole space (including oldforms)
 Make it possible to show a basis for the whole space, not "only" representatives for the Galois orbits
 Show sage/magma(?) commands
 produce zoomable images of fundamental domains
Available data
 Rewrite the whole backend for getting data for modular forms (spaces)
 Use (mostly) William's recently computed data
 Include Eisenstein series
 Add links to the lifts (e.g., SaitoKurokawa) of classical MFs
 Add more data to the homepages of spaces of newforms (e.g., Hecke polynomial (factored?))
 Add Satake parameters at "bad" primes
Elliptic Curves
Navigation pages
 Currently the search possibilities are rather limited, but if more search fields are introduced the search page will become very cluttered. At that point we should only have very common options on the main page, with a separate "advanced search" page. Examples include searching by [a1,a2,a3,a4,a6] or by [A,B]. Work on this is in progress.
Additional pages
 Add more to the Statistics page for elliptic curves over Q: currently this gives distributions of ranks, Shas and other things.
 Add page with list of elliptic curves over Q that have a point of canonical height at most some small'ish epsilon. (Noam Elkies has this data.)
 Add page with elliptic curves of smallest known conductor and MordellWeil group of the form F x Z^r for the 15 possible finite groups F and r = 0,1,2,...
Additional data (over Q)
 Currently the database contains all elliptic curves defined over $\mathbb{Q}$ of conductor at most 360000.
In future it is planned to add the much larger SteinWatkins database, which includes curves with larger conductor. This would not be complete. Currently we do not have as much data about all the SteinWatkins curves as for those in the Cremona database: for example we do not have generators for all the curve of rank 1. Also Mike Bennett's 2015 data, Randall Rathbun's data on congruent number curves for parameters up to 1000000.  More data about each curve:
 Fisher has explicit models of genus one curves representing every element of order 3 and order 5 in Sha, and one could easily do similar for elements of order 2. These could be shown on a curve's home page.
 Automatic construction of quadratic twists of each curve from its home page; if the twist is not in the database we can still create a home page for it, with limited data.
 Base change: the user could select an extension field and see the page for the basechanged curve. This would be easy for cases where the basechange curve is already in the database.
Elliptic curves over number fields
 Work has started to include elliptic curves over many number fields (real and imaginary quadratic and more). So far we have all curves over real quadratic fields whose corresponding Hilbert modular form is in the database; curves over the first 5 imaginary quadratic fields, which will be similarly linked to Bianchi newforms; and the GunnellsYasaki curves over the cubic field of discriminant 23.
 The above is currently only accessible from beta.lmfdb.org.
 Data available for each curve should be extended to include MordellWeil generators, which have in most cases not yet been computed.
 Automatic base change from the home page of every elliptic curve over $\Q$ to a number field, specified by its label. Whether or not the resulting curve is in the database, its home page can be displayed.
Dirichlet characters
Home pages
 improve speed
 add magma commands (and pari when characters are released)
 show a graph of characters above and below (parents and sons)
 show minimal polynomial and galois orbit
Search
 group characters by Galois orbits
 search from values
Available data
 add Dirichlet characters over number fields
 add Hecke characters on ideals and link with Hecke Lfunctions
Genus 2 curves
Navigation
 In the search by torsion order, it would be nice to know what possible torsion groups occur among the curves in the database.
Home pages
 List automorphic friends that arise via base change.
 Compute Kummer surface.
Hilbert modular forms
Navigation pages
 Spaces of Hilbert modular forms need web pages. Could list HeckeSturm bound data, list tables of dimensions, associated Shimura curve if relevant, etc.
Home pages
 Compute Eisenstein ideal (torsion).
 Simplify presentation of Hecke field (adjoin all elements to get an order, or mimic classical modular forms by working with the dual).
 For the CM forms, prove the CM and identify explicitly the Hecke character.
 For the base change forms, prove this rigorously (if possible) and identify explicitly the base change form.
Possible new subcategories
 Higher weight, odd weight, halfintegral weight, and characters.
 Maass forms.
Maass forms
Navigation pages
 Make browsing more intuitive (how do people actually want to browse / search Maass forms?)
 Is it possible to keep information about where in the browse list we are, when going to the page for a specific Maass form?
 Maass forms arising from quadratic fields.
Home pages
 Add download of Fourier coefficients
 Make it possible to manipulate Fourier coefficients (i.e. test Hecke relations etc.)
 Get statics for Fourier coefficients, for example plots of value distribution (to see SatoTate) etc.
 Add plots of the Maass form in the fundamental domain.
 Add next / previous buttons (like on the Dirichlet character pages).
Available Data
 All Maass form data (for GL(2)/Q) is currently recomputed using sage/psage/cython. These computations should continue until we have data for all levels, at least up to 100.
 Compute large sets of coefficients for all Maass forms in the database.
Technical Issues
 The option to show "all" records does not work on the browse page.
 Fix the search so that it is also done with serverside dataTables so we can remove the limit of 200 records (currently the client crash if we return more records)
 Try to make "tabbed" dataTables to make it possible to flip between different levels when browsing.
Possible new subcategories
 Maass forms on the Picard group (these already exist in the database, but the corresponding pages needs to be revised).
 Harmonic weak Maass forms.
 Maass forms of halfintegral weight.
Siegel Modular Forms
Home Page
 Provide data for and make links to the Lfunction section.
 Improve the presentation of many and big Fourier coefficients: the user should be able to ask for a range, reduction modulo arbitrary ideals of the number field generated by the coefficients should be possible (currently one can only reduce modula rational primes).
Database
 Compute generators and examples for the rings $\Gamma_0(2)$, $\Gamma_0(3)$, $\Gamma_0(4)$ (possibly with characters).
 Implement the data of Ibukiyama and Hayashida on half integral weight and vector valued modular forms.
 Let mongodb or sqlite keep the date.
 Normalise the Fourier coefficients so that they are integral and primitive.
Artin representations
Navigation pages
The search page will be expanded to include searches based on information depending on the Artin field (for instance the Galois group).
Home pages
 A link from degree 1 Artin representations to the corresponding Dirichlet character needs to be added (and similarly for Lfunctions)

Non integral root numbers need to be calculated (for 1dimensional, based on Gauss sums, for higher dimensional reps, based on analytic approximation)

A link from degree 2 Artin representations to the corresponding modular form will be added.
Available data
Very limited data is available right now (37000 representations). We will soon add more data, coming from computations of Tim Dokchitser.
Global number fields
Currently have fields with small absolute discriminant in degrees $\leq 11$, and other complete sets of fields with degrees up to $15$. See completeness of global number field data for more details
Home pages
 Data on narrow class numbers
 Include information on local algebras for ramified primes
 Pretty print names for abelian fields (e.g. cyclotomic fields, totally real subfield, biquadratic fields)
Available Data
 Add fields with bigger degrees and discriminants, and with more diverse Galois groups.
Local number fields
Currently have all degree $n$ fields over $\mathbb{Q}_p$ for $p<150$ and $1\leq n\leq11$.
Navigation
 Allow searching by inputing a prime and a polynomial (return components of the local algebra)
 Allow download of fields coming from a search
Home pages
Give components of interesting local resolvent algebras, such as twins for sextics and $D_4$ quartics
Available data
 Hope to add higher degree fields once data has been computed for wildly ramified fields of degree 12
Galois groups
Currently the database contains all transitive subgroups of $S_n$ up to conjugation for $n\leq 23$.
Navigation
 Add more human readable names for groups
Home Pages
 Better handling of character tables
Future areas
Objects which will be added to the LMFDB include:
 hyperelliptic curves
 Galois representations
 GL(N) automorphic forms of cohomological type
 K3 surfaces and CalabiYau 3folds
 Examples of every type of Lfunction of degree $\le 4$
 Automorphic representations
 Bianchi modular forms