
// Make newform 2400.2.a.y in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2400_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_2400_2_a_y();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_2400_2_a_y();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_2400_a();"
function MakeCharacter_2400_a()
    N := 2400;
    order := 1;
    char_gens := [1951, 901, 1601, 577];
    v := [1, 1, 1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_2400_a_Hecke(Kf)
    return MakeCharacter_2400_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[0], [1], [0], [0], [-4], [-2], [2], [-8], [4], [-6], [0], [-2], [-6], [4], [-12], [6], [-12], [14], [-12], [0], [-2], [8], [-4], [2], [14], [-14], [8], [-12], [14], [-6], [-16], [20], [2], [-16], [-6], [-8], [14], [-20], [12], [14], [-12], [22], [-16], [22], [-26], [-16], [8], [24], [-4], [-18], [-6], [8], [2], [-12], [-6], [20], [10], [-8], [30], [2], [-4], [-26], [28], [0], [14], [-18], [0], [22], [-4], [-2], [-14], [16], [0], [22], [16], [-20], [-6], [6], [10], [-22], [12], [-2], [8], [-26], [32], [-28], [10], [-26], [-14], [-32], [36], [40], [8], [-12], [-8], [-4], [-38], [-30], [-4], [30], [28], [-18], [36], [34], [-32], [-2], [36], [42], [-16], [10], [8], [14], [42], [16], [32], [2], [12], [20], [14], [-20], [-26], [6], [6], [12], [-16], [10], [-2], [-32], [40], [-10], [8], [-36], [-32], [-34], [-30], [-14], [-10], [20], [-34], [-22], [-24], [-30], [16], [-36], [30], [-48], [30], [10], [32], [4], [14], [-6], [44], [-20], [-36], [8], [8], [42], [-42], [34], [-4], [-30], [-32], [-28], [18], [-4], [-40], [6], [-14], [-42], [52], [-26], [24], [-18], [-56], [-30], [24], [-6], [-16], [-10], [-56], [-28], [-34], [18], [12], [10], [-18], [-4], [10], [0], [-26], [20], [-24], [2], [20], [-46], [-30], [-18], [-54], [-4], [-62], [0], [-42], [-14], [-60], [-66], [56], [4], [50], [-64], [-10], [42], [-8], [-36], [-8], [10], [-32], [42], [36], [-2], [-34], [-32], [-30], [-32], [52], [14], [-14], [0], [-56], [-36], [-66], [-16], [-56], [2], [28], [-20], [18], [-10], [-76], [0], [20], [-16], [64], [-18], [-14], [-8], [32], [28], [24], [-76], [6], [2], [60], [26], [30], [60], [38], [20], [54], [14], [8], [-36], [62], [70], [-54], [-40], [50], [42], [-68], [-58], [-42], [-52], [46], [24], [-42], [8], [28], [46], [-54], [36], [84], [40], [-12], [-18], [12], [-64], [-34], [22], [80], [-62], [-30], [-28], [-54], [28], [-26], [-46], [16], [22], [36], [12], [-66], [14], [-32], [-84], [32], [62], [4], [-26], [24], [6], [-76], [-30], [-38], [4], [20], [10], [-28], [48], [-10], [-62], [80], [-50], [90], [48], [26], [2], [32], [52], [84], [6], [-26], [62], [88], [-4], [56], [52], [-26], [-30], [74], [-16], [-58], [-54], [-30], [-72], [78], [-44], [14], [28], [0], [54], [-72], [38], [58], [40], [-42], [-54], [32], [60], [-6], [68], [-58], [-86], [68], [-44], [36], [6], [30], [-24], [-86], [60], [-16], [-52], [98], [-24], [38], [36], [-8], [-82], [-38], [38], [2], [-30], [-80], [34], [-48], [-60], [64], [-34], [-60], [-36], [-78], [6], [-44], [-28], [-40], [86], [32], [10], [64], [10], [-10], [90], [32], [-46], [50], [64], [70], [42], [-84], [-68], [-66], [-10], [12], [-80], [-66], [58], [96], [-80], [-54], [20], [50], [-26], [68], [12], [14], [78], [12], [90], [64], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2400_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2400_2_a_y();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_2400_2_a_y(:prec:=1)
    chi := MakeCharacter_2400_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_2400_2_a_y();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_2400_2_a_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2400_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![4, 1]>,<13,R![2, 1]>,<19,R![8, 1]>],Snew);
    return Vf;
end function;