
// Make newform 4800.2.a.f in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4800_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_4800_2_a_f();".
// 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_4800_2_a_f();".
// 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_4800_a();"
function MakeCharacter_4800_a()
    N := 4800;
    order := 1;
    char_gens := [4351, 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_4800_a_Hecke(Kf)
    return MakeCharacter_4800_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], [-4], [4], [-2], [6], [-4], [0], [-2], [-4], [-2], [2], [-4], [8], [10], [-4], [-6], [-4], [16], [6], [-4], [-12], [10], [14], [6], [-12], [4], [-14], [-2], [-20], [4], [-18], [-20], [-18], [-12], [-10], [4], [8], [-6], [-12], [10], [0], [-18], [-22], [-4], [20], [-4], [-12], [10], [6], [0], [-14], [-12], [-2], [-24], [-18], [-12], [22], [-6], [-28], [-6], [12], [24], [-10], [18], [28], [-2], [-20], [26], [14], [-16], [-12], [-34], [-28], [32], [30], [22], [10], [10], [-20], [-38], [-24], [14], [28], [12], [10], [-26], [6], [-20], [-28], [24], [-4], [28], [-12], [0], [-18], [-30], [12], [2], [-12], [-22], [-12], [-46], [44], [-18], [36], [30], [16], [10], [-20], [-18], [-42], [12], [20], [10], [-12], [48], [10], [36], [-14], [-2], [-14], [12], [-52], [-18], [-22], [-8], [12], [14], [20], [24], [-4], [6], [-30], [-14], [42], [-12], [-6], [-14], [-28], [-26], [-4], [20], [34], [-32], [-18], [14], [4], [32], [22], [18], [20], [24], [-20], [0], [36], [-6], [22], [-58], [12], [-18], [-12], [52], [22], [24], [-52], [-18], [34], [34], [12], [-54], [8], [-10], [-20], [58], [28], [30], [20], [26], [52], [28], [22], [22], [48], [-34], [-18], [-4], [-22], [0], [-34], [20], [-44], [38], [28], [-42], [-62], [-50], [-2], [-40], [6], [20], [-10], [18], [-44], [18], [52], [4], [58], [-44], [-2], [30], [36], [-60], [64], [42], [4], [-38], [-8], [-54], [-38], [-4], [-6], [-52], [-12], [-14], [-42], [24], [-28], [28], [-50], [28], [4], [-54], [44], [-8], [2], [18], [-60], [-16], [-12], [12], [-36], [-6], [14], [-16], [-28], [-12], [-4], [-64], [30], [-46], [-48], [42], [50], [36], [-54], [52], [42], [-58], [44], [60], [58], [22], [6], [20], [-10], [-22], [44], [66], [10], [28], [-26], [12], [14], [36], [-4], [-38], [58], [-28], [-24], [44], [-16], [18], [-68], [0], [14], [2], [-44], [-14], [-42], [-84], [54], [44], [6], [54], [52], [-54], [36], [-28], [-58], [74], [-52], [-52], [-76], [-34], [-36], [66], [48], [14], [-48], [86], [74], [52], [24], [10], [36], [-88], [14], [-30], [-12], [-74], [-2], [68], [-66], [-62], [-84], [-84], [-32], [-62], [-46], [74], [-28], [76], [-52], [-4], [42], [14], [-70], [60], [14], [14], [38], [-68], [-46], [-36], [42], [68], [88], [26], [28], [-26], [62], [-36], [18], [6], [48], [60], [-26], [-56], [-2], [-6], [16], [4], [-20], [38], [18], [-52], [58], [12], [20], [-48], [-42], [-44], [-82], [-84], [32], [94], [42], [-90], [-42], [-50], [-52], [30], [76], [40], [60], [-10], [-28], [40], [-94], [82], [12], [-28], [56], [-42], [20], [-30], [20], [46], [-70], [102], [-68], [-18], [38], [-44], [14], [-22], [36], [-4], [-34], [2], [12], [-36], [-58], [14], [-64], [44], [86], [-16], [22], [-34], [32], [-60], [70], [-46], [36], [-22], [-92], [-96]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4800_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4800_2_a_f();".
// 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_4800_2_a_f(:prec:=1)
    chi := MakeCharacter_4800_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_4800_2_a_f();".
// 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_4800_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, 1]>,<11,R![-4, 1]>,<13,R![2, 1]>,<19,R![4, 1]>,<23,R![0, 1]>],Snew);
    return Vf;
end function;