
// Make newform 4800.2.a.ch in Magma, downloaded from the LMFDB on 19 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_ch();".
// 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_ch();".
// 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], [2], [-2], [2], [-6], [-8], [4], [-8], [0], [-10], [2], [-12], [0], [10], [6], [-2], [-8], [-4], [-4], [-8], [4], [6], [-8], [0], [2], [4], [6], [-2], [18], [18], [-10], [-4], [12], [16], [-14], [16], [-12], [-18], [22], [14], [12], [-4], [-6], [-8], [-4], [-6], [-4], [-6], [-6], [20], [-10], [2], [6], [-12], [-24], [-16], [6], [-18], [0], [-22], [20], [12], [-4], [-18], [-16], [-28], [-28], [22], [-6], [16], [-14], [2], [4], [24], [-12], [6], [22], [-10], [-6], [26], [16], [12], [24], [-36], [-18], [0], [12], [22], [36], [-28], [-34], [34], [40], [-16], [8], [-26], [-16], [-10], [-20], [18], [-36], [-42], [-16], [-16], [4], [-34], [16], [-30], [22], [14], [10], [20], [-40], [18], [24], [8], [6], [-30], [-2], [20], [30], [20], [-48], [8], [-46], [40], [30], [-42], [0], [0], [8], [38], [-2], [34], [38], [32], [30], [18], [-12], [0], [-26], [-36], [14], [24], [-10], [34], [-28], [12], [-14], [-2], [-8], [-12], [-28], [48], [-24], [14], [56], [4], [36], [-42], [26], [-30], [2], [-56], [-32], [42], [-10], [-42], [-50], [18], [0], [4], [32], [-50], [-12], [-20], [-34], [42], [-2], [42], [6], [6], [24], [20], [-14], [32], [-14], [-12], [20], [-12], [20], [-36], [-12], [62], [46], [2], [38], [-56], [-36], [40], [-38], [-50], [62], [34], [-56], [52], [-34], [32], [-44], [60], [34], [20], [-4], [22], [-22], [-10], [-32], [34], [34], [-32], [66], [-66], [-28], [34], [-42], [56], [-58], [30], [26], [-36], [24], [14], [-56], [12], [-34], [18], [46], [-36], [-20], [40], [-34], [-74], [-30], [-20], [58], [6], [0], [4], [78], [66], [-12], [-26], [2], [18], [-74], [-8], [-70], [12], [10], [-20], [-22], [70], [-42], [-48], [-44], [50], [0], [-66], [2], [4], [28], [64], [-12], [34], [28], [6], [6], [-14], [-4], [40], [16], [-22], [-4], [-60], [-40], [78], [64], [34], [20], [-76], [-46], [-42], [-66], [-36], [8], [6], [-2], [68], [-52], [-74], [-16], [12], [-8], [48], [36], [26], [-36], [2], [-84], [24], [46], [-20], [-56], [-42], [-42], [44], [-36], [50], [-16], [-68], [-32], [6], [-62], [38], [-52], [-84], [88], [-26], [26], [-94], [-32], [-4], [12], [20], [50], [-22], [-6], [-78], [34], [-46], [-64], [-56], [-78], [30], [2], [12], [-84], [38], [20], [-16], [-72], [-54], [46], [82], [64], [22], [-82], [20], [-70], [-30], [8], [30], [-56], [16], [-30], [-46], [-22], [-66], [-28], [-24], [40], [-32], [-58], [-50], [24], [40], [46], [40], [-84], [-34], [26], [-70], [100], [60], [48], [-50], [52], [-60], [18], [-6], [-82], [-56], [56], [-64], [-32], [-6], [-48], [44], [-78], [38], [-54], [22], [72], [56], [-82], [-58], [-16], [-22], [8], [-78], [-84], [16], [-4], [104], [24], [50], [14], [-52], [-68], [2], [-96], [-30], [84], [66], [20], [-22], [-56], [104]];
    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_ch();".
// 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_ch(: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_ch();".
// 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_ch( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-2, 1]>,<11,R![2, 1]>,<13,R![-2, 1]>,<19,R![8, 1]>,<23,R![-4, 1]>],Snew);
    return Vf;
end function;