
// Make newform 4800.2.a.bf in Magma, downloaded from the LMFDB on 20 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_bf();".
// 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_bf();".
// 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], [0], [-4], [0], [4], [6], [-4], [-8], [-10], [-4], [-4], [-12], [4], [-2], [4], [0], [8], [12], [-4], [-10], [-8], [2], [-4], [-12], [2], [-12], [4], [12], [-12], [16], [2], [20], [8], [-20], [12], [-4], [4], [10], [-24], [-16], [-4], [12], [0], [-20], [20], [-26], [-20], [-8], [-2], [-12], [-12], [4], [-6], [-4], [-32], [6], [28], [12], [-12], [-24], [-16], [4], [-8], [8], [-12], [14], [28], [-24], [4], [24], [8], [-28], [34], [-8], [-14], [26], [-28], [-10], [-8], [-16], [12], [-36], [18], [-40], [-6], [12], [-12], [16], [-12], [-36], [24], [36], [-42], [-38], [28], [2], [-28], [-12], [-20], [26], [40], [32], [36], [36], [24], [38], [-28], [-8], [44], [-16], [28], [14], [-36], [12], [4], [28], [22], [0], [4], [44], [-32], [-26], [6], [-8], [-12], [-16], [40], [36], [28], [16], [42], [-18], [-12], [-28], [28], [6], [40], [14], [12], [-28], [18], [56], [8], [12], [-40], [-12], [56], [30], [12], [-36], [-12], [-48], [12], [18], [-32], [10], [-12], [4], [20], [-12], [12], [-28], [-28], [-24], [-34], [-36], [-12], [18], [-32], [-8], [-4], [-42], [-16], [-30], [-20], [-14], [-44], [-44], [32], [36], [20], [30], [16], [28], [-22], [56], [-24], [44], [16], [6], [-12], [36], [-18], [0], [36], [20], [54], [36], [48], [46], [-4], [-36], [36], [-4], [-26], [-32], [-32], [30], [44], [36], [-16], [70], [68], [-14], [-52], [4], [-42], [-4], [30], [-4], [-60], [-54], [-28], [8], [68], [20], [-16], [40], [-20], [-42], [-52], [76], [-30], [44], [12], [48], [-36], [8], [28], [62], [44], [-48], [-60], [36], [-40], [52], [-48], [2], [-52], [58], [-28], [12], [-22], [-44], [20], [64], [-36], [-28], [38], [-40], [36], [8], [-42], [-22], [-4], [-12], [34], [4], [-56], [44], [-40], [-52], [4], [-50], [38], [12], [36], [28], [-4], [-22], [4], [-8], [24], [-76], [4], [-46], [-26], [-12], [60], [36], [-8], [-6], [-60], [60], [44], [36], [-32], [-12], [52], [-84], [-64], [8], [4], [-50], [-64], [-72], [36], [30], [-30], [60], [76], [38], [-44], [32], [-40], [66], [-40], [-32], [6], [76], [12], [14], [24], [12], [-20], [-68], [-34], [4], [-44], [-20], [48], [36], [2], [-84], [22], [52], [-24], [-20], [82], [28], [-44], [12], [-42], [4], [80], [12], [-40], [-24], [54], [12], [70], [12], [48], [-12], [20], [-92], [-8], [-6], [60], [92], [-60], [80], [-12], [12], [-90], [-20], [56], [-44], [18], [-12], [0], [44], [32], [48], [-30], [40], [-54], [-44], [-76], [-68], [24], [-12], [12], [-64], [12], [-68], [-46], [36], [-60], [52], [80], [-88], [-20], [-26], [32], [78], [2], [20], [-28], [-60], [66], [76], [-48], [18], [28], [36], [-32], [-84], [-36], [-80], [-40], [10], [16], [-12], [-28], [-12], [-6], [56], [76], [-108], [64], [12], [28], [58], [0], [40]];
    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_bf();".
// 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_bf(: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_bf();".
// 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_bf( : 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![0, 1]>,<19,R![0, 1]>,<23,R![-4, 1]>],Snew);
    return Vf;
end function;