
// Make newform 8024.2.a.e in Magma, downloaded from the LMFDB on 05 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8024_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_8024_2_a_e();".
// 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_8024_2_a_e();".
// 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_8024_a();"
function MakeCharacter_8024_a()
    N := 8024;
    order := 1;
    char_gens := [2007, 4013, 3777, 3129];
    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_8024_a_Hecke(Kf)
    return MakeCharacter_8024_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], [-1], [-1], [-6], [-4], [1], [5], [-6], [-3], [0], [4], [-7], [-8], [-8], [-3], [-1], [-10], [-14], [-16], [4], [13], [-6], [-6], [-18], [-2], [6], [11], [-16], [12], [11], [-10], [-17], [16], [-16], [10], [-14], [4], [-13], [22], [22], [9], [0], [13], [26], [-1], [22], [24], [-20], [-14], [-30], [-3], [-25], [7], [-19], [-31], [10], [15], [-5], [15], [6], [11], [-17], [11], [14], [6], [-11], [8], [18], [-26], [-8], [-9], [-10], [-26], [-3], [24], [-10], [20], [-6], [-6], [-10], [30], [-34], [-17], [8], [22], [-17], [-22], [-2], [-10], [-32], [-32], [-33], [43], [23], [-24], [-38], [18], [-17], [-22], [28], [13], [-24], [-30], [30], [-15], [14], [-35], [3], [-20], [-17], [6], [19], [17], [0], [-30], [19], [49], [19], [16], [-3], [-26], [42], [48], [46], [-10], [-3], [30], [16], [-26], [-42], [-28], [34], [-29], [1], [-12], [4], [8], [-30], [26], [-26], [28], [-10], [-12], [11], [0], [41], [-14], [-14], [-24], [-19], [-54], [3], [-28], [3], [-49], [14], [-30], [-46], [-6], [-57], [2], [6], [35], [12], [-46], [42], [17], [-24], [0], [-36], [-44], [59], [32], [27], [-9], [-55], [-10], [-16], [-55], [-29], [-13], [-50], [15], [-16], [0], [-14], [-18], [28], [-10], [56], [-60], [38], [-18], [-57], [-54], [30], [-18], [-44], [34], [50], [-59], [34], [64], [3], [-36], [-8], [-44], [-16], [30], [-28], [13], [1], [31], [3], [-22], [8], [-38], [48], [42], [10], [6], [4], [63], [18], [50], [-11], [-30], [8], [-5], [-42], [30], [-72], [28], [-2], [57], [56], [66], [6], [8], [-69], [-70], [-67], [75], [51], [-48], [-2], [-60], [-37], [51], [19], [-54], [-48], [43], [31], [12], [-27], [-28], [-4], [-2], [-24], [-27], [39], [-14], [-81], [-10], [-49], [-78], [-61], [-5], [62], [4], [8], [-3], [-41], [70], [-36], [-3], [30], [-5], [4], [-38], [18], [-32], [42], [74], [24], [59], [10], [27], [10], [-28], [-26], [-30], [41], [66], [0], [-81], [-12], [-56], [-39], [48], [-58], [-40], [3], [-32], [83], [66], [42], [-54], [-59], [-85], [39], [-52], [-23], [58], [62], [11], [-26], [50], [52], [72], [-75], [-33], [51], [-16], [62], [20], [8], [-76], [88], [-18], [-2], [59], [28], [17], [-25], [-58], [-50], [7], [-34], [-62], [48], [-24], [-8], [-72], [80], [51], [-52], [-34], [28], [7], [39], [2], [-38], [-36], [96], [35], [-57], [9], [76], [-11], [3], [-66], [-12], [-16], [82], [-67], [-68], [-40], [68], [-68], [-42], [2], [-41], [-76], [-20], [-50], [-60], [17], [93], [-38], [56], [-40], [-67], [-58], [-43], [10], [20], [-72], [-50], [8], [30], [-88], [60], [76], [5], [7], [-79], [-21], [50], [78], [-1], [-58], [51], [32], [-36], [63], [-58], [-56], [1], [62], [-80], [21], [-25], [69], [-34], [76], [-4], [-61], [16], [6], [56], [-23], [21], [19], [-20], [66], [3], [56]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8024_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8024_2_a_e();".
// 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_8024_2_a_e(:prec:=1)
    chi := MakeCharacter_8024_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_8024_2_a_e();".
// 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_8024_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8024_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 1]>,<5,R![1, 1]>,<7,R![1, 1]>],Snew);
    return Vf;
end function;