
// Make newform 4788.2.a.d in Magma, downloaded from the LMFDB on 05 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_4788_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_4788_2_a_d();".
// 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_4788_2_a_d();".
// 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_4788_a();"
function MakeCharacter_4788_a()
    N := 4788;
    order := 1;
    char_gens := [2395, 533, 4105, 1009];
    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_4788_a_Hecke( : Kf := Kf)
    return MakeCharacter_4788_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]];
    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], [0], [2], [-1], [2], [2], [-2], [1], [-6], [0], [8], [6], [0], [8], [6], [0], [0], [10], [4], [-8], [6], [-8], [-6], [8], [6], [-2], [-8], [4], [10], [4], [8], [-2], [10], [4], [18], [-16], [18], [4], [-24], [16], [12], [2], [-18], [-14], [14], [-4], [20], [24], [12], [14], [14], [6], [-18], [-6], [12], [-26], [0], [4], [6], [0], [0], [0], [-4], [-34], [6], [12], [-12], [-2], [2], [14], [6], [6], [-28], [22], [-4], [20], [6], [-2], [28], [-10], [-34], [-30], [8], [-14], [8], [34], [16], [-10], [-10], [4], [18], [26], [16], [-30], [-12], [10], [0], [0], [-4], [2], [-12], [42], [-28], [-40], [40], [-18], [18], [-22], [-20], [-14], [-8], [2], [26], [20], [-40], [-28], [32], [-42], [-6], [-40], [22], [-14], [-12], [4], [-8], [-6], [-38], [6], [28], [50], [36], [-16], [8], [-18], [-2], [-18], [-32], [-36], [-32], [30], [-28], [-6], [24], [20], [50], [-52], [-42], [-40], [20], [-16], [-10], [-42], [12], [-24], [-52], [-52], [0], [34], [-38], [-40], [18], [-16], [-16], [-60], [4], [16], [-40], [-38], [30], [14], [4], [-26], [58], [26], [-24], [-6], [4], [26], [-40], [-50], [-36], [-36], [-6], [28], [42], [6], [-46], [4], [46], [14], [-42], [-54], [20], [52], [18], [-56], [34], [34], [-22], [18], [56], [-8], [10], [6], [-26], [-42], [-20], [-4], [26], [-20], [-42], [18], [-44], [-48], [4], [-14], [52], [32], [16], [46], [42], [-16], [20], [8], [36], [58], [-36], [8], [-48], [18], [-58], [-28], [-24], [4], [-4], [70], [-34], [-42], [-22], [12], [60], [-56], [-28], [46], [28], [-6], [-56], [-36], [60], [66], [22], [34], [50], [-18], [34], [42], [-22], [12], [20], [14], [-40], [-48], [26], [46], [42], [-60], [-72], [50], [-68], [38], [62], [4], [-38], [32], [-10], [56], [-42], [46], [-46], [2], [4], [16], [66], [-82], [-8], [-78], [-30], [20], [-32], [-84], [-30], [66], [-40], [-16], [-18], [-22], [-40], [-54], [-20], [-84], [30], [-72], [60], [20], [-20], [38], [32], [2], [66], [-10], [26], [30], [-76], [-28], [42], [-42], [14], [12], [-34], [-22], [-44], [10], [36], [-16], [58], [14], [20], [84], [8], [-54], [70], [80], [-80], [-66], [-80], [82], [-10], [0], [-46], [36], [-10], [18], [36], [48], [0], [-48], [22], [-4], [-96], [22], [52], [-26], [46], [-64], [94], [16], [-14], [-78], [18], [24], [-26], [30], [0], [36], [-4], [58], [-10], [88], [26], [30], [-4], [90], [0], [76], [2], [76], [82], [18], [2], [54], [-44], [6], [52], [30], [-4], [-20], [28], [46], [4], [-16], [22], [60], [-102], [-36], [-72], [-82], [32], [-44], [84], [-18], [38], [-66], [88], [92], [-56], [44], [2], [-88], [-84], [38], [66], [86], [16], [24], [22], [70], [92], [-104], [46], [-44], [-24], [-74], [18], [-28], [-86], [8], [84], [6], [4], [-58]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4788_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4788_2_a_d();".
// 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_4788_2_a_d(:prec:=1)
    chi := MakeCharacter_4788_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_4788_2_a_d();".
// 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_4788_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4788_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<11,R![-2, 1]>,<13,R![-2, 1]>],Snew);
    return Vf;
end function;