
// Make newform 4788.2.a.e in Magma, downloaded from the LMFDB on 19 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_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_4788_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_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], [-4], [4], [-6], [1], [-4], [-6], [4], [-10], [-4], [-8], [0], [-10], [4], [14], [-6], [6], [-2], [-10], [4], [-12], [-12], [6], [12], [6], [-10], [-10], [10], [12], [-6], [-12], [22], [10], [10], [-4], [16], [4], [18], [-16], [16], [-6], [-26], [8], [-26], [28], [-24], [18], [-22], [-4], [8], [-12], [16], [-24], [16], [16], [10], [10], [-28], [12], [16], [-24], [-14], [14], [-30], [-2], [12], [10], [-6], [4], [-24], [-22], [-30], [24], [-18], [18], [-14], [-28], [4], [-14], [6], [8], [-32], [-32], [-10], [-18], [22], [4], [12], [24], [-34], [4], [-24], [-16], [-8], [-16], [-28], [-18], [26], [26], [36], [26], [24], [2], [-12], [6], [22], [0], [28], [6], [-22], [-44], [40], [18], [-12], [24], [-30], [-2], [-4], [38], [8], [30], [52], [-30], [26], [-8], [16], [-2], [16], [-14], [30], [22], [42], [-30], [-20], [-44], [-4], [-30], [12], [-30], [-4], [30], [-4], [0], [14], [-44], [4], [38], [38], [-54], [16], [36], [26], [30], [24], [54], [-2], [16], [24], [-26], [-48], [20], [10], [-36], [-22], [2], [26], [14], [18], [-52], [-36], [6], [-36], [-42], [-24], [42], [-32], [-18], [-8], [-20], [14], [-28], [-40], [30], [34], [24], [38], [56], [24], [-32], [38], [-16], [24], [64], [-50], [26], [-58], [8], [-10], [52], [8], [40], [12], [-58], [-24], [42], [-54], [64], [-50], [50], [16], [12], [-48], [36], [24], [24], [-30], [2], [22], [-40], [-42], [-40], [-60], [-46], [24], [70], [52], [36], [10], [56], [-74], [38], [52], [-16], [-50], [42], [-64], [16], [-66], [20], [-24], [-22], [8], [-16], [40], [24], [-50], [-24], [2], [-50], [40], [56], [-30], [-12], [54], [-4], [-32], [14], [-46], [-26], [-42], [-24], [-82], [-16], [54], [-2], [2], [-18], [52], [18], [46], [-76], [52], [-56], [-4], [58], [-54], [-36], [-84], [-28], [48], [-48], [36], [28], [-42], [22], [8], [0], [42], [-68], [18], [-12], [-62], [34], [-40], [26], [-20], [-36], [6], [-78], [16], [18], [60], [-58], [-10], [-4], [40], [38], [24], [-66], [50], [22], [-60], [76], [-68], [-54], [-14], [-26], [44], [-62], [24], [74], [-74], [-24], [-74], [48], [38], [-22], [-14], [42], [56], [12], [68], [44], [86], [36], [50], [0], [-38], [14], [-36], [62], [-6], [-18], [-50], [-82], [-12], [-42], [-92], [-70], [34], [84], [-14], [-24], [-48], [-12], [-66], [-18], [-6], [26], [-90], [-78], [-20], [-66], [30], [-30], [10], [72], [36], [60], [-18], [-32], [38], [-100], [-12], [-78], [-66], [-8], [24], [-62], [-36], [14], [-16], [-72], [12], [-14], [-100], [-24], [86], [84], [32], [-28], [-54], [-66], [40], [-60], [-74], [42], [64], [-66], [-50], [-72], [96], [64], [50], [6], [-20], [-36], [-16], [-34], [50], [64], [-62], [70], [6], [-56], [-14], [100], [-30], [-76], [-36], [16], [56], [-36], [34], [54], [20], [56]];
    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_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_4788_2_a_e(: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_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_4788_2_a_e( : 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![4, 1]>,<13,R![-4, 1]>],Snew);
    return Vf;
end function;