
// Make newform 4788.2.a.f 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_f();".
// 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_f();".
// 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], [-6], [-2], [1], [-6], [8], [-8], [-10], [-8], [-8], [-2], [8], [0], [-6], [-4], [0], [6], [0], [-14], [16], [-2], [-18], [-8], [12], [-6], [12], [16], [6], [-6], [4], [2], [8], [-14], [4], [8], [8], [4], [26], [14], [2], [-18], [12], [-4], [-8], [12], [-2], [-18], [22], [22], [2], [-28], [-10], [-24], [-28], [-26], [-8], [-16], [-24], [28], [22], [22], [-12], [28], [-34], [-14], [-2], [6], [6], [4], [6], [36], [4], [-26], [14], [-12], [-34], [-10], [-14], [16], [-6], [-40], [18], [-24], [22], [-26], [-28], [-6], [2], [-24], [18], [-12], [2], [24], [-8], [-4], [2], [28], [-6], [20], [32], [-24], [-2], [-38], [26], [20], [-38], [40], [34], [-22], [4], [-40], [12], [-32], [-18], [42], [-32], [46], [-46], [28], [44], [8], [42], [-6], [-18], [-4], [-30], [20], [8], [-48], [-18], [30], [-2], [24], [-4], [-24], [30], [20], [42], [8], [-36], [26], [-36], [-42], [32], [-28], [24], [22], [-42], [-20], [24], [4], [-12], [-48], [-46], [10], [-16], [50], [-24], [16], [20], [28], [32], [32], [26], [30], [-34], [-4], [-2], [42], [26], [-40], [42], [52], [42], [-8], [14], [44], [44], [-6], [-12], [42], [54], [-46], [-12], [-34], [-42], [14], [10], [-20], [12], [18], [-48], [-30], [-30], [-6], [-6], [0], [40], [2], [-66], [-18], [-58], [-52], [52], [42], [-20], [22], [50], [20], [16], [-12], [-38], [52], [-24], [-24], [-50], [58], [-64], [-20], [24], [-12], [-6], [36], [-16], [32], [-30], [-26], [-60], [-32], [60], [-4], [-18], [-2], [-42], [26], [-68], [4], [-40], [52], [46], [-60], [66], [56], [-36], [20], [-62], [-10], [18], [-30], [6], [-62], [74], [10], [-4], [12], [-18], [-16], [-72], [10], [-74], [-38], [20], [-32], [2], [4], [6], [-58], [60], [42], [32], [-50], [56], [38], [-82], [34], [10], [4], [-32], [26], [-58], [8], [-30], [34], [-52], [-32], [20], [18], [-70], [-16], [48], [30], [10], [8], [-54], [60], [-4], [-34], [-32], [76], [-84], [12], [38], [72], [26], [-46], [22], [-78], [30], [-36], [-4], [74], [46], [-10], [-60], [-18], [-6], [20], [74], [-52], [-24], [42], [-42], [44], [36], [-16], [26], [38], [-24], [-32], [22], [-32], [58], [-10], [-56], [18], [-60], [-26], [-30], [12], [-56], [24], [88], [-10], [4], [48], [38], [52], [-90], [-50], [-48], [-18], [40], [-22], [42], [-62], [-32], [-26], [30], [-56], [-36], [-4], [-6], [-90], [-64], [10], [-2], [12], [-54], [-72], [-52], [-30], [60], [50], [-14], [-14], [30], [60], [70], [-60], [-66], [-84], [-4], [-84], [30], [-60], [-48], [6], [20], [90], [44], [16], [14], [-64], [44], [28], [-18], [-50], [-66], [64], [-60], [-80], [-84], [2], [-16], [60], [94], [-22], [54], [40], [8], [86], [-42], [-28], [24], [46], [84], [0], [-66], [34], [4], [2], [0], [108], [6], [-76], [-66]];
    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_f();".
// 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_f(: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_f();".
// 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_f( : 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![6, 1]>],Snew);
    return Vf;
end function;