
// Make newform 4788.2.a.c in Magma, downloaded from the LMFDB on 10 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_c();".
// 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_c();".
// 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], [0], [-1], [2], [6], [8], [1], [2], [-6], [0], [10], [8], [-12], [-4], [-2], [8], [-10], [-12], [10], [-2], [-8], [4], [8], [2], [4], [0], [-2], [14], [-10], [16], [12], [18], [-4], [22], [4], [6], [8], [12], [24], [2], [-18], [-10], [14], [-10], [8], [-8], [-24], [16], [10], [-30], [-6], [10], [0], [-24], [-6], [-20], [8], [10], [-22], [-20], [-8], [-12], [-28], [-6], [-18], [24], [2], [-10], [-2], [24], [30], [32], [14], [8], [-8], [34], [6], [-6], [-6], [24], [10], [10], [-18], [-32], [-18], [30], [-22], [12], [36], [16], [-12], [-28], [-42], [20], [8], [-20], [-4], [28], [2], [-28], [14], [-24], [42], [-40], [-38], [8], [32], [34], [6], [32], [38], [-6], [-20], [-4], [26], [-20], [-36], [42], [-34], [10], [14], [-24], [-6], [-20], [-46], [-10], [32], [32], [-14], [-32], [-30], [-44], [-10], [28], [-6], [0], [28], [-56], [-18], [-44], [6], [28], [-18], [-34], [-12], [30], [24], [-20], [-2], [14], [0], [36], [-36], [16], [-54], [16], [36], [-22], [44], [-46], [54], [-36], [60], [-6], [0], [4], [14], [-26], [-12], [30], [-14], [6], [-26], [40], [-24], [0], [10], [0], [-50], [28], [-52], [-14], [4], [10], [60], [-34], [-20], [-26], [-56], [42], [42], [-32], [24], [-54], [36], [-2], [-14], [-44], [12], [6], [40], [54], [10], [-12], [36], [8], [30], [-18], [-52], [50], [-8], [-48], [-20], [-52], [30], [8], [-12], [-10], [-62], [-58], [-56], [14], [32], [-20], [-46], [0], [22], [48], [30], [14], [4], [44], [-66], [-60], [32], [62], [54], [-10], [-40], [-46], [-44], [-56], [18], [-64], [-44], [32], [-48], [-4], [-6], [-26], [-48], [78], [10], [-68], [-30], [-6], [20], [24], [-70], [16], [18], [18], [-14], [56], [-28], [-50], [0], [-36], [66], [-70], [-24], [-2], [64], [-70], [56], [-22], [2], [-58], [56], [-24], [68], [48], [-30], [-20], [-18], [-46], [22], [-8], [-8], [6], [-68], [-38], [-48], [-26], [-36], [16], [-8], [-40], [-4], [-14], [-38], [12], [-10], [64], [82], [-22], [-34], [-42], [-38], [40], [58], [18], [40], [18], [-14], [-20], [-18], [46], [34], [20], [22], [-24], [-12], [14], [-14], [-20], [-92], [66], [-6], [70], [-22], [-24], [24], [-24], [-48], [-66], [32], [-18], [-56], [46], [-14], [-12], [-60], [-62], [30], [-38], [-88], [72], [20], [-68], [-2], [54], [-24], [-50], [-12], [-48], [-84], [-10], [66], [-26], [-60], [42], [46], [92], [-42], [72], [56], [6], [62], [-76], [-10], [62], [-48], [-30], [-4], [26], [-70], [80], [42], [-72], [62], [72], [22], [92], [0], [-40], [66], [-8], [12], [-66], [12], [90], [68], [18], [10], [80], [-52], [20], [-78], [-46], [70], [88], [64], [-28], [56], [18], [6], [-28], [56], [-10], [2], [66], [-100], [-42], [-88], [30], [-16], [-32], [28], [66], [-38], [-82], [68], [46], [72], [-54], [-62], [4], [-60]];
    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_c();".
// 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_c(: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_c();".
// 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_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4788_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<11,R![-2, 1]>,<13,R![-6, 1]>],Snew);
    return Vf;
end function;