
// Make newform 78.4.a.c in Magma, downloaded from the LMFDB on 02 January 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_78_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_78_4_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_78_4_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_78_a();"
function MakeCharacter_78_a()
    N := 78;
    order := 1;
    char_gens := [53, 67];
    v := [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_78_a_Hecke(Kf)
    return MakeCharacter_78_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 := 4;
    raw_aps := [[-2], [3], [10], [-8], [40], [13], [130], [-20], [0], [-18], [-184], [-74], [-362], [76], [-452], [382], [464], [358], [-700], [-748], [1058], [-976], [-1008], [-386], [-614], [518], [112], [-372], [934], [1914], [1296], [-892], [2326], [1932], [882], [-1776], [-2410], [3212], [1668], [3598], [1068], [-4786], [-1312], [-350], [-342], [-3368], [-2004], [-5608], [-1928], [-3938], [2562], [7164], [-6182], [-1396], [6906], [-6848], [-6034], [4832], [-4082], [3350], [7796], [3922], [5956], [2352], [8442], [-5550], [140], [-6174], [-2988], [-162], [-10754], [3588], [11272], [-10914], [8100], [6180], [-7522], [6078], [1830], [12434], [-14188], [8638], [4292], [-5982], [256], [12556], [5574], [1266], [7554], [-6752], [7924], [-11084], [4432], [-1140], [1764], [16976], [9474], [14114], [20284], [-14362], [-20956], [-4134], [-16228], [2514], [-11612], [6354], [-13240], [-1146], [10464], [6650], [-6664], [2134], [-714], [29228], [-13536], [17218], [15044], [25176], [-16034], [25356], [18310], [24802], [-22706], [-14792], [-1148], [14870], [-6354], [9288], [-21544], [19990], [532], [-25452], [6440], [-786], [-1498], [14738], [-3822], [-11900], [-21274], [-27566], [-11244], [13554], [14384], [-2488], [-20858], [23116], [934], [12642], [-22796], [-76], [-46130], [6682], [47404], [33672], [-14540], [-7840], [47720], [7502], [22058], [23338], [-30488], [9522], [-7616], [51316], [-48666], [17388], [11496], [48862]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_78_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_78_4_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_78_4_a_c(:prec:=1)
    chi := MakeCharacter_78_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_78_4_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_78_4_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_78_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![-10, 1]>,<7,R![8, 1]>],Snew);
    return Vf;
end function;