
// Make newform 49.6.a.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_49_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_49_6_a_a();".
// 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_49_6_a_a();".
// 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_49_a();"
function MakeCharacter_49_a()
    N := 49;
    order := 1;
    char_gens := [3];
    v := [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_49_a_Hecke(Kf)
    return MakeCharacter_49_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+1)]];
    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 := 6;
    raw_aps := [[-10], [14], [56], [0], [232], [140], [1722], [98], [1824], [3418], [7644], [-10398], [17962], [10880], [-9324], [2262], [2730], [-25648], [-48404], [-58560], [-68082], [31784], [20538], [50582], [58506], [-38696], [-53060], [-146324], [92898], [-83354], [60384], [61586], [-204462], [35406], [-20226], [70904], [-293524], [13192], [-493612], [-240716], [294932], [336980], [358264], [-989554], [-990050], [840756], [1150732], [824264], [-74382], [-1131956], [-198726], [482904], [-805910], [-430738], [1176910], [1290976], [1277556], [-1650544], [-1064090], [-22342], [2495738], [1931776], [459074], [-667128], [111034], [-68778], [-564448], [2077294], [-53248], [2272004], [-4006450], [73784], [-1404312], [-1603234], [-4770120], [2230788], [4840242], [-995820], [-3316050], [-3072734], [-2814378], [3058022], [1937496], [-3947902], [7417704], [1402692], [-590574], [-2904842], [922684], [7182352], [612570], [-2603300], [5463088], [1640900], [2997964], [6894048], [-2304764], [12096042], [-5484430], [-6717994], [-5002352], [9019606], [-12405106], [6488038], [-10228504], [-2653378], [14304374], [10026534], [-7522920], [-3386250], [6908608], [-9688958], [-7847418], [10197194], [-8362576], [1102830], [-17135426], [54964], [-485166], [-2721360], [2145248], [2927962], [13499220], [-5429724], [-20827982], [23514066], [-19574670], [26115180], [-15412628], [16986816], [2015112], [-15138072], [7214008], [-10969726], [-19244190], [-8211854], [-18618656], [-26250070], [10037300], [14088390], [-18143342], [-21366922], [17801736], [16292124], [2084992], [22784972], [22697500], [-25289950], [10394734], [43339944], [37165930], [-9047850], [32967860], [16109884], [-44728564], [-6605184], [-30892984], [4872154], [32500398], [2640400], [-40817920], [-6719830], [-2789792], [-33359382], [-7600326], [5797596], [12682496], [-14440048]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_49_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_49_6_a_a();".
// 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_49_6_a_a(:prec:=1)
    chi := MakeCharacter_49_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_49_6_a_a();".
// 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_49_6_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_49_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<2,R![10, 1]>,<3,R![-14, 1]>],Snew);
    return Vf;
end function;