
// Make newform 70.4.a.c in Magma, downloaded from the LMFDB on 07 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_70_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_70_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_70_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_70_a();"
function MakeCharacter_70_a()
    N := 70;
    order := 1;
    char_gens := [57, 31];
    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_70_a_Hecke(Kf)
    return MakeCharacter_70_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], [-1], [-5], [7], [-65], [13], [-73], [-142], [130], [111], [256], [-266], [-424], [534], [-269], [-132], [-224], [-572], [-108], [560], [586], [57], [252], [-184], [-605], [-750], [-337], [634], [-827], [-1862], [48], [-582], [-344], [-1894], [618], [233], [-2218], [-2298], [311], [-1081], [-908], [-460], [4679], [-2528], [4404], [-1240], [-2959], [5559], [-5109], [3184], [4552], [-1819], [-4470], [-5258], [6566], [2730], [6762], [-6276], [-3610], [-3821], [-931], [-1773], [7469], [-1174], [7945], [-2578], [28], [398], [2142], [-7282], [5739], [904], [9451], [-10888], [1276], [588], [2577], [8575], [5217], [-5594], [-6036], [-14915], [3219], [-6942], [-1978], [6254], [14287], [-6860], [8552], [2876], [-9287], [11466], [17898], [3333], [-14141], [-17735], [-1246], [14346], [15596], [23], [12076], [-23528], [-420], [3234], [19372], [-5011], [-5904], [-14883], [-7021], [-10474], [9609], [-934], [-25358], [-11486], [12459], [5442], [-11513], [-29744], [27254], [-11861], [-5320], [1688], [23061], [14804], [-1196], [-18237], [16351], [-13902], [-4992], [38505], [-5649], [-25308], [-335], [7304], [31082], [-16994], [-33083], [-41269], [40431], [-31991], [-18498], [9601], [-22640], [28818], [20384], [-28842], [-11618], [-14926], [-2764], [-4332], [-24274], [5136], [6280], [-20312], [42618], [47808], [-34211], [19084], [-381], [11196], [28908], [4068], [-9026], [38844], [-1302], [31337], [-24296], [46931]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_70_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_70_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_70_4_a_c(:prec:=1)
    chi := MakeCharacter_70_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_70_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_70_4_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_70_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![1, 1]>],Snew);
    return Vf;
end function;