
// Make newform 2010.2.a.a in Magma, downloaded from the LMFDB on 29 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2010_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_2010_2_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_2010_2_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_2010_a();"
function MakeCharacter_2010_a()
    N := 2010;
    order := 1;
    char_gens := [671, 1207, 1141];
    v := [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_2010_a_Hecke(Kf)
    return MakeCharacter_2010_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 := 2;
    raw_aps := [[-1], [-1], [-1], [0], [4], [2], [6], [-4], [4], [-10], [-4], [-2], [-2], [4], [4], [10], [12], [6], [1], [0], [-6], [-12], [8], [2], [6], [2], [8], [16], [6], [-2], [-8], [4], [6], [8], [14], [16], [-2], [4], [-12], [-14], [4], [22], [8], [10], [-6], [0], [-12], [8], [8], [22], [6], [0], [2], [20], [14], [-4], [-10], [-20], [-18], [22], [4], [-6], [28], [24], [-18], [-22], [24], [-34], [28], [-2], [30], [0], [8], [-6], [-16], [-8], [-34], [14], [-18], [-30], [-36], [-26], [32], [-2], [16], [12], [-30], [-6], [6], [32], [0], [-24], [24], [4], [24], [-8], [-18], [-2], [4], [22], [12], [-38], [4], [-30], [-44], [14], [36], [30], [-16], [26], [-16], [-34], [46], [-12], [12], [-42], [-12], [0], [-30], [-12], [-2], [-26], [42], [28], [20], [-6], [-26], [-40], [-32], [34], [24], [12], [0], [42], [-38], [-22], [-14], [-28], [2], [38], [24], [30], [24], [0], [46], [-40], [6], [-42], [28], [12], [-2], [42], [4], [36], [-52], [-32], [52], [54], [14], [10], [-32], [6], [-8], [36], [-42], [16], [-52], [-42], [34], [-54], [12], [-34], [-32], [-42], [-4], [-18], [-8], [38], [-56], [14], [-32], [36], [-10], [22], [-56], [-10], [58], [-20], [-30], [0], [-6], [56], [-32], [-14], [4], [-42], [-14], [34], [46], [-20], [-66], [32], [10], [66], [36], [66], [64], [-8], [-54], [8], [-22], [-30], [-16], [4], [-32], [-46], [32], [-62], [-32], [10], [-26], [-8], [-18], [16], [-52], [-42], [-42], [-24], [-40], [-36], [-22], [40], [-16], [-10], [4], [-24], [-14], [58], [44], [16], [72], [-48], [64], [-10], [30], [-16], [-48], [-60], [-16], [48], [-2], [-6], [-8], [10], [2], [12], [-34], [-52], [-30], [-38], [-56], [-72], [70], [10], [62], [76], [34], [30], [20], [-22], [14], [4], [14], [-24], [50], [8], [12], [46], [10], [4], [-12], [-32], [-32], [-10], [-76], [-24], [18], [50], [-76], [-34], [-42], [36], [-18], [-84], [-54], [78], [-52], [-54], [-28], [-68], [-58], [74], [-72], [-8], [-28], [38], [40], [14], [72], [-6], [48], [-58], [-38], [-44], [-68], [-22], [36], [24], [34], [-34], [-36], [10], [-18], [-40], [-18], [34], [-36], [-20], [48], [-6], [-82], [42], [68], [36], [28], [72], [-18], [-42], [10], [0], [-50], [46], [66], [-16], [-70], [52], [38], [36], [-72], [42], [-44], [14], [-90], [-80], [14], [-90], [-56], [76], [-2], [-24], [-50], [90], [92], [-20], [-4], [62], [42], [64], [34], [52], [68], [12], [34], [4], [-86], [-52], [16], [74], [38], [58], [18], [54], [24], [-2], [48], [-56], [-44], [22], [-76], [-32], [2], [74], [-92], [-28], [88], [-62], [-8], [26], [40], [2], [46], [-18], [8], [70], [-54], [76], [-86], [2], [-84], [84], [82], [-102], [96], [52], [62], [-66], [80], [-32], [62], [76], [2], [-50], [56], [-44], [-18], [18], [72], [-62], [-52], [24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2010_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2010_2_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_2010_2_a_a(:prec:=1)
    chi := MakeCharacter_2010_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_2010_2_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_2010_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2010_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![-4, 1]>,<13,R![-2, 1]>,<17,R![-6, 1]>],Snew);
    return Vf;
end function;