
// Make newform 6013.2.a.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6013_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_6013_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_6013_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_6013_a();"
function MakeCharacter_6013_a()
    N := 6013;
    order := 1;
    char_gens := [5155, 3438];
    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_6013_a_Hecke(Kf)
    return MakeCharacter_6013_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 := [[-2], [-2], [-1], [1], [0], [-2], [-3], [-4], [-6], [0], [-9], [2], [-7], [-11], [4], [-13], [-9], [11], [-2], [0], [-6], [13], [6], [6], [-6], [-4], [9], [-10], [-8], [-12], [13], [-9], [6], [-18], [5], [-15], [-16], [-20], [12], [-6], [-7], [21], [-11], [-22], [18], [-8], [-11], [-9], [21], [-5], [0], [-6], [-14], [6], [3], [26], [-2], [-1], [5], [-6], [-16], [4], [-17], [10], [-16], [1], [-32], [-22], [23], [0], [21], [-30], [1], [18], [-6], [22], [6], [-1], [-1], [1], [-14], [-25], [15], [32], [8], [-12], [-26], [34], [36], [-7], [-3], [-9], [-29], [-12], [31], [31], [18], [-27], [-4], [20], [26], [-3], [4], [-8], [-23], [-42], [-2], [-34], [-18], [28], [-18], [40], [7], [35], [-32], [-10], [34], [38], [14], [36], [-40], [-8], [-31], [27], [-20], [27], [-16], [22], [22], [-37], [8], [39], [-25], [-16], [-21], [22], [-1], [-8], [-5], [-27], [37], [0], [20], [-33], [40], [-44], [-14], [-31], [-1], [0], [6], [-19], [-7], [3], [26], [-13], [-52], [-6], [11], [4], [-3], [-36], [44], [19], [16], [29], [17], [-42], [43], [21], [-48], [40], [24], [34], [-36], [6], [-44], [-62], [-14], [-21], [40], [-43], [-30], [-6], [-27], [16], [-37], [26], [-49], [-24], [-10], [34], [24], [9], [54], [2], [-54], [2], [6], [51], [14], [46], [21], [-11], [44], [52], [-48], [52], [-66], [38], [-10], [-42], [41], [29], [40], [30], [-15], [-55], [-51], [-40], [26], [-56], [-73], [8], [14], [-58], [-54], [2], [-2], [12], [-46], [4], [16], [39], [4], [-6], [10], [-6], [8], [-9], [-36], [28], [4], [-36], [-66], [71], [-36], [49], [-5], [15], [-6], [-4], [12], [54], [-2], [-25], [-25], [-38], [-4], [-12], [56], [-8], [10], [-42], [33], [-51], [-60], [-72], [51], [44], [-32], [76], [-49], [-16], [51], [-7], [29], [62], [52], [-8], [16], [-68], [42], [52], [-60], [13], [-61], [10], [-14], [58], [-54], [-12], [-62], [82], [67], [17], [64], [-30], [-20], [-60], [10], [-61], [69], [-57], [44], [-80], [12], [40], [64], [-4], [69], [58], [12], [24], [7], [-1], [-4], [-30], [-45], [-4], [28], [-52], [-33], [-11], [75], [30], [30], [58], [-14], [-8], [6], [-18], [-55], [-26], [36], [-20], [78], [37], [-26], [-28], [-74], [91], [-33], [-68], [18], [-21], [-36], [42], [76], [-31], [7], [37], [-12], [-26], [-34], [-48], [-18], [-12], [-65], [-76], [88], [-87], [-31], [36], [-40], [-14], [96], [4], [50], [-18], [62], [-69], [30], [19], [4], [86], [6], [-53], [-60], [-62], [60], [26], [12], [-33], [26], [-62], [78], [7], [6], [60], [-81], [-42], [-76], [46], [20], [10], [12], [58], [67], [-33], [-21], [34], [28], [-72], [66], [17], [-36], [-18], [-91], [-14], [22], [33], [-76], [32], [-24], [-14], [-90], [84], [10], [-80], [72], [29], [-32], [-16], [-6], [94], [-68], [0], [25], [26]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6013_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6013_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_6013_2_a_a(:prec:=1)
    chi := MakeCharacter_6013_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_6013_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_6013_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6013_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![2, 1]>],Snew);
    return Vf;
end function;