
// Make newform 130.2.l.a in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_130_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_130_l_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_130_2_l_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_130_2_l_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
        poly := [1, 0, -1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_130_l();"
function MakeCharacter_130_l()
    N := 130;
    order := 6;
    char_gens := [27, 41];
    v := [6, 5];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_130_l_Hecke();"
function MakeCharacter_130_l_Hecke(Kf)
    N := 130;
    order := 6;
    char_gens := [27, 41];
    char_values := [[1, 0, 0, 0], [0, 0, 1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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 := [[0, 1, 0, 0], [1, 1, -1, -2], [0, 0, 0, 1], [0, -3, 0, 3], [0, -3, 0, 0], [4, 0, -1, 0], [0, 3, -3, 3], [-4, -3, 2, 3], [-6, 2, 6, -4], [6, 2, -6, -4], [-1, 0, 2, -3], [3, 6, 3, 0], [-6, 0, -6, 0], [0, 0, -2, 0], [0, 0, 0, -3], [-3, -4, 0, 2], [12, 0, -6, 0], [0, 3, -1, 3], [0, 0, 0, 0], [0, 6, 0, -6], [-5, 0, 10, -3], [1, 6, 0, -3], [3, 0, -6, 3], [-3, -12, -3, 0], [14, 3, -7, -3], [-9, 1, 9, -2], [-4, 6, 0, -3], [-9, 5, 9, -10], [-2, 0, 4, 12], [0, 4, 0, 4], [-16, -3, 16, 6], [-6, -14, 0, 7], [6, 3, -3, -3], [0, 3, 4, 3], [0, -6, 0, 6], [5, 0, -10, 15], [-13, 0, 0, 0], [-2, 9, 1, -9], [0, -3, 0, 0], [0, 0, 15, 0], [6, -2, -6, 4], [-1, -18, 0, 9], [0, -1, 21, -1], [-6, 6, -6, 0], [9, -6, 9, 0], [0, -6, -4, -6], [2, -9, -2, 18], [-2, 3, -2, 0], [12, -6, -6, 6], [-1, 0, 2, 3], [3, -2, 0, 1], [-3, 0, 6, 3], [-10, 0, 5, 0], [0, 3, -12, 3], [0, -8, 0, 16], [12, 9, -12, -18], [0, -7, 15, -7], [2, -6, 2, 0], [0, 0, 1, 0], [6, 0, -12, 0], [20, 6, -20, -12], [6, -6, -3, 6], [2, 0, -4, 24], [3, 14, 0, -7], [16, -12, 0, 6], [3, 0, -6, 18], [8, -6, -4, 6], [1, -6, 0, 3], [0, -1, 27, -1], [-11, -15, -11, 0], [-3, -15, -3, 0], [6, 0, -12, -12], [16, 6, -16, -12], [0, -6, -10, -6], [12, -9, 12, 0], [12, -12, -6, 12], [21, -2, 0, 1], [-6, 6, 3, -6], [-3, 0, -3, 0], [2, 18, -1, -18], [0, -10, 0, 20], [7, 0, -14, -15], [-3, 3, -3, 0], [0, 6, 2, 6], [19, 9, -19, -18], [-12, 12, 0, -6], [18, 0, -9, 0], [-5, -27, -5, 0], [-18, 15, 9, -15], [-4, 0, 8, -6], [-24, 16, 0, -8], [-15, 15, -15, 0], [-28, -9, 14, 9], [-12, 5, 12, -10], [-8, 0, 16, -12], [0, -5, 18, -5], [-6, 12, -6, 0], [3, -4, 0, 2], [-8, -6, 8, 12], [-11, 0, 22, -3], [14, 24, 0, -12], [3, 12, 3, 0], [0, -2, -18, -2], [3, -14, -3, 28], [-2, 42, 0, -21], [17, 0, -34, -3], [0, -30, 0, 0], [12, 0, -24, 0], [-6, 16, 0, -8], [17, -12, -17, 24], [0, -9, 22, -9], [19, -6, 19, 0], [0, -18, 0, 18], [6, 0, -12, 21], [-24, 0, 12, 0], [0, -4, -39, -4], [-2, 9, 1, -9], [6, 3, -6, -6], [9, 6, -9, -12], [0, -10, -12, -10], [19, -3, 19, 0], [-14, 6, 14, -12], [-12, 16, 0, -8], [36, 0, -18, 0], [-16, 9, -16, 0], [3, -38, 0, 19], [30, 15, -15, -15], [0, 8, -12, 8], [-2, -42, 0, 21], [-19, 0, 38, 18], [24, 9, 24, 0], [6, 24, 6, 0], [10, -6, -10, 12], [17, -12, -17, 24], [-6, -24, 3, 24], [-22, 0, -22, 0], [-18, -18, 9, 18], [-46, 3, 23, -3], [0, -4, 6, -4], [-24, 0, 24, 0], [-6, 0, 12, -33], [3, -15, 3, 0], [0, -3, 22, -3], [9, 0, -18, 27], [20, 0, -20, 0], [-6, -27, 3, 27], [-8, 0, 16, 0], [-15, 2, 0, -1], [16, 42, 0, -21], [-18, 0, 36, -12], [-4, 24, 2, -24], [-9, 10, 9, -20], [-11, -18, 0, 9], [-6, -27, 6, 54], [11, 9, -11, -18], [-18, 4, 0, -2], [0, 12, -10, 12], [24, 24, -12, -24], [-20, -12, 0, 6], [12, 0, -24, -24], [21, 21, 21, 0], [0, -9, -9, -9], [24, 0, -48, 3], [0, 27, 0, 27], [6, -6, 6, 0], [-24, 0, 48, -3], [-20, -18, 20, 36], [0, 24, 1, 24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_130_l_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_130_2_l_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_130_2_l_a(:prec:=4)
    chi := MakeCharacter_130_l();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_130_2_l_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_130_2_l_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_130_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, 4, 6, -2, 1]>],Snew);
    return Vf;
end function;