
// Make newform 1470.2.d.a in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1470_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1470_d_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_1470_2_d_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_1470_2_d_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 := [9, -3, -2, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, -2, 2, 1], [3, 2, 0, -1]];
    Rf_basisdens := [1, 1, 6, 2];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_1470_d();"
function MakeCharacter_1470_d()
    N := 1470;
    order := 2;
    char_gens := [491, 1177, 1081];
    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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1470_d_Hecke();"
function MakeCharacter_1470_d_Hecke(Kf)
    N := 1470;
    order := 2;
    char_gens := [491, 1177, 1081];
    char_values := [[-1, 0, 0, 0], [-1, 0, 0, 0], [-1, 0, 0, 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 := [[-1, 0, 0, 0], [0, -1, 0, 0], [1, 1, -1, 0], [0, 0, 0, 0], [1, 1, -3, -1], [2, 0, 0, 0], [-2, 4, 0, -4], [-2, 0, 4, 0], [-2, 1, 1, 1], [1, -2, 0, 2], [-3, 3, 3, -3], [-2, 6, -2, -6], [-4, -1, -1, -1], [2, 3, -7, -3], [4, -2, -6, 2], [1, 1, 1, 1], [-3, -3, -3, -3], [-6, 3, 9, -3], [0, 3, -3, -3], [4, -2, -6, 2], [2, 0, 0, 0], [-1, 3, 3, 3], [-5, 4, 6, -4], [2, -1, -1, -1], [5, 3, 3, 3], [14, -1, -1, -1], [8, -3, -3, -3], [-13, 2, 2, 2], [2, -3, -3, -3], [8, 2, 2, 2], [-3, 3, 3, -3], [7, 1, 1, 1], [2, 2, 2, 2], [-2, 6, -2, -6], [-2, 7, -3, -7], [-1, 3, 3, 3], [8, 0, 0, 0], [-2, 0, 4, 0], [-8, 1, 15, -1], [-8, 4, 12, -4], [10, -8, -12, 8], [-6, -3, 15, 3], [-8, 4, 12, -4], [9, -3, -15, 3], [6, -6, -6, -6], [2, 6, -10, -6], [-4, 6, 6, 6], [11, -3, -3, -3], [7, -11, -3, 11], [8, 0, -16, 0], [14, -4, -4, -4], [10, -8, -12, 8], [7, -9, -5, 9], [-9, 3, 3, 3], [-2, -2, 6, 2], [-4, -1, -1, -1], [1, -8, -8, -8], [1, -9, 7, 9], [-8, -6, 22, 6], [-2, 10, -6, -10], [-16, 0, 0, 0], [13, 1, -27, -1], [14, 3, 3, 3], [0, 6, 6, 6], [-7, 3, 3, 3], [-9, 3, 3, 3], [-4, -6, -6, -6], [1, 3, -5, -3], [-12, 3, 3, 3], [8, -15, -1, 15], [-8, -2, 18, 2], [-14, 4, 24, -4], [-7, 6, 6, 6], [8, -12, -4, 12], [8, -6, -6, -6], [4, -11, 3, 11], [-8, -8, 24, 8], [8, 0, 0, 0], [10, 1, -21, -1], [3, 12, -18, -12], [20, -4, -4, -4], [8, -3, -3, -3], [-8, 4, 12, -4], [-34, 0, 0, 0], [-3, -3, 9, 3], [-9, 0, 0, 0], [4, -17, 9, 17], [3, 3, -9, -3], [10, 4, 4, 4], [-8, -9, 25, 9], [16, -11, -21, 11], [-4, 2, 2, 2], [-7, 3, 11, -3], [-17, 1, 33, -1], [14, -6, -6, -6], [4, 1, -9, -1], [-5, -2, -2, -2], [-8, -2, -2, -2], [-10, 6, 6, 6], [8, 9, 9, 9], [-4, 3, 5, -3], [-11, -5, -5, -5], [1, 4, -6, -4], [-2, 10, -6, -10], [20, -6, -6, -6], [-25, -3, -3, -3], [-5, -11, 21, 11], [-14, 4, 24, -4], [-2, -2, 6, 2], [-7, 9, 5, -9], [-1, -12, -12, -12], [-10, 12, 8, -12], [20, 8, 8, 8], [4, -12, 4, 12], [23, -3, -3, -3], [-2, 7, -3, -7], [2, -6, -6, -6], [22, -5, -39, 5], [13, 1, 1, 1], [-2, 16, -12, -16], [-6, 3, 9, -3], [23, -9, -37, 9], [7, 13, -27, -13], [-5, -8, -8, -8], [0, -12, 12, 12], [19, -8, -30, 8], [2, 3, 3, 3], [38, 2, 2, 2], [-19, 0, 0, 0], [8, -12, -12, -12], [-22, 6, 6, 6], [2, -7, -7, -7], [35, -3, -3, -3], [12, -6, -18, 6], [34, -8, -8, -8], [19, 3, -41, -3], [4, -2, -6, 2], [20, -3, -3, -3], [1, 1, -3, -1], [22, -5, -39, 5], [6, 0, -12, 0], [-5, -5, 15, 5], [-12, -9, 33, 9], [-33, 0, 0, 0], [2, 6, -10, -6], [22, 10, 10, 10], [2, -12, -12, -12], [16, 4, -36, -4], [-16, 24, 8, -24], [-10, -1, -1, -1], [8, -18, 2, 18], [30, -9, -9, -9], [12, -30, 6, 30], [4, 19, -27, -19], [4, 15, -23, -15], [-20, 4, 36, -4], [-4, 6, 6, 6], [-24, -3, -3, -3], [29, -9, -9, -9], [3, -9, -9, -9], [-36, 3, 3, 3], [48, 0, 0, 0], [-5, 24, -14, -24], [-31, -1, -1, -1], [12, -12, -12, -12], [-14, 1, 27, -1], [-7, 9, 9, 9], [2, -12, -12, -12], [-19, 0, 0, 0], [7, -11, -3, 11], [-11, 13, 9, -13], [-2, 3, 1, -3], [-2, 4, 0, -4], [-17, 21, 13, -21], [9, -9, -9, 9], [-4, -1, -1, -1], [8, -6, -6, -6], [13, -11, -15, 11], [20, -3, -3, -3], [6, -27, 15, 27], [9, -18, 0, 18], [-17, 7, 7, 7], [-2, 18, -14, -18], [10, 10, -30, -10], [18, 6, 6, 6], [29, -4, -4, -4], [8, 0, -16, 0], [-34, 6, 6, 6], [29, 0, 0, 0], [10, -14, -14, -14], [11, 9, 9, 9], [-5, 13, 13, 13], [-34, -6, -6, -6], [-30, 9, 9, 9], [1, 19, 19, 19], [-2, -2, 6, 2], [26, 3, 3, 3], [-12, -12, 36, 12], [-8, 10, 6, -10], [-20, 22, 18, -22], [-23, 22, 24, -22], [-10, 12, 8, -12], [-58, 0, 0, 0], [-2, -3, 7, 3], [12, -6, -6, -6], [7, 13, -27, -13], [10, -24, 4, 24], [25, -5, -5, -5], [-8, 25, -9, -25], [30, -18, -42, 18], [-20, 6, 34, -6], [-25, -10, -10, -10], [23, -12, -34, 12], [19, -32, -6, 32], [-30, 6, 6, 6], [24, -24, -24, 24], [3, -6, 0, 6], [-10, -1, -1, -1], [-10, -10, -10, -10], [2, -10, -10, -10], [8, -15, -15, -15], [10, 6, -26, -6], [22, -35, -9, 35], [7, 9, -23, -9], [-11, -8, 30, 8], [8, -9, -9, -9], [-2, -2, 6, 2], [16, 16, -48, -16], [35, 12, 12, 12], [-26, -8, 60, 8], [36, -6, -66, 6], [32, -18, -46, 18], [-31, -9, -9, -9], [4, 19, -27, -19], [14, 3, 3, 3], [-20, 7, 33, -7], [11, 6, -28, -6], [35, -1, -1, -1], [-2, 28, -24, -28], [18, -6, -6, -6], [56, -4, -4, -4], [16, -12, -20, 12], [-31, -9, -9, -9], [-46, -3, -3, -3], [-14, 22, 6, -22], [12, -12, -12, -12], [5, 18, 18, 18], [36, -6, -6, -6], [8, 12, 12, 12], [0, 12, 12, 12], [22, -30, -14, 30], [-4, -19, -19, -19], [18, 3, 3, 3], [14, 9, -37, -9], [-23, -5, 51, 5], [-5, 13, -3, -13], [-4, 3, 3, 3], [-28, 21, 21, 21], [7, 19, -33, -19], [47, 3, 3, 3], [4, -9, 1, 9], [-29, 16, 16, 16], [18, -6, -30, 6], [-4, 6, 6, 6], [16, 4, -36, -4], [-42, 6, 78, -6], [-11, 1, 21, -1], [-14, -8, -8, -8], [-26, 21, 31, -21], [-57, -3, -3, -3], [2, -21, 17, 21], [-34, 24, 44, -24], [65, -3, -3, -3], [5, -9, -9, -9], [5, 3, 3, 3], [8, 0, 0, 0], [43, 1, 1, 1], [-22, 15, 15, 15], [14, 9, 9, 9], [60, -6, -6, -6], [10, -11, -9, 11], [47, -3, -3, -3], [10, -20, 0, 20], [14, -27, -1, 27], [14, -3, -3, -3], [-14, 22, 6, -22], [-37, 9, 65, -9], [-13, -7, -7, -7], [-23, 15, 31, -15], [0, -18, -18, -18], [-2, -11, 15, 11], [-11, -14, 36, 14], [-44, 10, 10, 10], [-28, 20, 20, 20], [-10, 0, 20, 0], [9, 18, 18, 18], [-6, 36, -24, -36], [16, -38, 6, 38], [44, -4, -4, -4], [20, 0, 0, 0], [-10, 0, 0, 0], [-31, 5, 5, 5], [-49, -3, -3, -3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1470_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1470_2_d_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_1470_2_d_a(:prec:=4)
    chi := MakeCharacter_1470_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1470_2_d_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_1470_2_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1470_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![16, 0, 19, 0, 1]>,<13,R![-2, 1]>,<23,R![-6, 3, 1]>],Snew);
    return Vf;
end function;