
// Make newform 1512.2.s.l in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1512_s();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1512_s_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_1512_2_s_l();".
// 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_1512_2_s_l();".
// 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 := [3, -12, 19, -15, 10, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [2, -1, 1, 0, 0, 0], [6, -18, 5, -8, 1, -1], [3, -5, 6, -2, 1, 0], [9, -21, 19, -16, 5, -2], [-9, 30, -22, 19, -5, 2]];
    Rf_basisdens := [1, 1, 3, 1, 3, 3];
    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_1512_s();"
function MakeCharacter_1512_s()
    N := 1512;
    order := 3;
    char_gens := [1135, 757, 785, 1081];
    v := [3, 3, 3, 2];
    // 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_1512_s_Hecke();"
function MakeCharacter_1512_s_Hecke(Kf)
    N := 1512;
    order := 3;
    char_gens := [1135, 757, 785, 1081];
    char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 1], [-1, -1, 1, 0, 0, 1], [-1, 0, 1, 0, 1, -1], [1, 1, 0, 0, 0, 0], [0, 0, -1, 0, 0, 1], [0, 0, 1, 1, 2, 0], [0, 0, 1, 1, 1, 0], [-1, -3, 0, -1, 0, 0], [-1, 0, 2, 0, 1, 3], [0, 1, 3, 3, 4, -1], [1, 0, 0, 3, 0, 0], [1, 5, 0, 1, 0, 0], [0, 5, -2, -2, 0, -5], [3, 0, -2, 0, -3, -1], [4, 0, 1, 0, -4, 4], [0, 0, 2, 2, 5, 0], [5, 0, 3, 0, -5, 2], [-2, -2, 0, -3, 0, 0], [-1, 0, 3, 0, 1, 3], [0, 0, 3, 3, 5, 0], [0, 3, 0, -4, 0, 0], [0, 2, 0, 0, -5, -2], [4, 0, 0, -2, 0, 0], [-4, 0, -2, 0, 4, -1], [0, 0, 2, 2, 9, 0], [0, 2, 1, 1, 8, -2], [8, 0, 2, 0, -8, 6], [5, -4, 0, 6, 0, 0], [-8, -4, 0, -1, 0, 0], [0, 1, 7, 7, 8, -1], [-6, 0, -2, 0, 6, -4], [9, 3, 0, 2, 0, 0], [0, -10, 3, 3, -8, 10], [8, 0, 1, 0, -8, -3], [-6, 0, 0, 0, 6, -9], [0, 4, -5, -5, -6, -4], [1, -5, 0, 0, 0, 0], [0, 1, -3, -3, 5, -1], [0, 0, 6, 0, 0, -1], [3, 7, 0, 0, 0, 0], [0, 1, -7, -7, 3, -1], [11, 0, -5, 0, -11, -4], [14, -1, 0, 4, 0, 0], [-7, 0, -6, 0, 7, 4], [-6, 1, 0, 1, 0, 0], [-1, -7, 0, -2, 0, 0], [2, 0, 7, 0, -2, -2], [0, -6, -7, -7, 4, 6], [0, 0, -4, -4, -5, 0], [-11, 2, 0, 5, 0, 0], [-11, 0, -4, 0, 11, -10], [10, 9, 0, 1, 0, 0], [0, -6, 4, 4, 9, 6], [-4, 0, -1, 0, 4, -10], [-17, 0, -1, 0, 17, -5], [0, -2, -9, -9, -1, 2], [17, 0, -3, 0, -17, -3], [8, 2, 0, -5, 0, 0], [-12, 0, 3, 0, 12, -1], [2, 8, 0, 7, 0, 0], [-3, -13, 0, -5, 0, 0], [18, 0, -4, 0, -18, -6], [0, 10, -8, -8, 5, -10], [0, 11, -2, -2, -7, -11], [0, -5, -4, -4, 8, 5], [-14, -13, 0, 1, 0, 0], [2, 0, 2, 0, -2, 14], [14, -3, 0, 8, 0, 0], [-4, 0, 0, 0, 4, -7], [0, 1, -1, -1, 7, -1], [4, 0, -2, 0, -4, -6], [0, 8, -12, -12, -6, -8], [-18, -9, 0, 4, 0, 0], [0, -14, -7, -7, -2, 14], [2, 0, 5, 0, -2, 6], [0, -5, 2, 2, 18, 5], [0, 0, 9, 9, -3, 0], [7, 0, -9, 0, -7, 10], [11, 0, 0, 4, 0, 0], [-15, 8, 0, 1, 0, 0], [12, 0, -11, 0, -12, -9], [8, 4, 0, -1, 0, 0], [0, -2, 2, 2, 13, 2], [0, -5, 9, 9, 17, 5], [33, 5, 0, -3, 0, 0], [0, -2, -7, -7, -9, 2], [-3, 0, 0, -3, 0, 0], [-12, 4, 0, -4, 0, 0], [0, -9, 5, 5, -21, 9], [13, 0, -12, 0, -13, -1], [22, 0, -5, 0, -22, -7], [-29, -5, 0, -1, 0, 0], [0, -1, 11, 11, 10, 1], [5, -5, 0, 1, 0, 0], [0, 3, -1, -1, 33, -3], [-29, 0, 7, 0, 29, 1], [0, -2, -5, -5, 17, 2], [0, 1, 3, 3, 18, -1], [2, 4, 0, 3, 0, 0], [12, 0, 1, 0, -12, -2], [30, 0, 4, 0, -30, 1], [0, 10, -3, -3, -8, -10], [14, 0, 5, 0, -14, 1], [2, 0, 4, 0, -2, 11], [12, 2, 0, 8, 0, 0], [0, -3, -4, -4, -27, 3], [-5, 0, -12, 0, 5, -10], [5, 1, 0, -8, 0, 0], [0, 1, 3, 3, 18, -1], [0, 0, 11, 0, 0, 4], [8, 8, 0, -10, 0, 0], [8, 0, 10, 0, -8, -1], [-10, -3, 0, -4, 0, 0], [-7, 0, -2, 0, 7, 15], [13, -3, 0, -6, 0, 0], [8, 0, -12, 0, -8, -13], [0, 5, -17, -17, -21, -5], [-2, 14, 0, 9, 0, 0], [-10, 0, -2, 0, 10, -5], [-16, -8, 0, -10, 0, 0], [0, -6, 6, 6, 21, 6], [21, 0, 9, 0, -21, 9], [0, -3, 2, 2, 13, 3], [19, -3, 0, -8, 0, 0], [0, -9, 11, 11, -7, 9], [0, 2, 2, 2, -34, -2], [4, 17, 0, 2, 0, 0], [0, 10, -3, -3, 15, -10], [14, 0, 3, 0, -14, 18], [-35, -4, 0, -1, 0, 0], [0, -2, 4, 4, 8, 2], [7, 18, 0, 0, 0, 0], [0, 22, -7, -7, 22, -22], [22, 0, 0, 17, 0, 0], [-27, 0, 1, 0, 27, 4], [-13, 0, 2, 0, 13, 6], [-10, 8, 0, -14, 0, 0], [-18, 0, 2, 0, 18, -9], [3, 15, 0, 2, 0, 0], [0, -7, 17, 17, 5, 7], [-10, 0, 12, 0, 10, -5], [10, -5, 0, 10, 0, 0], [22, 0, 6, 0, -22, 9], [0, -3, 0, 6, 0, 0], [25, 0, 0, -1, 0, 0], [-19, 0, 11, 0, 19, -7], [0, -14, 9, 9, -3, 14], [0, -13, -8, -8, 21, 13], [0, 16, 0, 0, 14, -16], [5, 13, 0, 12, 0, 0], [-21, 8, 0, -5, 0, 0], [0, 12, -8, -8, 24, -12], [-2, 0, 6, 0, 2, 0], [-22, -20, 0, 3, 0, 0], [0, -24, 4, 4, -13, 24], [0, -5, 21, 21, 12, 5], [-14, 1, 0, -2, 0, 0], [-15, 0, 15, 0, 15, -9], [0, -14, 3, 3, 3, 14], [46, 3, 0, -2, 0, 0], [-6, -3, 0, 10, 0, 0], [0, 0, -7, -7, -40, 0], [-32, 0, -3, 0, 32, -11], [25, 0, 7, 0, -25, 25], [6, 0, 4, 0, -6, -3], [-8, 0, -7, 0, 8, 17], [9, -6, 0, 1, 0, 0], [0, -2, -5, -5, 36, 2], [16, 0, 0, 0, -16, -16], [-16, -7, 0, -21, 0, 0], [0, 11, -14, -14, -19, -11], [-12, 0, 0, 0, 12, -23], [3, 0, 5, 0, -3, -3], [-31, 6, 0, -13, 0, 0], [4, -10, 0, -13, 0, 0], [44, 0, 6, 0, -44, 11], [10, -3, 0, -11, 0, 0], [0, -20, 12, 12, -12, 20], [0, -12, -9, -9, 14, 12], [17, 9, 0, 5, 0, 0], [-13, 1, 0, -14, 0, 0], [0, -1, -3, -3, -44, 1], [-14, 0, 15, 0, 14, -2], [14, 0, -10, 0, -14, 11], [2, 0, -21, 0, -2, 7], [36, 0, -12, 0, -36, -4], [0, -7, 11, 11, 5, 7], [-8, 0, 3, 0, 8, -8], [0, 16, -16, -16, 5, -16], [-11, 4, 0, 0, 0, 0], [0, 17, 0, 0, -3, -17], [0, 4, -4, -4, 19, -4], [20, 0, -14, 0, -20, -6], [6, 0, -21, 0, -6, -15], [4, 0, -5, 0, -4, 11], [0, 13, 3, 3, -34, -13], [-8, 3, 0, 12, 0, 0], [0, 19, 16, 16, -12, -19], [8, 0, -11, 0, -8, -5], [-36, -21, 0, 7, 0, 0], [0, 15, -11, -11, -2, -15], [-16, 0, -5, 0, 16, 18], [2, 5, 0, -15, 0, 0], [-34, 0, -6, 0, 34, 14], [0, 11, 17, 17, -3, -11], [0, 19, 3, 3, -2, -19], [6, -23, 0, 1, 0, 0], [-7, 0, -11, 0, 7, -15], [0, -11, -8, -8, 10, 11], [7, -8, 0, 11, 0, 0], [10, -6, 0, -4, 0, 0], [0, 12, 12, 12, 12, -12], [36, 0, 1, 0, -36, 8], [0, -1, -18, -18, -28, 1], [-21, 0, -6, 0, 21, -9], [20, 0, 0, 0, -20, 25], [0, -7, 1, 1, 11, 7], [-23, -21, 0, -4, 0, 0], [0, -7, -18, -18, -1, 7], [-16, -17, 0, 11, 0, 0], [0, -15, 0, 0, -38, 15], [0, -2, -7, -7, -20, 2], [10, 21, 0, -4, 0, 0], [19, -10, 0, 27, 0, 0], [0, -1, -20, -20, -25, 1], [20, 0, 11, 0, -20, 10], [0, -14, -4, -4, -3, 14], [0, -5, -14, -14, -2, 5], [-46, 0, 2, 0, 46, -6], [0, 0, -2, 0, 0, 17], [-20, -3, 0, -16, 0, 0], [-25, 0, -4, 0, 25, 15], [-20, 9, 0, -18, 0, 0], [4, 0, 6, 0, -4, -5], [0, -12, 15, 15, -23, 12], [0, 0, 10, 10, -3, 0], [-36, -7, 0, -7, 0, 0], [-10, -4, 0, -6, 0, 0], [-24, 0, 15, 0, 24, 14], [0, -19, -11, -11, -24, 19], [-5, 0, 0, 0, 5, -24], [0, -16, -13, -13, -5, 16], [-15, -21, 0, 9, 0, 0], [0, -18, -7, -7, 8, 18], [5, -5, 0, -11, 0, 0], [21, 0, -13, 0, -21, -13], [8, 0, -14, 0, -8, -4], [29, 33, 0, -7, 0, 0], [1, -13, 0, -13, 0, 0], [0, -17, 18, 18, -22, 17], [-17, 0, 16, 0, 17, -4], [26, -1, 0, -4, 0, 0], [-25, 0, -17, 0, 25, -18], [0, 26, -14, -14, 11, -26], [-22, 0, -4, 0, 22, -23], [29, 0, -5, 0, -29, 10], [19, 7, 0, 22, 0, 0], [0, -4, -14, -14, -40, 4], [-30, 0, -15, 0, 30, 8], [-36, 0, 0, -9, 0, 0], [-12, 0, 17, 0, 12, -6], [-31, 10, 0, -2, 0, 0], [52, 0, 8, 0, -52, 16], [0, -8, -2, -2, -13, 8], [-2, 30, 0, -9, 0, 0], [-9, 11, 0, -11, 0, 0], [5, -28, 0, -8, 0, 0], [36, 0, -16, 0, -36, -3], [0, 25, 8, 8, -15, -25], [-26, 0, 15, 0, 26, 11], [-24, 0, -10, 0, 24, -16], [0, -20, 3, 3, -14, 20], [4, 16, 0, 8, 0, 0], [0, -5, -4, -4, -40, 5], [0, -3, -7, -7, -8, 3], [22, 0, -7, 0, -22, 23], [0, 4, -11, -11, 23, -4], [0, 9, -18, -18, -12, -9], [19, 0, 10, 0, -19, 25], [-16, 0, -13, 0, 16, 1], [16, 16, 0, -12, 0, 0], [15, -3, 0, 13, 0, 0], [0, 6, -2, -2, -23, -6], [0, 15, 10, 10, 12, -15], [-38, 0, -7, 0, 38, 10], [5, 22, 0, 19, 0, 0], [33, 0, 13, 0, -33, -3], [-10, 11, 0, 2, 0, 0], [18, 0, 15, 0, -18, -18], [6, 0, -19, 0, -6, 4], [0, 15, 0, 0, 1, -15], [-21, 8, 0, 5, 0, 0], [8, -3, 0, -14, 0, 0], [6, 0, -15, 0, -6, -20], [0, 5, 5, 5, -28, -5], [-9, -14, 0, -8, 0, 0], [0, 13, -3, -3, -20, -13], [-43, -21, 0, 15, 0, 0], [0, -4, 23, 23, 30, 4], [0, -17, -17, -17, 1, 17], [2, 0, 21, 0, -2, 23]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1512_s_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1512_2_s_l();".
// 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_1512_2_s_l(:prec:=6)
    chi := MakeCharacter_1512_s();
    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_1512_2_s_l();".
// 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_1512_2_s_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1512_s();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, -4, 17, 2, 5, -1, 1]>,<11,R![81, -108, 135, -30, 13, 1, 1]>,<13,R![3, -3, -2, 1]>],Snew);
    return Vf;
end function;