
// Make newform 3528.2.s.bl in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3528_s();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3528_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_3528_2_s_bl();".
// 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_3528_2_s_bl();".
// 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 := [4, 0, 2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 2, 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_3528_s();"
function MakeCharacter_3528_s()
    N := 3528;
    order := 3;
    char_gens := [2647, 1765, 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_3528_s_Hecke();"
function MakeCharacter_3528_s_Hecke(Kf)
    N := 3528;
    order := 3;
    char_gens := [2647, 1765, 785, 1081];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [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, 0, 0, 0], [0, 0, 0, 0], [0, -1, -2, -1], [0, 0, 0, 0], [-2, 2, -2, 0], [0, 0, 0, 3], [6, 1, 6, 0], [0, 2, 4, 2], [0, 2, 2, 2], [0, 0, 0, 2], [0, 2, 0, 0], [0, -4, 4, -4], [-6, 0, 0, 3], [0, 0, 0, 8], [0, 6, -4, 6], [-2, 0, -2, 0], [0, 6, 0, 0], [0, 5, 4, 5], [0, -8, 0, 0], [2, 0, 0, -6], [-12, 3, -12, 0], [0, 4, 8, 4], [4, 0, 0, 0], [0, 3, -10, 3], [4, 0, 0, 3], [2, -9, 2, 0], [0, 10, 0, 10], [0, -2, -10, -2], [-8, 8, -8, 0], [10, 0, 0, 0], [20, 0, 0, 0], [0, 0, 4, 0], [-12, 2, -12, 0], [-4, 0, 0, 12], [0, -8, 10, -8], [4, 4, 4, 0], [4, -3, 4, 0], [0, 4, 0, 4], [12, 0, 0, -2], [0, -1, 18, -1], [2, -6, 2, 0], [0, 0, 0, 5], [0, -10, 2, -10], [-2, 0, -2, 0], [14, 0, 0, 8], [0, 4, 0, 0], [-4, 0, 0, -4], [8, 0, 0, -4], [-8, 2, -8, 0], [0, -1, -24, -1], [0, 2, 12, 2], [10, 0, 0, -2], [-4, -7, -4, 0], [-8, 0, 0, -10], [0, 1, 2, 1], [-6, -10, -6, 0], [-10, 7, -10, 0], [0, 10, -8, 10], [6, 16, 6, 0], [-4, 0, 0, -2], [12, 10, 12, 0], [6, 0, 0, 3], [-20, 0, 0, -6], [4, 14, 4, 0], [0, 9, 16, 9], [0, 0, 22, 0], [0, 8, -4, 8], [16, 0, 0, 4], [6, -6, 6, 0], [12, 0, 0, 1], [2, -21, 2, 0], [0, -2, -26, -2], [-16, 8, -16, 0], [0, 8, -6, 8], [-12, 0, 0, 8], [0, 12, -8, 12], [-16, -14, -16, 0], [0, -3, -12, -3], [0, 2, -4, 2], [4, 3, 4, 0], [-16, 0, 0, -6], [-14, 0, 0, 8], [6, -10, 6, 0], [8, 0, 0, 5], [0, -8, 24, -8], [0, -2, 14, -2], [-6, 0, 0, -16], [0, -16, -2, -16], [2, 0, 0, 13], [-4, 0, 0, -4], [0, 2, 16, 2], [20, 2, 20, 0], [12, -12, 12, 0], [-22, 0, 0, -2], [0, -12, -8, -12], [8, 0, 0, -8], [0, 13, 18, 13], [10, 3, 10, 0], [0, -12, 12, -12], [0, 0, -18, 0], [-20, 0, 0, -12], [10, 16, 10, 0], [16, -10, 16, 0], [0, -10, 20, -10], [36, 4, 36, 0], [8, -15, 8, 0], [16, 0, 0, -14], [0, -1, -46, -1], [-2, 18, -2, 0], [-8, 0, 0, 3], [0, 12, 8, 12], [16, -4, 16, 0], [-4, 0, 0, -6], [-28, -4, -28, 0], [-20, 0, 0, -12], [4, 22, 4, 0], [12, 0, 0, -14], [-20, 14, -20, 0], [0, -10, -32, -10], [-30, 0, 0, -6], [-12, 5, -12, 0], [-12, 0, 0, 8], [0, -21, 2, -21], [18, -14, 18, 0], [0, -8, 20, -8], [-8, 0, 0, -10], [0, 16, -16, 16], [0, 16, -16, 16], [-24, 0, 0, -10], [0, -17, 16, -17], [-16, -12, -16, 0], [2, 0, 0, -2], [0, 20, -8, 20], [8, 0, 0, -24], [0, 7, 22, 7], [8, 0, 0, -19], [-2, 25, -2, 0], [-4, -8, -4, 0], [38, 0, 0, -1], [18, -16, 18, 0], [20, 0, 0, -4], [0, -8, 22, -8], [8, 12, 8, 0], [-26, 0, 0, 10], [-16, -13, -16, 0], [-4, 0, 0, 14], [12, 0, 0, 5], [10, -15, 10, 0], [0, -18, 28, -18], [0, -34, -6, -34], [0, -20, 0, -20], [-6, 0, 0, 17], [-16, 0, 0, -4], [0, 14, 20, 14], [-48, -4, -48, 0], [2, 0, 0, 6], [0, 8, 20, 8], [0, 7, 6, 7], [16, 0, 0, -25], [-2, -9, -2, 0], [0, 26, 22, 26], [-2, 0, 0, 0], [-20, 0, 0, 8], [0, 0, 20, 0], [12, -14, 12, 0], [24, -16, 24, 0], [-4, 16, -4, 0], [36, 13, 36, 0], [-10, 0, 0, -16], [0, -15, -10, -15], [-30, -6, -30, 0], [36, 0, 0, -5], [0, 6, 50, 6], [20, 4, 20, 0], [32, -16, 32, 0], [-26, 0, 0, -21], [-4, 0, 0, 8], [8, -2, 8, 0], [-48, 0, 0, 2], [0, 31, 12, 31], [0, 8, 40, 8], [52, 0, 0, 0], [12, 0, 0, 0], [0, 5, 46, 5], [14, -26, 14, 0], [6, -15, 6, 0], [-10, -8, -10, 0], [20, -4, 20, 0], [0, 44, 4, 44], [-16, 24, -16, 0], [0, 19, -28, 19], [-14, 0, 0, -18], [0, -12, -12, -12], [0, 13, -26, 13], [-14, 2, -14, 0], [-18, 29, -18, 0], [12, 24, 12, 0], [0, -16, -4, -16], [6, 0, 0, 5], [0, 30, 12, 30], [22, -32, 22, 0], [8, 0, 0, 28], [0, -7, 24, -7], [-12, 9, -12, 0], [-36, 0, 0, -4], [22, 25, 22, 0], [0, 0, 56, 0], [0, -18, -14, -18], [42, 0, 0, 16], [-4, -42, -4, 0], [0, -8, 0, -8], [10, 0, 0, -5], [44, 0, 0, -12], [0, -28, 28, -28], [-20, -14, -20, 0], [0, 27, 16, 27], [4, -16, 4, 0], [-30, -13, -30, 0], [0, 30, 18, 30], [16, 0, 0, -14], [0, 16, 40, 16], [32, 0, 0, 6], [0, -24, -2, -24], [0, 12, 0, 12], [36, 0, 0, 4], [20, 0, 0, -4], [0, 15, -18, 15], [-14, 2, -14, 0], [0, -10, -48, -10], [0, -22, 14, -22], [-16, -24, -16, 0], [20, -2, 20, 0], [-28, 0, 0, 32], [46, 16, 46, 0], [12, 0, 0, -18], [16, -36, 16, 0], [0, 3, -36, 3], [0, 16, 2, 16], [-6, 0, 0, -42], [8, 0, 0, 44], [-34, 18, -34, 0], [0, 8, 4, 8], [16, -8, 16, 0], [0, -16, -18, -16], [-22, 0, 0, 15], [0, -10, -28, -10], [-56, 0, 0, -14], [48, -10, 48, 0], [12, -12, 12, 0], [14, 0, 0, -10], [-6, 0, 0, 0], [0, 25, -10, 25], [-10, 34, -10, 0], [36, 0, 0, -3], [26, -21, 26, 0], [0, 22, 6, 22], [28, -8, 28, 0], [20, 34, 20, 0], [14, 0, 0, -37], [0, 11, 32, 11], [-48, 0, -48, 0], [-6, 0, 0, 18], [-8, 35, -8, 0], [-20, 0, 0, 13], [-26, -3, -26, 0], [0, 16, 4, 16], [26, 0, 0, -16], [2, 0, 0, -7], [-28, 0, 0, 16], [-24, 38, -24, 0], [0, 35, 20, 35], [-36, -8, -36, 0], [12, -3, 12, 0], [0, 36, 16, 36], [-36, 0, 0, -19], [0, 14, -8, 14], [0, 2, 6, 2], [38, -24, 38, 0], [0, 0, 10, 0], [0, -2, 0, -2], [-48, 8, -48, 0], [12, -28, 12, 0], [-24, 0, 0, 36], [4, 0, 0, 51], [0, 4, 44, 4], [0, -6, 2, -6], [-10, -32, -10, 0], [6, 0, 0, -32], [-72, 6, -72, 0], [6, 0, 0, 33], [38, 16, 38, 0], [20, 44, 20, 0], [0, -10, -44, -10], [-64, 0, 0, -2], [-2, 0, 0, 24], [-22, -1, -22, 0], [0, -16, 8, -16], [22, 0, 0, -3], [0, -18, -32, -18], [-28, 0, 0, -24], [0, -3, -8, -3], [0, 26, -48, 26], [36, 20, 36, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3528_s_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3528_2_s_bl();".
// 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_3528_2_s_bl(:prec:=4)
    chi := MakeCharacter_3528_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_3528_2_s_bl();".
// 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_3528_2_s_bl( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3528_s();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![4, -8, 14, -4, 1]>,<11,R![16, -16, 20, 4, 1]>,<13,R![-18, 0, 1]>,<23,R![16, -16, 20, 4, 1]>],Snew);
    return Vf;
end function;