
// Make newform 42.3.h.a in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_42_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_42_h_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_42_3_h_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_42_3_h_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 := [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_42_h();"
function MakeCharacter_42_h()
    N := 42;
    order := 6;
    char_gens := [29, 31];
    v := [3, 4];
    // 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_42_h_Hecke();"
function MakeCharacter_42_h_Hecke(Kf)
    N := 42;
    order := 6;
    char_gens := [29, 31];
    char_values := [[-1, 0, 0, 0], [-1, 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 := 3;
    raw_aps := [[0, 1, 0, 0], [0, 2, 1, -2], [0, -6, 0, 0], [-7, 0, 7, 0], [0, 0, 0, 0], [-1, 0, 0, 0], [0, -6, 0, 6], [31, 0, -31, 0], [0, 6, 0, 0], [0, 0, 0, 12], [0, 0, 7, 0], [1, 0, -1, 0], [0, 0, 0, 24], [-31, 0, 0, 0], [0, -30, 0, 0], [0, 18, 0, -18], [0, -6, 0, 6], [-50, 0, 50, 0], [0, 0, -65, 0], [0, 0, 0, -42], [0, 0, 97, 0], [103, 0, -103, 0], [0, 0, 0, 30], [0, -84, 0, 0], [-166, 0, 0, 0], [0, 96, 0, -96], [-65, 0, 65, 0], [0, 114, 0, 0], [0, 0, -167, 0], [0, 0, 0, 114], [113, 0, 0, 0], [0, 60, 0, 0], [0, 132, 0, -132], [113, 0, 0, 0], [0, -48, 0, 0], [0, 0, 58, 0], [0, 0, 118, 0], [106, 0, -106, 0], [0, 0, 0, 102], [0, -54, 0, 0], [0, 114, 0, -114], [215, 0, 0, 0], [0, -126, 0, 0], [0, 0, 97, 0], [0, 0, 0, 258], [0, 0, 106, 0], [62, 0, 0, 0], [-202, 0, 0, 0], [0, -12, 0, 12], [1, 0, -1, 0], [0, -234, 0, 0], [0, 0, 0, -324], [0, 0, 22, 0], [0, 0, 0, -126], [0, 198, 0, 0], [0, -72, 0, 72], [0, -300, 0, 300], [250, 0, -250, 0], [0, 0, -383, 0], [0, 0, 0, -126], [0, 0, -305, 0], [0, 0, 0, -96], [-199, 0, 0, 0], [0, 312, 0, -312], [-119, 0, 119, 0], [0, 144, 0, 0], [-521, 0, 521, 0], [311, 0, 0, 0], [0, -78, 0, 78], [50, 0, 0, 0], [0, -114, 0, 114], [0, 288, 0, 0], [0, 0, 535, 0], [385, 0, -385, 0], [-55, 0, 0, 0], [0, 294, 0, 0], [0, -270, 0, 270], [481, 0, -481, 0], [0, -54, 0, 0], [0, 0, 49, 0], [0, 0, 0, -228], [-313, 0, 0, 0], [0, -246, 0, 246], [-97, 0, 0, 0], [-374, 0, 374, 0], [0, -426, 0, 0], [0, 0, 0, 30], [-455, 0, 455, 0], [0, 0, 0, -156], [-7, 0, 0, 0], [0, -438, 0, 0], [0, 474, 0, -474], [0, 0, 463, 0], [0, 0, 0, 252], [535, 0, -535, 0], [0, 0, 0, -444], [0, -450, 0, 0], [0, -456, 0, 456], [-689, 0, 689, 0], [-575, 0, 575, 0], [302, 0, 0, 0], [0, 96, 0, -96], [0, 426, 0, -426], [0, -468, 0, 0], [0, 0, -113, 0], [0, 0, 121, 0], [0, 0, 0, -48], [0, 348, 0, 0], [0, 72, 0, -72], [791, 0, 0, 0], [-113, 0, 113, 0], [0, 0, 574, 0], [0, 0, 0, 468], [0, 0, -953, 0], [758, 0, 0, 0], [0, -762, 0, 762], [-1111, 0, 0, 0], [0, -240, 0, 240], [0, -456, 0, 0], [0, 0, 0, 618], [0, 0, 865, 0], [-505, 0, 0, 0], [0, -228, 0, 0], [0, 558, 0, -558], [-185, 0, 185, 0], [0, 0, 0, 66], [-194, 0, 194, 0], [0, -276, 0, 0], [425, 0, 0, 0], [-407, 0, 407, 0], [0, 0, -761, 0], [0, 0, 0, 408], [-689, 0, 689, 0], [-142, 0, 0, 0], [0, 786, 0, 0], [1127, 0, 0, 0], [0, 324, 0, -324], [0, 0, 586, 0], [0, 0, 0, -342], [0, 570, 0, -570], [398, 0, 0, 0], [0, 1146, 0, 0], [0, 0, 1162, 0], [0, 0, 0, -102], [0, 0, -1247, 0], [0, 0, 0, 708], [-337, 0, 0, 0], [0, 792, 0, -792], [1090, 0, -1090, 0], [0, 816, 0, 0], [46, 0, -46, 0], [0, 0, 0, -654], [329, 0, 0, 0], [0, -996, 0, 0], [0, 0, 175, 0], [0, 0, 0, -1026], [-1097, 0, 1097, 0], [0, 402, 0, 0], [-1, 0, 0, 0], [0, -270, 0, 270], [0, -1188, 0, 0], [0, 0, 0, 138], [-223, 0, 0, 0], [0, 402, 0, 0], [0, 1140, 0, -1140], [0, -1218, 0, 1218], [0, 0, 895, 0], [0, 0, -407, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_42_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_42_3_h_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_42_3_h_a(:prec:=4)
    chi := MakeCharacter_42_h();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_42_3_h_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_42_3_h_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_42_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![5184, 0, -72, 0, 1]>],Snew);
    return Vf;
end function;