
// Make newform 74.2.e.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_74_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_74_e_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_74_2_e_b();".
// 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_74_2_e_b();".
// 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_74_e();"
function MakeCharacter_74_e()
    N := 74;
    order := 6;
    char_gens := [39];
    v := [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_74_e_Hecke();"
function MakeCharacter_74_e_Hecke(Kf)
    N := 74;
    order := 6;
    char_gens := [39];
    char_values := [[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], [0, -1, -1, -1], [2, 0, -1, 0], [0, 0, 2, 0], [-3, -2, 0, 1], [-4, 0, 2, 0], [1, 6, 1, 0], [-2, 3, 1, -3], [1, 0, -2, -3], [5, 0, -10, 0], [1, 0, -2, 3], [0, -6, -1, 6], [0, -4, -3, -4], [-4, 0, 8, -6], [3, 2, 0, -1], [-6, 2, 6, -4], [-2, 6, -2, 0], [2, 0, -1, 0], [0, 3, 5, 3], [0, -2, 0, -2], [4, 0, 0, 0], [2, -15, -1, 15], [-3, 5, 3, -10], [3, -12, 3, 0], [1, 0, -2, -6], [9, 4, 0, -2], [2, 0, -4, 12], [0, 6, 0, 6], [5, -6, 5, 0], [-2, 0, -2, 0], [13, -3, -13, 6], [-3, 15, -3, 0], [9, -12, 0, 6], [5, 9, -5, -18], [-9, -8, 0, 4], [0, -3, 1, -3], [-1, 0, 1, 0], [4, 6, 4, 0], [-16, 0, 8, 0], [-9, 2, 9, -4], [-2, 0, 4, 6], [0, 6, -1, 6], [3, 0, -6, 3], [3, 0, -6, -6], [-3, -2, 3, 4], [-6, 0, 12, 0], [8, -12, 0, 6], [11, 6, 0, -3], [-2, 3, 1, -3], [0, -6, -17, -6], [15, 0, 0, 0], [-10, 0, -10, 0], [4, 12, -2, -12], [-10, 0, 20, 0], [-3, 12, -3, 0], [0, -2, -18, -2], [6, -12, 0, 6], [-13, 9, 13, -18], [6, 6, -3, -6], [-9, -12, -9, 0], [-16, 6, 8, -6], [0, 6, -9, 6], [-22, 0, 0, 0], [0, -18, 0, 0], [7, 12, 7, 0], [3, -6, -3, 12], [0, 0, 0, 0], [0, 18, 1, 18], [7, 0, -14, 3], [-1, -6, 1, 12], [-1, -18, -1, 0], [18, 20, 0, -10], [0, -6, -2, -6], [0, 0, -7, 0], [14, 12, -14, -24], [-4, 0, 2, 0], [-30, 0, 15, 0], [-23, 0, 0, 0], [-6, 0, 12, 0], [13, -12, 13, 0], [18, -8, -18, 16], [-7, 0, 14, 0], [-10, -9, 5, 9], [-5, 24, 0, -12], [-28, 6, 14, -6], [-18, 4, 0, -2], [-4, 24, 2, -24], [-30, -12, 15, 12], [-2, -36, -2, 0], [0, -30, 0, 30], [2, 0, -4, 6], [2, 6, 2, 0], [11, 0, -22, 3], [21, 6, 0, -3], [-6, -15, 3, 15], [-3, -15, -3, 0], [3, 2, -3, -4], [0, 4, -12, 4], [34, -9, -17, 9], [11, 0, -22, 0], [5, 0, -10, -33], [17, -6, 17, 0], [-9, 0, 18, 3], [7, 0, -14, 18], [-1, -21, 1, 42], [-4, 6, -4, 0], [14, 21, -7, -21], [9, 40, 0, -20], [0, -5, -39, -5], [23, -6, -23, 12], [-19, 9, -19, 0], [13, -12, -13, 24], [-30, 10, 30, -20], [20, 12, 0, -6], [-15, -9, -15, 0], [-15, 8, 15, -16], [1, 0, -2, 15], [-8, 30, 4, -30], [-30, -18, 15, 18], [0, 2, 6, 2], [2, 0, -1, 0], [0, 0, -16, 0], [27, 8, 0, -4], [2, -24, 2, 0], [-1, -15, 1, 30], [4, 12, 4, 0], [-16, 0, 32, -18], [-39, 1, 39, -2], [32, 12, -16, -12], [0, 24, -8, 24], [17, 18, 0, -9], [0, -13, 21, -13], [40, -12, 0, 6], [3, 12, 3, 0], [-15, 10, 15, -20], [-6, 0, 12, -24], [0, -4, -27, -4], [-10, -36, 0, 18], [12, -12, 12, 0], [32, -12, -16, 12], [0, -6, -10, -6], [0, 2, -6, 2], [-29, 3, 29, -6], [-16, 12, 8, -12], [6, 0, 6, 0], [27, 15, -27, -30], [1, -12, 1, 0], [5, 0, -10, 6], [0, 0, 0, -30], [6, -6, -6, 12], [7, 36, 0, -18], [0, 6, -39, 6], [-22, 3, 11, -3], [-27, -22, 0, 11], [-18, 27, 9, -27], [5, 0, -10, -27], [-7, 0, 14, 9], [0, 6, 15, 6], [-17, -6, 17, 12], [-9, -12, 9, 24], [18, 6, 18, 0], [-6, -8, 6, 16], [0, -48, 0, 48], [33, -15, -33, 30], [-2, 0, -2, 0], [-33, -7, 33, 14], [-20, 0, 40, -18], [-6, 0, -6, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_74_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_74_2_e_b();".
// 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_74_2_e_b(:prec:=4)
    chi := MakeCharacter_74_e();
    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_74_2_e_b();".
// 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_74_2_e_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_74_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![3, -3, 1]>],Snew);
    return Vf;
end function;