
// Make newform 78.2.i.b in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_78_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_78_i_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_78_2_i_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_78_2_i_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_78_i();"
function MakeCharacter_78_i()
    N := 78;
    order := 6;
    char_gens := [53, 67];
    v := [6, 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_78_i_Hecke();"
function MakeCharacter_78_i_Hecke(Kf)
    N := 78;
    order := 6;
    char_gens := [53, 67];
    char_values := [[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, 1, 0, 0], [1, 0, -1, 0], [1, 0, -2, 0], [-2, -3, 1, 3], [1, 3, 1, 0], [-2, -3, 2, 0], [0, -3, 0, -3], [-2, 3, 1, -3], [-3, 3, 3, -6], [3, 0, -3, 0], [2, 0, -4, 6], [0, 3, 0, 0], [2, -3, 2, 0], [0, 3, 1, 3], [1, 0, -2, 3], [3, 0, 0, 0], [-16, 0, 8, 0], [0, 3, -10, 3], [-1, -9, -1, 0], [6, 3, -3, -3], [-7, 0, 14, 0], [-2, -12, 0, 6], [5, 0, -10, -3], [-2, 6, -2, 0], [0, 6, 0, -6], [9, -6, -9, 12], [-1, 6, 0, -3], [-3, -3, 3, 6], [-6, 0, 12, 6], [0, -3, -6, -3], [-4, 0, 4, 0], [6, 12, 0, -6], [0, 9, 0, -9], [0, 0, 4, 0], [-14, -6, 7, 6], [-1, 0, 2, -9], [2, -6, 0, 3], [-4, -6, 2, 6], [2, -6, 2, 0], [0, 6, 6, 6], [-3, -3, 3, 6], [-4, 18, 0, -9], [0, -12, 0, -12], [3, -18, 3, 0], [-4, 0, -4, 0], [0, 9, -7, 9], [14, -6, -14, 12], [4, 12, 4, 0], [6, -15, -3, 15], [8, 0, -16, -6], [18, 0, 0, 0], [9, 0, -18, -9], [6, -6, -3, 6], [0, -6, 6, -6], [18, -3, -18, 6], [-3, 3, 3, -6], [0, 6, 18, 6], [0, 0, 0, 0], [0, 3, 10, 3], [2, 0, -4, -21], [-25, -3, 25, 6], [10, 6, -5, -6], [1, 0, -2, -9], [3, -6, 0, 3], [-14, -12, 0, 6], [-7, 0, 14, -6], [0, 12, 0, -12], [31, 0, 0, 0], [0, 9, -3, 9], [-2, 6, -2, 0], [-12, 15, -12, 0], [11, 0, -22, -3], [19, 3, -19, -6], [0, -15, 2, -15], [14, -6, 14, 0], [20, -6, -10, 6], [3, 12, 0, -6], [12, 6, -6, -6], [0, 21, 0, 0], [10, -12, -5, 12], [-6, -6, 6, 12], [-2, 0, 4, -3], [3, -33, 3, 0], [0, -12, -13, -12], [1, 9, -1, -18], [6, -12, 0, 6], [-4, -30, 2, 30], [-15, 6, -15, 0], [-30, 6, 15, -6], [-3, 0, 6, 21], [-21, -18, 0, 9], [-10, -18, -10, 0], [14, -3, -7, 3], [-15, 9, 15, -18], [0, 0, 0, 0], [0, 9, 3, 9], [5, -18, 5, 0], [-24, 6, 0, -3], [-17, 9, 17, -18], [8, 0, -16, -3], [-1, -6, 0, 3], [-1, 24, -1, 0], [0, 12, -12, 12], [-12, -12, 12, 24], [19, -6, 0, 3], [-1, 0, 2, -18], [0, 0, 0, 0], [-8, 0, 16, 33], [6, -12, 0, 6], [-1, -12, 1, 24], [0, 12, 28, 12], [8, 27, 8, 0], [-12, 21, 6, -21], [6, 0, -12, 18], [-16, -12, 8, 12], [0, 3, -36, 3], [-32, 0, 16, 0], [-18, 18, 18, -36], [18, 18, -18, -36], [0, -18, 6, -18], [0, 9, 0, 0], [1, 0, -1, 0], [-6, 12, 0, -6], [-32, 0, 16, 0], [13, -3, 13, 0], [6, -12, 0, 6], [-28, 21, 14, -21], [0, 6, 42, 6], [19, 6, 0, -3], [10, 0, -20, 3], [-4, -12, -4, 0], [-6, 6, -6, 0], [1, -15, -1, 30], [2, -12, -2, 24], [12, -6, -6, 6], [-16, -6, -16, 0], [-24, -30, 12, 30], [-4, 18, 2, -18], [0, 0, 6, 0], [42, 3, -42, -6], [-6, 0, 12, -6], [20, 6, 20, 0], [0, 0, -32, 0], [-22, 0, 44, -6], [14, -15, -14, 30], [0, 12, 0, -12], [0, 0, 0, -9], [18, 42, 0, -21], [-5, 18, 0, -9], [13, 0, -26, 27], [12, 33, -6, -33], [12, 15, -12, -30], [4, 24, 0, -12], [-48, 0, 48, 0], [-10, -18, 10, 36], [6, 36, 0, -18], [0, -6, -22, -6], [12, -45, -6, 45], [-5, 12, 0, -6], [-12, 0, 24, -18], [-4, -36, -4, 0], [0, 0, -24, 0], [-7, 0, 14, -27], [0, 18, -18, 18], [8, 3, 8, 0], [-12, 0, 24, 0], [7, -21, -7, 42], [0, -3, -8, -3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_78_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_78_2_i_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_78_2_i_b(:prec:=4)
    chi := MakeCharacter_78_i();
    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_78_2_i_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_78_2_i_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_78_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![3, 0, 1]>],Snew);
    return Vf;
end function;