
// Make newform 78.2.i.a in Magma, downloaded from the LMFDB on 19 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_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_78_2_i_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 := [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, -2], [2, -1, -1, 1], [-1, -3, -1, 0], [0, 3, 0, -4], [0, 1, -4, 1], [2, -3, -1, 3], [-1, -3, 1, 6], [1, -2, -1, 4], [-2, 0, 4, 2], [-2, -7, -2, 0], [6, 1, 6, 0], [0, 5, -1, 5], [3, 0, -6, -3], [-3, 4, 0, -2], [0, 8, 0, -8], [0, -3, 4, -3], [-7, 1, -7, 0], [2, -3, -1, 3], [1, 0, -2, -8], [-6, -4, 0, 2], [3, 0, -6, -5], [-2, 6, -2, 0], [0, 6, 0, -6], [-5, -4, 5, 8], [17, 2, 0, -1], [-5, 3, 5, -6], [-2, 0, 4, 2], [0, 5, 10, 5], [4, 8, -4, -16], [10, 4, 0, -2], [-8, 5, 4, -5], [0, 8, -4, 8], [6, -8, -3, 8], [1, 0, -2, 5], [-8, -18, 0, 9], [4, -10, -2, 10], [-2, 6, -2, 0], [0, -6, -6, -6], [3, 11, -3, -22], [-14, 6, 0, -3], [0, -4, 0, -4], [-1, 10, -1, 0], [8, -4, 8, 0], [0, 3, -9, 3], [-6, 6, 6, -12], [4, -20, 4, 0], [-6, 7, 3, -7], [-8, 0, 16, 2], [-6, -16, 0, 8], [-5, 0, 10, 1], [-18, -2, 9, 2], [0, -2, 10, -2], [18, 5, -18, -10], [-9, -11, 9, 22], [0, 2, -2, 2], [-8, 8, -8, 0], [0, 1, -4, 1], [10, 0, -20, -5], [17, -5, -17, 10], [-26, 8, 13, -8], [-9, 0, 18, -7], [-7, -10, 0, 5], [10, 4, 0, -2], [-5, 0, 10, -12], [0, -20, 0, 20], [7, -16, 0, 8], [0, 7, -21, 7], [10, 2, 10, 0], [12, -1, 12, 0], [-7, 0, 14, 11], [13, 1, -13, -2], [0, -1, 12, -1], [2, -2, 2, 0], [-4, -2, 2, 2], [9, 24, 0, -12], [-12, 10, 6, -10], [-4, -15, -4, 0], [2, -16, -1, 16], [14, -2, -14, 4], [16, 0, -32, -5], [-7, 1, -7, 0], [0, -8, -1, -8], [-9, -5, 9, 10], [26, 12, 0, -6], [-20, -6, 10, 6], [5, 10, 5, 0], [-2, -24, 1, 24], [11, 0, -22, -9], [-3, 18, 0, -9], [2, -30, 2, 0], [-14, -9, 7, 9], [7, -1, -7, 2], [0, 0, 0, -32], [0, 15, -15, 15], [-1, 12, -1, 0], [12, -34, 0, 17], [-7, -17, 7, 34], [-10, 0, 20, 23], [1, 6, 0, -3], [21, -6, 21, 0], [0, -4, 28, -4], [-4, -20, 4, 40], [5, 22, 0, -11], [-1, 0, 2, -2], [0, 16, 0, 0], [8, 0, -16, -23], [-6, -4, 0, 2], [-1, 4, 1, -8], [0, -12, 20, -12], [6, 1, 6, 0], [-28, -11, 14, 11], [2, 0, -4, 14], [32, 20, -16, -20], [0, 15, 0, 15], [-16, 0, 8, 0], [2, 14, -2, -28], [14, -2, -14, 4], [0, -14, -14, -14], [6, -1, 6, 0], [21, -4, -21, 8], [-42, 4, 0, -2], [-16, -24, 8, 24], [-5, 35, -5, 0], [-14, -20, 0, 10], [16, 27, -8, -27], [0, 2, -26, 2], [5, -30, 0, 15], [0, 0, 0, -19], [4, 4, 4, 0], [6, -38, 6, 0], [-17, 19, 17, -38], [-14, 4, 14, -8], [-20, -6, 10, 6], [8, -30, 8, 0], [-8, 42, 4, -42], [20, 22, -10, -22], [0, 0, -34, 0], [38, -9, -38, 18], [-18, 0, 36, 14], [4, 6, 4, 0], [0, -24, 0, -24], [-2, 0, 4, 30], [0, 7, 0, -14], [-16, 28, 8, -28], [-18, 0, 36, -23], [-10, 34, 0, -17], [-35, 14, 0, -7], [7, 0, -14, 5], [8, -17, -4, 17], [12, 23, -12, -46], [-36, 8, 0, -4], [-24, 8, 24, -16], [26, -6, -26, 12], [-6, -4, 0, 2], [0, 30, 6, 30], [-28, 15, 14, -15], [19, 28, 0, -14], [-20, 0, 40, 22], [-4, -28, -4, 0], [0, -24, 0, -24], [7, 0, -14, 31], [0, 14, -6, 14], [-4, 39, -4, 0], [12, 0, -24, 0], [-7, 9, 7, -18], [0, -5, -30, -5]];
    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_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_78_2_i_a(: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_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_78_2_i_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_78_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, 0, 14, 0, 1]>],Snew);
    return Vf;
end function;