
// Make newform 342.2.u.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_342_u();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_342_u_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_342_2_u_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_342_2_u_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, 0, -1, 0, 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_342_u();"
function MakeCharacter_342_u()
    N := 342;
    order := 9;
    char_gens := [191, 325];
    v := [9, 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_342_u_Hecke();"
function MakeCharacter_342_u_Hecke(Kf)
    N := 342;
    order := 9;
    char_gens := [191, 325];
    char_values := [[1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 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, 0], [0, 0, 0, 0, 0, 0], [-1, 0, 1, -1, 0, 1], [1, 0, -1, -1, 1, 1], [0, 0, 1, 0, 1, 0], [-1, 0, 0, 1, 1, 2], [2, 2, -1, -3, 0, 3], [-2, -1, 0, -2, 2, 2], [2, 3, -2, 3, 2, 0], [-2, -2, -2, 5, -3, 2], [3, -3, 3, -3, 0, 0], [-3, 6, 6, 0, -4, -2], [1, 0, -1, 3, 3, 4], [-1, 5, -3, 3, -5, 1], [0, 0, 1, 3, -3, -1], [2, -8, -1, -8, 2, 0], [-6, 4, -3, 3, 0, -3], [1, -1, -11, -1, 1, 0], [4, 4, 4, -1, -3, -4], [-7, 5, -1, 1, -5, 7], [7, 0, -7, -2, 1, 5], [-5, 0, 5, -5, -5, -10], [9, 3, -2, -9, -1, -1], [7, -3, 0, -4, -4, 5], [-1, -4, -1, 0, 0, 0], [-4, -4, 0, 8, 8, -1], [0, 6, -7, 1, -7, 6], [7, 4, 3, -7, -7, -7], [-2, -13, -1, 1, 13, 2], [4, 0, 0, 0, 5, -5], [-2, -4, 0, 6, 6, -8], [3, 2, 5, 2, 0, -2], [8, 0, -1, 0, 8, 0], [-3, -1, 0, 4, 4, 5], [-3, 0, 3, 5, 8, 2], [8, 5, 5, 0, 9, -14], [2, -8, 5, -5, 8, -2], [0, 4, 1, 16, 1, 4], [-5, 0, 4, 0, -5, 0], [-5, 1, -3, 2, 0, -2], [0, -8, 0, -13, 0, -8], [13, 13, -4, -9, -4, 4], [0, 9, 9, 0, -5, -4], [-4, 0, 4, 5, 5, 1], [1, -11, 7, -1, 4, 4], [-2, -2, 10, -5, 7, -10], [0, 11, -3, -3, 0, 3], [3, -1, -1, 1, 1, -3], [3, 5, 5, 0, -2, -3], [-14, -4, -4, 0, -4, 8], [-2, 8, -5, 5, -8, 2], [0, 14, -6, -4, -6, 14], [10, -8, 0, -2, -2, 5], [-2, -1, -13, -1, -2, 0], [15, 15, 4, -8, -7, -4], [10, 0, -10, -7, -1, 3], [-11, 0, 11, 5, 15, -6], [-3, 3, 10, -10, -3, 3], [0, -3, -6, 11, -6, -3], [-3, 0, 5, 0, -3, 0], [-5, 1, -11, -6, 0, 6], [0, 8, -13, -1, -13, 8], [-5, 0, 5, 3, 13, -2], [-9, -3, 8, 9, -5, -5], [13, 13, 3, 7, -20, -3], [-3, 5, 0, -2, -2, -5], [0, -4, -16, 9, -16, -4], [-5, 2, -16, 16, -2, 5], [2, -5, 0, 0, 5, -2], [-5, -5, 0, 5, 5, 5], [0, 0, -6, 27, -6, 0], [4, 8, 4, 0, 0, 0], [-6, 1, 0, 5, 5, 6], [13, 17, -10, -13, -7, -7], [-6, 7, 7, 0, -7, 0], [0, 0, 0, -2, 9, -2], [-8, -8, 1, -3, 11, -1], [11, 0, 7, -4, 0, 4], [-15, 5, 2, 17, 0, -17], [-14, -14, -10, 6, 8, 10], [7, 4, 4, 0, -19, 15], [-20, 0, 20, 7, 10, -13], [-13, 0, 0, 13, 13, 1], [20, -10, 0, -10, 20, 0], [5, -23, -3, -8, 0, 8], [11, -3, 0, -8, -8, 17], [12, 10, -14, -12, 4, 4], [-14, 25, 25, 0, -6, -19], [8, 15, 7, -7, -15, -8], [-3, 3, 6, 3, -9, -9], [0, -3, 9, 26, 9, -3], [-4, -7, -18, -7, -4, 0], [-13, -9, -2, 13, 11, 11], [4, 0, -4, -16, 11, -12], [9, 5, 24, -24, -5, -9], [-3, -3, 1, 15, -12, -1], [6, -27, 1, -27, 6, 0], [0, -1, 13, -4, 13, -1], [8, 8, -3, -21, 13, 3], [19, 19, -7, -1, -18, 7], [-7, -7, 17, -7, -7, 0], [2, -3, 0, 1, 1, -15], [-5, -15, -2, 5, 17, 17], [3, 16, 16, 0, -8, -8], [21, -9, -9, 0, -2, 11], [26, 7, -11, -26, 4, 4], [-22, 8, -19, 3, 0, -3], [-14, 11, -11, 11, -14, 0], [-11, -11, -2, 3, 8, 2], [-5, 6, 13, 5, -19, -19], [8, -10, -10, 0, -7, 17], [7, -16, -1, 1, 16, -7], [8, 8, 0, -16, 8, 0], [0, -10, 16, 0, 16, -10], [7, 6, 3, 6, 7, 0], [21, 6, 16, -16, -6, -21], [-2, 0, 2, -2, -29, -4], [7, 13, 13, 0, 2, -15], [5, 16, -21, -5, 5, 5], [-6, -13, 0, 19, 19, -13], [5, 2, 10, 2, 5, 0], [0, -21, -4, -1, -4, -21], [14, -1, -8, -14, 9, 9], [30, 8, 8, 0, -4, -4], [-5, 4, -12, 5, 8, 8], [-13, 13, 7, 20, 0, -20], [-6, 25, 0, -19, -19, -6], [-5, 0, 5, -19, -2, -24], [7, 11, -10, 10, -11, -7], [0, -16, -1, 11, -1, -16], [2, -22, 4, 2, 0, -2], [10, -21, 5, -5, 0, 5], [2, 2, 6, 0, -2, -6], [6, 0, -6, -20, -8, -14], [-3, -27, -27, 0, 26, 1], [-14, -14, -1, 15, -1, 1], [-10, -12, 0, 22, 22, 5], [0, 23, -2, -3, -2, 23], [-21, 4, 4, 0, -9, 5], [0, 9, -21, 12, -21, 9], [-25, 27, 0, -2, -2, -5], [3, -19, 25, -19, 3, 0], [11, 15, 0, -26, -26, 13], [-20, -20, -10, -9, 29, 10], [13, 0, 0, -13, 0, 0], [-8, 0, 8, 6, 13, -2], [8, 22, 14, 6, 0, -6], [-12, 8, -31, -19, 0, 19], [10, -7, 18, -7, 10, 0], [0, -27, 11, -12, 11, -27], [1, 0, -1, 17, -2, 18], [-4, 22, -9, 4, -13, -13], [-18, -18, -3, 0, 18, 3], [11, -15, 0, 4, 4, 2], [8, 9, 6, -6, -9, -8], [11, -21, -21, 0, 17, 4], [-14, 6, -1, 14, -5, -5], [26, 3, 31, 5, 0, -5], [24, -3, 0, -21, -21, 7], [4, 4, -12, -3, -1, 12], [10, 0, -10, 13, -13, 23], [8, 0, -8, 20, 6, 28], [10, -17, 24, 14, 0, -14], [18, 14, -10, -28, 0, 28], [0, -1, 6, 23, 6, -1], [-4, 34, -7, 7, -34, 4], [-22, 0, 22, 10, -1, -12], [-17, -17, -20, -11, 28, 20]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_342_u_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_342_2_u_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_342_2_u_a(:prec:=6)
    chi := MakeCharacter_342_u();
    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_342_2_u_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_342_2_u_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_342_u();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![81, 162, 162, 90, 36, 9, 1]>],Snew);
    return Vf;
end function;