
// Make newform 342.2.u.d in Magma, downloaded from the LMFDB on 23 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_d();".
// 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_d();".
// 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, 2, 1, -1, -2, -1], [1, -2, 1, -1, 1, 1], [0, 0, 1, -4, 1, 0], [-3, 4, 0, -1, -1, 0], [0, -4, -1, -1, 0, 1], [0, 1, 2, 2, -2, 2], [2, 1, 4, 1, 2, 0], [-4, -4, -2, 3, 1, 2], [1, -1, -1, -1, 2, 2], [1, 0, 0, 0, 2, -2], [3, 0, -3, -3, 3, 0], [-3, -1, -3, 3, 1, 3], [4, 4, -5, -1, -3, 5], [-4, -2, 1, -2, -4, 0], [-2, 0, -7, -5, 0, 5], [-3, 3, -1, 3, -3, 0], [-4, -4, 2, 3, 1, -2], [-3, 3, -3, 3, -3, 3], [7, 0, -7, -10, 1, -3], [5, 0, -5, -5, 3, 0], [1, 11, -6, -1, -5, -5], [5, -11, 0, 6, 6, 1], [-1, 4, 7, 8, 0, -8], [-2, 2, 0, 0, 0, 5], [0, 6, -1, -7, -1, 6], [-5, -12, 7, 5, 5, 5], [-2, -7, -1, 1, 7, 2], [2, 8, 8, 0, 1, -9], [-2, 4, 0, -2, -2, 4], [7, 2, 1, -6, 0, 6], [-8, 6, 1, 6, -8, 0], [7, -5, 0, -2, -2, 1], [9, 0, -9, -7, -2, 2], [2, 1, 1, 0, -11, 10], [0, 14, -3, 3, -14, 0], [0, -6, -1, -4, -1, -6], [-11, 10, 0, 10, -11, 0], [1, 5, 7, 6, 0, -6], [0, 8, 4, -5, 4, 8], [1, 1, 16, 1, -2, -16], [4, -3, -3, 0, 1, 2], [0, 0, 0, 1, -11, 1], [7, 7, -1, -7, -6, -6], [0, 0, -6, -7, 7, 6], [0, 3, -1, -1, 0, 1], [1, 23, 5, -5, -23, -1], [3, -11, -11, 0, -2, 13], [10, -4, -4, 0, 4, 0], [-10, 2, -9, 9, -2, 10], [0, 2, -2, 4, -2, 2], [2, 8, 0, -10, -10, 13], [14, -9, -1, -9, 14, 0], [-5, -5, 4, 2, 3, -4], [-4, 0, 4, -11, 1, -15], [3, 0, -3, 1, -13, 4], [-3, -1, -12, 12, 1, 3], [0, 5, -4, -17, -4, 5], [7, 0, 1, 0, 7, 0], [11, 7, 7, -4, 0, 4], [0, 0, 1, -13, 1, 0], [9, 0, -9, 9, 5, 18], [5, -1, -2, -5, 3, 3], [-3, -3, -5, -9, 12, 5], [-17, 17, 0, 0, 0, 5], [0, -12, 10, -9, 10, -12], [-1, 2, -8, 8, -2, 1], [2, 3, -8, 8, -3, -2], [-7, -7, -6, 7, 13, 13], [0, -8, 4, -3, 4, -8], [-4, -8, 8, 12, 0, -12], [2, -7, 0, 5, 5, 16], [-5, 15, 0, 5, -15, -15], [-4, -17, -17, 0, 7, 10], [-2, 0, 2, -8, 9, -10], [-2, -2, -17, 5, -3, 17], [9, 10, 9, 0, 0, 0], [3, -19, 8, 5, 0, -5], [-6, -6, 10, 6, 0, -10], [-1, 12, 12, 0, -7, -5], [12, 0, -12, -15, -4, -3], [-7, 6, 0, 1, 1, -13], [8, -10, 8, -10, 8, 0], [11, -3, 3, -8, 0, 8], [15, -7, 0, -8, -8, -11], [8, -18, 2, -8, 16, 16], [-14, 5, 5, 0, 10, -15], [6, 9, -15, 15, -9, -6], [-3, 13, 6, 3, -19, -19], [0, 5, 5, -6, 5, 5], [0, 3, 12, 3, 0, 0], [7, -7, 14, -7, -7, -7], [8, 0, -8, 0, -1, 8], [7, -15, 12, -12, 15, -7], [-5, -5, -7, -21, 26, 7], [-2, 3, 1, 3, -2, 0], [0, 7, 5, 14, 5, 7], [-6, -6, -11, 15, -9, 11], [-3, -3, -19, -5, 8, 19], [-15, -9, 5, -9, -15, 0], [-4, 19, 0, -15, -15, -11], [-5, 5, -6, 5, 1, 1], [5, 14, 14, 0, 4, -18], [-17, 1, 1, 0, 12, -13], [22, 7, -11, -22, 4, 4], [-10, -16, 9, 19, 0, -19], [-4, 3, 11, 3, -4, 0], [-23, -23, -6, 13, 10, 6], [-17, -6, -3, 17, 9, 9], [20, 4, 4, 0, 7, -11], [5, 2, -11, 11, -2, -5], [-4, -4, -12, 20, -16, 12], [0, -10, 8, -8, 8, -10], [-7, 12, -5, 12, -7, 0], [-23, 0, -10, 10, 0, 23], [-6, 0, 6, -4, -3, -10], [17, -15, -15, 0, 8, 7], [13, -10, 21, -13, -11, -11], [-22, -5, 0, 27, 27, -17], [-17, 4, -22, 4, -17, 0], [0, 3, -8, 15, -8, 3], [20, -1, 10, -20, -9, -9], [34, -4, -4, 0, 0, 4], [1, -2, -20, -1, 22, 22], [1, 17, 1, 0, 0, 0], [-12, -5, 0, 17, 17, -30], [-7, 0, 7, 5, 6, -2], [7, 15, 0, 0, -15, -7], [0, 22, -23, 1, -23, 22], [-2, 14, 0, 2, 0, -2], [-12, -9, -9, 3, 0, -3], [-18, -18, 14, 4, 14, -14], [-6, 0, 6, 0, 32, -6], [-9, 23, 23, 0, -16, -7], [6, 6, 3, -13, 7, -3], [-12, 14, 0, -2, -2, -5], [0, -11, 10, 1, 10, -11], [17, -14, -14, 0, 3, 11], [0, 9, -1, -14, -1, 9], [5, -1, 0, -4, -4, 7], [3, 3, -33, 3, 3, 0], [-23, 19, 0, 4, 4, 19], [0, 0, -22, -13, 13, 22], [9, -8, 6, -9, 2, 2], [6, 0, -6, 8, 25, 14], [-20, -22, -18, 2, 0, -2], [-6, 26, -7, -1, 0, 1], [4, -1, -26, -1, 4, 0], [0, 19, -33, -14, -33, 19], [-29, 0, 29, 5, -2, -24], [2, 0, -19, -2, 19, 19], [8, 8, 27, 4, -12, -27], [23, -9, 0, -14, -14, 8], [2, -25, 16, -16, 25, -2], [15, -27, -27, 0, 21, 6], [0, 2, 31, 0, -33, -33], [18, 7, 25, 7, 0, -7], [4, -15, 0, 11, 11, -1], [-6, -6, 4, -17, 23, -4], [-22, 0, 22, 21, 11, -1], [-24, 0, 24, 12, 18, -12], [-8, 5, -24, -16, 0, 16], [-18, 2, 2, 20, 0, -20], [0, -15, 26, -5, 26, -15], [6, 20, 17, -17, -20, -6], [-4, 0, 4, -22, -7, -26], [-13, -13, -24, 19, -6, 24]];
    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_d();".
// 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_d(: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_d();".
// 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_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_342_u();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![9, 0, 36, 30, 0, -3, 1]>],Snew);
    return Vf;
end function;