
// Make newform 343.2.g.d in Magma, downloaded from the LMFDB on 02 October 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_343_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_343_g_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_343_2_g_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_343_2_g_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, -1, 0, 1, -1, 0, 1, 0, -1, 1, 0, -1, 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_343_g();"
function MakeCharacter_343_g()
    N := 343;
    order := 21;
    char_gens := [3];
    v := [16];
    // 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_343_g_Hecke();"
function MakeCharacter_343_g_Hecke( : Kf := false)
    N := 343;
    order := 21;
    char_gens := [3];
    char_values := [[0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0], [-1, -1, 1, -2, 0, 1, 0, -1, 1, 0, -1, 1], [2, 0, -2, 0, 0, -1, 1, 1, -1, -1, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3, 0, -3, 3, -3, -3, 2, 1, -2, 1, 1, -4], [1, -1, 0, 2, -1, 0, 1, -1, -1, 1, 1, -1], [2, -2, 1, 5, -2, 1, 3, 0, -2, 2, 2, -2], [4, -2, -1, 3, -1, -3, 2, 2, -2, 4, 0, -2], [-1, 2, 1, -1, 1, 0, -1, 0, 1, -1, 0, 2], [2, 1, -6, 3, -1, -3, 0, 4, -1, -3, 3, -4], [0, 0, 0, 3, 0, -2, -2, -1, -2, -2, 0, 3], [7, -3, -2, 6, -3, -3, 5, 4, -7, 1, 0, -3], [-1, 5, 1, -5, 3, 3, -5, 1, 5, -1, 0, 0], [0, 4, 2, -1, 0, -1, 2, -1, 0, -1, 2, 4], [0, -2, 5, -1, -2, -1, -2, -1, 5, -2, 0, 0], [-4, 2, 2, -2, 0, 0, 0, -2, 4, -2, -4, 0], [0, -1, 1, -3, -1, 0, 0, 0, 0, 4, -4, 0], [-1, 4, 2, -4, 3, 1, -5, 1, 2, -1, 2, -2], [1, -1, -1, 0, 1, -2, -3, 6, -5, -1, 2, -2], [-9, -1, 0, 0, 2, 2, -6, -6, 3, -5, -3, 8], [0, 2, -4, 0, 6, -1, -1, 6, 0, -4, 2, 0], [0, 2, 3, 1, 1, 4, 0, 1, 1, -2, 2, -1], [2, -4, -1, 5, -1, -2, 1, 4, 0, -3, 6, 1], [-6, 1, -1, -9, -1, 3, -6, 1, 7, -3, -4, 2], [0, -3, 4, 1, -1, -4, 3, 0, -3, 4, -2, -3], [1, -11, 6, -1, -5, 5, 1, -5, 0, 5, -1, -5], [4, -1, -3, 1, -2, -3, 3, 1, -11, -1, 2, -4], [-9, 5, -1, -7, -1, 4, -8, 0, 8, -4, -4, 4], [-2, 6, -5, 0, 1, 1, -2, 1, 3, -4, 2, 0], [-3, 4, 2, -6, 1, 4, -2, 4, 5, -2, -3, 9], [-3, 0, 6, -3, -4, 2, 2, 0, 6, 5, -4, -1], [2, -7, 4, -2, 3, 0, 2, -2, 3, -1, -7, 8], [-8, -4, 5, -9, 3, 12, -10, -10, 3, -6, -4, 17], [2, 2, -13, 7, -1, -6, 2, -3, -3, -3, 10, -5], [-3, 2, 2, 0, 6, -5, 6, 0, 2, 2, -3, 0], [-6, -1, 1, 0, -1, 1, 6, -6, -1, 7, -2, 6], [-1, 2, 0, 3, 1, -1, 0, 0, 1, -3, 3, -1], [-7, -5, 15, -12, -1, 10, 3, -13, 8, 9, -9, 10], [3, -2, -2, 1, 5, -1, 2, -4, -8, 1, 1, 4], [7, -14, 2, 9, -8, -2, 6, -1, -9, 6, 2, -5], [10, 0, 0, 6, -1, -6, 6, 6, -2, 6, 6, -6], [-10, 6, 7, -13, 3, 14, -7, -8, 6, -11, -14, 9], [-3, -2, 5, 2, -4, -3, 2, -10, 2, -3, 0, -5], [5, 6, 5, -3, -1, 1, 3, 0, 0, 0, -3, -1], [-8, 11, -1, -9, 7, 5, -13, -2, 15, -3, -2, 9], [-11, 1, 9, -12, 6, 2, -4, -12, 8, 4, -4, 14], [8, -13, -2, 15, -6, -4, 2, -1, -16, 0, 1, -11], [14, -9, 0, 14, -5, -13, 10, 8, -4, 9, 4, -13], [-4, 4, 4, -5, 0, 12, -5, -6, 3, 8, -4, 3], [0, 0, -6, 9, -6, 3, 3, -6, -6, -2, 1, -9], [-4, 9, 1, -9, 7, 2, -8, 1, 6, -6, 5, 7], [0, -2, -7, -3, -1, -1, -3, -1, -1, -3, -7, -2], [-16, -4, 16, -9, -7, 13, -2, -7, 5, -2, -11, 0], [0, 1, 11, 6, 6, 0, -8, 0, 6, 6, 11, 1], [4, 2, 4, 2, -2, 0, -2, -2, 0, 6, -2, -15], [-6, 6, 6, -3, 8, -2, -2, -3, 10, -8, 2, 9], [-7, -1, -1, 0, -7, -8, -8, -7, 0, -1, -1, -7], [6, -12, 3, 10, 1, 0, 14, 8, -6, 13, 4, -3], [0, 5, -2, -2, 11, 10, -8, -1, 12, -6, -6, 6], [17, -11, 0, 15, -16, 1, 15, 1, -9, 10, -3, -14], [-18, 13, 3, -14, 16, 16, -9, -10, 18, -11, -8, 23], [-15, 9, -6, -4, 5, 4, -8, 1, 7, -5, 2, 5], [-2, 18, 14, -3, 4, 2, -6, -1, 8, -6, -3, 8], [7, 6, -1, -7, 1, -13, 2, 9, 1, 3, 3, -3], [-24, -4, 8, -18, 2, 14, -16, -18, 6, -12, -8, 14], [3, -6, -3, -3, 1, -3, 2, -1, -6, 7, -6, -6], [-7, -17, 9, -1, -4, 5, -5, 2, -6, 13, -9, 0], [0, -1, -4, 7, 1, 1, -1, 1, 1, 7, -4, -1], [10, -4, 8, 1, -9, 3, -8, 9, -5, 9, -1, -1], [0, 16, -5, -13, 12, 4, -17, 4, 12, -13, -5, 16], [8, 5, -24, 6, 6, -16, 12, 12, -13, -6, 18, -11], [4, -4, -2, 5, -2, -1, 3, -2, 0, -8, -9, 1], [7, 3, -9, -6, 2, -3, -3, 2, -6, -9, 3, 7], [-4, 21, -19, -6, 12, -7, -3, 8, 12, -24, 6, 6], [-5, -5, 4, -15, -4, 2, 2, 8, -5, 7, -12, 0], [-15, -4, -2, -8, -2, 2, -4, -9, 2, -2, 0, 11], [-10, 14, 5, -9, 5, 9, -19, -1, 14, -13, 5, 19], [-5, 2, 3, -2, 4, 11, -2, -7, 14, -6, 0, 13], [-3, 2, -3, 8, 2, 5, -10, 5, -11, -5, 16, 0], [-16, -2, -1, -7, -13, 7, -8, -9, -2, -7, -7, 7], [-3, 1, 4, -13, 5, 5, 1, -8, 4, -5, -4, 9], [3, -4, 2, -1, -4, 9, -5, 6, 4, -4, 7, -3], [-5, 10, 1, -13, 9, 1, -3, -3, 4, 8, -11, 2], [-7, -20, 6, -9, -1, 3, -5, -2, -16, -3, -8, 4], [-9, -3, -2, -6, 10, 3, 10, -6, -2, -3, -9, 0], [-10, 2, -3, -8, -4, 8, -4, -8, -3, 2, -10, 0], [-10, 16, -6, -16, 9, 6, -19, 0, 13, -13, 6, -3], [-2, 0, 0, -1, 2, 1, -1, 0, 3, 11, 12, 4], [2, 3, -10, 5, -12, -11, 6, 10, 3, 1, -1, -17], [11, 4, -11, 10, 4, -3, 4, 14, -7, -8, 7, 1], [-11, 22, 1, -15, 5, 5, -14, -3, 16, -17, -4, 12], [-18, 7, 6, -8, 13, 13, -16, -6, 26, 6, -1, 10], [-6, -17, -4, 5, -6, 0, 7, -8, -6, 14, -1, -12], [16, -3, -5, -4, 8, -3, 7, 4, -11, -1, 0, 0], [8, 7, 13, 12, -9, -6, 12, 5, 1, 6, -7, -11], [-4, -1, 6, -1, -23, 7, 16, -10, 7, 0, -4, -9], [-6, -7, 18, -10, -9, 13, 3, -11, -5, 10, -11, 6], [10, -10, -10, 8, 1, -17, -3, 20, -23, -7, 11, -12], [27, -4, -11, 17, -8, -21, 6, 15, -19, 11, 16, -21], [11, 7, 5, 4, 2, -6, 2, 4, 5, 7, 11, 0], [-18, 14, -8, -5, 6, 9, -2, -14, 17, -15, -6, 3], [-12, 12, 12, -13, -7, 19, -6, -13, 18, 7, -19, 11], [-13, 2, -5, 10, -18, -3, -3, -18, 10, -5, 2, -13], [6, -12, 0, -16, -13, -2, -8, 4, -2, 4, -7, 5], [9, -10, -2, 33, -6, -1, 15, 7, -25, 17, 17, -17], [18, -14, 4, 2, -10, -10, 17, -2, 1, 25, -10, -3], [-32, 0, 1, -4, 4, -1, 0, 0, 0, 1, -4, 0], [5, 1, -6, -1, 2, 4, -8, 13, 13, 6, -7, 5], [-17, 9, 9, -19, 18, 18, -23, 0, 10, -17, -1, 13], [-4, -20, 7, 1, -6, 3, 3, -8, -21, 24, 2, 0], [3, -1, -15, 21, 12, -11, -9, 7, 4, -8, 17, -13], [-1, 12, -21, -1, -2, -10, -10, -2, -1, -21, 12, -1], [5, -1, -8, 4, -9, -16, 19, 3, -6, 15, -8, -13], [15, -15, -15, 17, 3, -11, 8, 17, -22, 4, 18, -13], [0, 0, 17, 12, -7, -8, 8, -8, -7, 12, 17, 0], [-3, -11, 21, 3, -24, 8, 9, -6, -14, 7, -9, -20], [0, 14, -8, -22, 22, 5, 0, 5, 22, -22, -8, 14], [8, 4, -14, 16, 7, -14, 11, 11, -14, -8, 19, -8], [-20, 1, -13, 5, 15, 12, -8, -4, 15, -19, 2, 22], [17, -20, -5, 6, -9, -25, 12, 10, -5, 9, 5, -14], [19, 2, -4, 11, 13, 4, 4, 13, 11, -4, 2, 19], [5, -9, -8, 19, -12, -2, 14, -12, -5, 3, 16, -16], [4, 7, 7, 21, -19, -4, 8, 15, -10, 4, 6, -19], [17, -23, -8, 11, -8, -4, 13, 13, -12, -1, -1, -20], [11, -3, -8, 3, -6, -9, 14, -3, -28, -8, 11, -12], [-22, 14, 0, -14, 22, 0, 0, 9, 6, -9, -8, 31], [12, -24, -19, 17, -21, -18, 16, 4, -30, 3, 5, -31], [-15, 2, 25, -29, 6, 11, -4, -7, 22, 11, -10, 24], [16, 0, 12, 8, 0, -6, 2, 24, -6, 10, 12, -6], [1, 3, -7, 5, -12, -10, -12, 5, -7, 3, 1, 0], [36, -7, -19, 23, -4, -16, 10, 13, -30, 8, 9, -19], [-2, 5, -8, 1, 3, -1, -3, 0, 1, -6, 8, -4], [1, 4, 18, -18, -4, -1, 0, 1, 22, 3, -15, 18], [1, 9, -3, 13, 14, 4, -4, 10, -5, -12, 5, 7], [6, -12, -11, 21, -7, -15, 16, 10, -21, 5, 15, -17], [-17, 1, -2, 1, 19, -4, 2, -11, -3, -5, 0, 4], [-25, 15, 3, -11, 18, 18, -20, -18, 16, -17, -11, 16], [17, -3, -21, 23, -1, -19, 17, 18, -20, 2, 3, -21], [-30, 14, 1, -26, 11, 9, -20, -13, 31, -22, -5, 29], [4, 4, 9, -3, 10, -14, -14, 10, -3, 9, 4, 4], [13, -11, -15, 20, -24, -20, 4, 18, -9, 18, 9, -27], [-3, 5, 5, 3, -3, 0, -2, -14, 11, 8, -3, -9], [42, -29, -4, 16, -25, 4, 19, 19, -23, 18, 1, -15], [0, 4, -9, -17, 12, -11, -10, -11, 12, -17, -9, 4], [13, 13, 3, 3, -6, -16, -7, 10, 10, -13, 7, -22], [0, -14, -10, 0, -15, 2, 5, 2, -15, 0, -10, -14], [16, -22, 12, -6, -6, -2, -9, -2, -12, -3, -14, 0], [-5, -13, -2, -19, -2, 10, 10, -2, -19, -2, -13, -5], [5, -10, -10, -13, -10, 14, -1, -6, 9, 0, -18, -15], [4, -27, 8, 12, -2, -7, 3, -11, -15, 30, -16, 1], [2, 3, -5, -3, 6, -7, -7, 8, 2, 0, -4, -4], [21, 3, -23, 29, -15, -5, 11, 14, -25, -9, 15, -28], [-12, -9, 7, -16, 5, 15, -2, -11, 13, -4, -14, 13], [-16, 7, 9, -7, 14, 12, -20, -10, -7, -8, -13, 19], [-1, 18, 4, 14, 4, -5, -5, 4, 14, 4, 18, -1], [6, 7, -26, 13, 0, -7, 8, 1, -18, -5, 19, -13], [13, -10, -6, 16, -2, -14, 8, 19, -21, -4, 12, -16], [27, 2, -28, 37, 11, -14, 11, 37, -28, 2, 27, 0], [0, -15, 17, 28, -8, -13, 23, -13, -8, 28, 17, -15], [26, 11, -17, 18, 5, -16, 21, 21, -15, 7, 14, -27], [-4, -10, -4, 6, 6, 6, 2, -8, -19, 2, 4, -11], [-26, 18, 12, -17, 16, 24, -20, -12, 6, -10, -6, 16], [-10, -6, -7, -10, 23, -2, -21, 2, -3, 1, 2, 19], [-11, 4, -13, -17, 21, 8, -16, -16, 13, -8, -5, 24], [-1, -19, 3, -1, -21, 11, 21, -14, -10, 24, -11, -31], [-5, -14, -2, 17, 4, 2, -29, 3, 6, -9, 23, 10], [-19, 28, -5, -27, 23, 23, -13, -7, 27, -24, -11, 33], [-12, 10, 15, -32, 14, 18, -10, -15, 38, -25, -25, 25]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_343_g_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_343_2_g_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_343_2_g_d(:prec:=12)
    chi := MakeCharacter_343_g();
    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_343_2_g_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_343_2_g_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_343_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -8, 28, 71, 48, -21, 22, 0, -15, 1, 0, -1, 1]>,<3,R![49, -98, 294, -147, 147, 49, -35, 35, 63, -7, -14, 0, 1]>],Snew);
    return Vf;
end function;