
// Make newform 735.2.a.g in Magma, downloaded from the LMFDB on 10 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_735_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_735_2_a_g();".
// 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_735_2_a_g();".
// 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 := [-3, 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_735_a();"
function MakeCharacter_735_a()
    N := 735;
    order := 1;
    char_gens := [491, 442, 346];
    v := [1, 1, 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;

function MakeCharacter_735_a_Hecke( : Kf := Kf)
    return MakeCharacter_735_a();
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 := [[-1, 1], [-1, 0], [1, 0], [0, 0], [-1, -1], [4, 1], [5, 1], [1, -2], [-3, 1], [1, 3], [3, 2], [2, 3], [1, 1], [-2, -3], [2, 0], [2, -6], [5, 3], [4, 0], [-6, 5], [1, -3], [4, -5], [3, 6], [3, -7], [3, -7], [8, -4], [-9, -1], [-4, 3], [-3, -3], [11, 0], [2, -4], [10, 3], [12, -2], [-3, -3], [-1, -4], [-8, -8], [-2, 4], [-4, -6], [8, -8], [-9, 5], [18, 2], [-10, 0], [-7, 10], [-2, 4], [-4, -3], [-9, -5], [22, 0], [14, 4], [-10, 6], [7, 5], [-3, 0], [0, -10], [14, 4], [10, -2], [-9, 9], [3, 5], [-2, -6], [-16, -2], [10, 4], [-6, 5], [0, -8], [-12, 7], [12, -4], [-20, 7], [-3, -7], [-4, 5], [1, 17], [15, 4], [-8, -15], [-28, -4], [22, 0], [9, 7], [-3, -1], [6, -3], [4, -13], [-11, -10], [-6, 10], [21, 9], [-26, 3], [-6, 6], [17, 8], [-32, 2], [7, 4], [0, -10], [10, -3], [4, 2], [6, -2], [-22, 8], [-8, 5], [-9, -15], [24, -1], [-14, -8], [-12, -12], [-16, -9], [-14, 14], [-17, -2], [4, -6], [8, 2], [-2, 2], [-26, 1], [-15, -12], [-2, 14], [-6, -12], [-18, 0], [3, -17], [29, 6], [20, -5], [39, -1], [-5, -19], [-14, 14], [-7, -4], [-2, -3], [-20, 4], [-12, 14], [-37, -4], [28, -12], [9, 3], [-18, -13], [5, 19], [-29, 9], [-4, -4], [7, -2], [-36, -5], [-21, 7], [-9, 11], [-5, -8], [7, 9], [-8, -4], [-24, -2], [-22, -5], [-10, -5], [7, -12], [17, 19], [25, 4], [-14, 8], [45, -1], [35, -10], [21, -13], [10, -2], [-23, -11], [-2, -16], [0, 2], [-3, 13], [8, -18], [-28, -14], [-15, -6], [-11, -17], [-6, 15], [17, -7], [4, -2], [4, -6], [-18, -8], [-4, -8], [38, -9], [-27, -1], [18, -9], [-13, 1], [-17, 8], [-23, 17], [30, -1], [-11, 11], [3, 5], [2, 22], [2, 7], [-24, 0], [-1, 9], [15, 3], [17, 22], [-30, 15]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_735_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_735_2_a_g();".
// 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_735_2_a_g(:prec:=2)
    chi := MakeCharacter_735_a();
    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_735_2_a_g();".
// 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_735_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_735_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-2, 2, 1]>,<13,R![13, -8, 1]>],Snew);
    return Vf;
end function;