
// Make newform 650.2.o.e in Magma, downloaded from the LMFDB on 20 October 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_650_o();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_650_o_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_650_2_o_e();".
// 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_650_2_o_e();".
// 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_650_o();"
function MakeCharacter_650_o()
    N := 650;
    order := 6;
    char_gens := [27, 301];
    v := [3, 4];
    // 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_650_o_Hecke();"
function MakeCharacter_650_o_Hecke( : Kf := false)
    N := 650;
    order := 6;
    char_gens := [27, 301];
    char_values := [[-1, 0, 0, 0], [0, 0, -1, 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, 0, 0], [1, 1, 1, 0], [0, 0, 0, 0], [-4, -1, 2, 1], [0, -1, 0, 2], [0, 2, -3, -2], [2, 3, -1, -3], [0, 3, 2, 3], [-2, 0, -2, 0], [6, -2, -6, 4], [5, 2, 0, -1], [-3, 4, -3, 0], [0, 4, 0, -8], [8, -2, -4, 2], [-2, 0, 4, -3], [1, 0, -2, 0], [0, 2, 6, 2], [0, 1, -5, 1], [-6, -2, -6, 0], [0, 2, -12, 2], [1, 0, -2, 1], [-5, -2, 0, 1], [-5, 0, 10, -3], [-9, 0, 9, 0], [-2, -7, 1, 7], [-9, 3, 9, -6], [2, 0, -4, 13], [-1, -15, -1, 0], [-2, 16, 0, -8], [0, 0, 0, 0], [6, -5, 6, 0], [6, -6, 0, 3], [10, 3, -5, -3], [0, 1, -4, 1], [0, -10, 0, -10], [-7, -6, 0, 3], [7, 0, -14, 2], [-6, -11, 3, 11], [2, -15, 2, 0], [6, 6, -3, -6], [6, 6, -6, -12], [5, -18, 0, 9], [0, 9, -3, 9], [4, -4, 4, 0], [1, 24, 1, 0], [0, 0, 2, 0], [22, -3, -22, 6], [-12, -1, -12, 0], [20, 6, -10, -6], [-11, -18, 0, 9], [-3, 0, 6, 3], [15, -2, 0, 1], [0, 0, 7, 0], [0, 3, 12, 3], [-8, -12, -8, 0], [2, 9, 2, 0], [0, 11, 3, 11], [-14, 10, 14, -20], [-30, 2, 15, -2], [-12, 8, 0, -4], [-8, -10, -8, 0], [-6, 12, 3, -12], [14, 0, -28, -4], [27, -6, 0, 3], [6, 0, -12, 16], [1, 0, -2, 0], [0, 8, -20, 8], [3, 0, -6, 5], [2, -27, -1, 27], [-19, 1, 19, -2], [3, -3, 3, 0], [6, 16, 0, -8], [0, 16, 0, 0], [-8, 10, 4, -10], [8, -9, -8, 18], [12, -12, -6, 12], [3, -10, 0, 5], [10, -16, -5, 16], [-3, 0, 3, 0], [0, 6, 5, 6], [12, -10, -12, 20], [-7, -14, 0, 7], [-9, -7, 9, 14], [-20, 10, 10, -10], [-7, 17, 7, -34], [-16, 0, 32, -6], [0, 0, -21, 0], [-3, 13, -3, 0], [0, -1, -21, -1], [4, 0, -8, 10], [-12, 0, 24, 0], [3, 1, -3, -2], [0, -31, 0, 31], [24, -5, -24, 10], [-8, -36, 0, 18], [-8, -9, 4, 9], [0, 14, 0, -28], [-9, 20, 0, -10], [0, 2, 0, 0], [-13, -14, 0, 7], [-16, 0, 32, 2], [15, -6, 15, 0], [40, 0, -20, 0], [3, -6, -3, 12], [2, -6, 0, 3], [9, 0, -18, -1], [10, 24, 10, 0], [16, 0, -32, -12], [-42, 0, 0, 0], [19, 12, -19, -24], [24, 23, -12, -23], [7, -16, 7, 0], [16, -18, -8, 18], [-14, -14, 0, 7], [0, -6, -26, -6], [0, 8, 15, 8], [6, 7, -3, -7], [-4, -9, -4, 0], [7, 36, 7, 0], [0, 6, 24, 6], [7, -23, -7, 46], [10, 14, 10, 0], [6, 0, -12, -12], [4, 0, -2, 0], [-8, 21, 8, -42], [9, -6, 0, 3], [0, -11, -19, -11], [0, 8, -24, 8], [-8, 0, 16, 11], [-9, 0, 18, 28], [-4, -3, 4, 6], [-6, -12, -6, 0], [-2, 6, 2, -12], [15, 10, 15, 0], [0, 20, 3, 20], [32, 0, -32, 0], [-10, -24, 5, 24], [10, 5, -5, -5], [8, -12, -4, 12], [36, 0, -36, 0], [-10, 10, 0, -5], [9, 7, -9, -14], [-24, 19, 12, -19], [-17, 0, 34, -9], [-4, -8, 4, 16], [0, -13, -9, -13], [0, 0, 0, 16], [11, 0, -22, -3], [28, -22, 0, 11], [-10, 0, 20, 0], [-12, -16, 6, 16], [-15, -22, 15, 44], [-1, 0, 2, -5], [-20, -15, -20, 0], [-9, 13, -9, 0], [24, 8, 0, -4], [0, 18, 8, 18], [0, 0, 6, 0], [-18, 0, 36, -4], [30, -12, 0, 6], [-7, -21, -7, 0], [-42, -21, 21, 21], [-18, 0, 36, 11], [0, -3, 36, -3], [-14, 6, -14, 0], [14, 0, -28, 21], [22, 12, -22, -24], [-6, 38, 3, -38]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_650_o_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_650_2_o_e();".
// 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_650_2_o_e(:prec:=4)
    chi := MakeCharacter_650_o();
    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_650_2_o_e();".
// 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_650_2_o_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_650_o();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, -12, 14, -6, 1]>,<7,R![121, 132, 59, 12, 1]>],Snew);
    return Vf;
end function;