
// Make newform 950.2.l.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_950_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_950_l_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_950_2_l_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_950_2_l_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_950_l();"
function MakeCharacter_950_l()
    N := 950;
    order := 9;
    char_gens := [77, 401];
    v := [9, 7];
    // 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_950_l_Hecke();"
function MakeCharacter_950_l_Hecke(Kf)
    N := 950;
    order := 9;
    char_gens := [77, 401];
    char_values := [[1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1]];
    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, 1, 0], [0, 0, 1, 1, 0, -1], [0, 0, 0, 0, 0, 0], [2, 2, -2, -2, 0, 0], [0, -2, 1, -2, 1, -2], [-2, -2, -2, 0, 2, 2], [4, -1, 4, -4, 1, -4], [4, 2, -2, -2, -2, 1], [2, -2, 0, 0, 0, 0], [-2, -2, 4, -2, -2, 0], [2, -4, 2, -2, 2, 2], [2, 2, 2, 0, -4, 2], [1, 0, 0, -1, 0, 1], [2, 0, -2, -2, -3, 0], [-4, -2, 2, -2, -4, 0], [-6, 2, 0, 4, 4, 0], [1, -2, 3, -3, 2, -1], [0, -2, 0, 2, 2, -6], [3, -3, 6, -3, 3, 0], [-2, 0, 2, -2, -4, -4], [3, 4, 7, 4, 0, -4], [0, 6, 2, 2, 0, -2], [2, 1, 3, -2, -4, -4], [-4, -4, -3, 8, -4, 3], [0, 0, 1, -1, 0, 0], [2, 2, -2, -4, 2, 2], [0, 2, 4, -2, 4, 2], [-7, -4, 2, 7, 2, 2], [2, 0, -2, 0, 8, 2], [0, -3, -3, 0, 8, -5], [8, 8, -6, 2, -10, 6], [1, -8, 0, 0, 8, -1], [0, 5, 0, -5, -5, -4], [0, 0, 10, -5, 5, -10], [4, 12, 2, -2, 0, 2], [2, 4, 4, 0, -14, 10], [-2, 0, 2, -2, -4, -4], [0, 1, 4, 8, 4, 1], [0, 6, 0, -6, -6, 6], [2, -12, -2, 2, 12, -2], [0, -5, -7, 2, -7, -5], [-8, 8, 0, 8, -8, 0], [-2, 8, 8, 0, -10, 2], [-12, 5, -16, -4, 0, 4], [2, -10, -4, -2, 14, 14], [-6, 6, 8, 6, -6, 0], [-6, -9, -12, 12, 9, 6], [0, 0, 0, 6, -2, 6], [-6, 0, 0, 0, -5, 5], [-8, -8, -8, 0, 0, 8], [13, 0, -13, -9, 4, 4], [0, -2, 10, 0, 10, -2], [-7, -7, -4, -1, 8, 4], [8, 1, 0, -9, -9, 8], [-17, 16, 4, 16, -17, 0], [12, -4, -4, -16, 0, 16], [6, -4, 2, -4, 0, 4], [8, 0, -8, 6, 6, 14], [0, -6, 12, 4, 12, -6], [4, -3, 0, -1, -1, 0], [5, -2, 11, -11, 2, -5], [0, -8, -8, 0, -8, -8], [1, -2, 15, 14, 0, -14], [2, 8, 8, -2, -16, -16], [-3, 8, -16, 8, -3, 0], [6, 6, -4, -10, 4, 4], [0, -18, 7, -6, 7, -18], [-7, 0, 7, -1, -16, -8], [11, 0, -11, -8, 6, 3], [0, -6, -2, 0, 8, 8], [0, 11, 5, 0, 5, 11], [-12, -16, 2, -2, 16, 12], [-12, -12, 10, 6, 6, -10], [-2, 4, 12, 2, -16, -16], [0, 8, 8, 0, 8, -16], [2, 6, 12, 10, 0, -10], [2, 4, -20, 4, 2, 0], [2, -12, 6, -6, 12, -2], [9, -8, -3, 3, 8, -9], [-8, 17, 12, 17, -8, 0], [-1, -2, -2, 0, -18, 20], [10, -4, 6, -4, 0, 4], [6, 6, 20, -4, -2, -20], [-4, 8, 0, -4, -4, 14], [0, 16, 14, -14, -16, 0], [5, 5, 12, 7, -12, -12], [4, 8, -11, -4, 3, 3], [4, -4, -4, 0, 5, -1], [-2, 0, 2, -10, 4, -12], [14, -4, -6, -14, 10, 10], [0, -7, 3, 18, 3, -7], [-12, 0, 0, 12, 12, -4], [10, 28, -14, -10, -14, -14], [0, -5, 18, 18, 0, -18], [14, 0, -14, -15, -6, -1], [-2, -12, -8, -12, -2, 0], [-8, 2, 0, 6, 6, -16], [0, 17, -8, -16, -8, 17], [0, 12, 4, 12, 0, 0], [2, -8, 0, -8, 2, 0], [-4, 10, 0, -6, -6, 11], [-14, -14, -6, 24, -10, 6], [14, -17, 6, -14, 11, 11], [-3, -20, -20, 0, 12, 8], [4, 17, 17, 0, -5, -12], [16, 3, -4, -16, 1, 1], [6, 1, 4, -4, -1, -6], [1, 16, 0, -17, -17, -20], [10, 14, -16, 14, 10, 0], [-16, 1, 4, 16, -5, -5], [16, 4, 4, 0, 10, -14], [18, 0, -18, 0, 8, 18], [4, 13, 0, 13, 4, 0], [0, 6, -4, 27, -4, 6], [2, -8, 0, 6, 6, -4], [-9, 0, 9, 4, -16, -5], [0, 0, 17, 17, 0, -17], [14, -4, -4, 0, -12, 16], [4, -2, 2, -4, 0, 0], [6, 6, -1, 10, -16, 1], [10, -4, 0, -6, -6, 16], [0, -20, 28, 17, 28, -20], [-6, -20, 0, 6, 20, 20], [-19, -10, -10, 0, 14, -4], [-19, 0, -10, 19, 10, 10], [2, 2, 4, -4, -2, -2], [-2, -2, 12, -14, 16, -12], [20, 12, 8, -12, 0, 12], [-12, 0, 12, 18, -4, 6], [0, -10, -12, 14, -12, -10], [-28, -2, -3, 3, 2, 28], [22, 18, -2, 2, -18, -22], [-4, 2, -34, 2, -4, 0], [4, 2, -6, -10, 0, 10], [20, -1, -1, 0, 13, -12], [12, 4, -15, 4, 12, 0], [6, 6, -20, 10, -16, 20], [0, 10, -19, 18, -19, 10], [4, -22, -22, 0, 18, 4], [0, 4, -13, -4, -13, 4], [16, 16, -32, -4, -12, 32], [12, -12, 0, 0, 0, 6], [-16, -16, 0, -12, 28, 0], [6, 11, -10, 11, 6, 0], [14, -8, 24, -14, -16, -16], [20, 26, 14, -6, 0, 6], [-14, 2, 10, -10, -2, 14], [-3, 32, -4, 4, -32, 3], [17, 10, 0, -27, -27, -6], [0, -10, 12, -4, 12, -10], [28, 2, 0, -28, 0, 28], [-20, 15, -23, 20, 8, 8], [-29, 17, -2, 17, -29, 0], [-2, -2, -12, 12, -10, 12], [2, 0, -2, -11, -22, -9], [-34, -8, -8, 0, 12, -4], [36, -10, 10, -36, 0, 0], [5, 8, -11, 11, -8, -5], [-3, -3, 8, -9, 12, -8], [-24, 6, 4, 6, -24, 0], [16, 20, 4, -12, 0, 12], [-7, 20, 25, 32, 0, -32], [0, -16, -16, 16, 16, 0], [17, 24, -4, 4, -24, -17], [0, 19, -36, -12, -36, 19], [-18, 0, 18, 14, 10, -4], [14, -26, 12, -2, 0, 2], [20, -22, 2, -22, 20, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_950_l_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_950_2_l_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_950_2_l_d(:prec:=6)
    chi := MakeCharacter_950_l();
    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_950_2_l_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_950_2_l_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_950_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 3, 3, -8, 6, -3, 1]>],Snew);
    return Vf;
end function;