
// Make newform 350.4.j.d in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_350_j();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_350_j_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_350_4_j_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_350_4_j_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, -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_350_j();"
function MakeCharacter_350_j()
    N := 350;
    order := 6;
    char_gens := [127, 101];
    v := [3, 2];
    // 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_350_j_Hecke();"
function MakeCharacter_350_j_Hecke(Kf)
    N := 350;
    order := 6;
    char_gens := [127, 101];
    char_values := [[-1, 0, 0, 0], [0, 0, -1, 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 := 4;
    raw_aps := [[0, 2, 0, -2], [0, 1, 0, 0], [0, 0, 0, 0], [0, -18, 0, 19], [-35, 0, 35, 0], [0, 0, 0, 66], [0, 59, 0, 0], [0, 0, 137, 0], [0, -7, 0, 7], [-106, 0, 0, 0], [-75, 0, 75, 0], [0, -11, 0, 11], [-498, 0, 0, 0], [0, 0, 0, 260], [0, 171, 0, -171], [0, 417, 0, 0], [-17, 0, 17, 0], [0, 0, -51, 0], [0, 439, 0, 0], [-784, 0, 0, 0], [0, -295, 0, 0], [0, 0, -495, 0], [0, 0, 0, 932], [0, 0, -873, 0], [0, 0, 0, 290], [1085, 0, -1085, 0], [0, 1553, 0, -1553], [0, -129, 0, 129], [-965, 0, 965, 0], [0, 0, 0, -50], [0, 0, 0, -936], [0, 0, 755, 0], [0, -2357, 0, 0], [-28, 0, 0, 0], [0, 0, 2295, 0], [1109, 0, -1109, 0], [0, 1559, 0, 0], [0, -2251, 0, 2251], [0, 0, 0, -2788], [0, 1579, 0, -1579], [2451, 0, -2451, 0], [-1170, 0, 0, 0], [0, 0, 1275, 0], [0, -35, 0, 0], [0, 0, 0, 2734], [2243, 0, -2243, 0], [1172, 0, 0, 0], [0, 0, 0, 2024], [0, 2571, 0, 0], [0, 0, 895, 0], [0, 1787, 0, -1787], [5100, 0, 0, 0], [4177, 0, -4177, 0], [-4680, 0, 0, 0], [0, 1749, 0, -1749], [0, 4473, 0, 0], [1975, 0, -1975, 0], [0, 0, 8439, 0], [0, 527, 0, 0], [-202, 0, 0, 0], [0, 7949, 0, 0], [0, 0, 0, 318], [0, 0, 0, 8132], [929, 0, -929, 0], [0, -209, 0, 209], [0, -7131, 0, 7131], [0, 0, 6571, 0], [0, 0, 0, 11466], [0, -9777, 0, 0], [-11914, 0, 0, 0], [0, -9123, 0, 0], [0, 0, 8149, 0], [0, 9671, 0, 0], [0, -4109, 0, 4109], [3488, 0, 0, 0], [0, 8717, 0, -8717], [163, 0, -163, 0], [0, -999, 0, 999], [0, 0, 14757, 0], [-133, 0, 133, 0], [6420, 0, 0, 0], [10266, 0, 0, 0], [15213, 0, -15213, 0], [0, 0, 0, -1378], [0, 0, -2763, 0], [0, 5849, 0, -5849], [-4582, 0, 0, 0], [0, -11551, 0, 11551], [-9494, 0, 0, 0], [0, 0, 0, -10160], [0, 1307, 0, -1307], [18287, 0, -18287, 0], [0, -14953, 0, 0], [14352, 0, 0, 0], [0, 0, -5531, 0], [0, 0, 0, 8400], [0, 0, -2385, 0], [9153, 0, -9153, 0], [0, -13807, 0, 13807], [0, 0, -8175, 0], [0, 0, 0, -4656], [0, 7003, 0, 0], [0, 19753, 0, 0], [0, 0, -6897, 0], [-24915, 0, 24915, 0], [0, 127, 0, 0], [0, 0, 0, -9044], [0, -10701, 0, 10701], [20799, 0, -20799, 0], [-1402, 0, 0, 0], [0, -6525, 0, 6525], [0, -15051, 0, 0], [0, 0, 0, -11150], [3415, 0, -3415, 0], [-21184, 0, 0, 0], [10705, 0, -10705, 0], [0, 0, 0, 6860], [0, 14463, 0, 0], [0, 5979, 0, -5979], [6940, 0, 0, 0], [-13399, 0, 13399, 0], [0, 0, 0, 29510], [0, 26001, 0, -26001], [0, 8805, 0, 0], [0, 0, -28685, 0], [-3146, 0, 0, 0], [0, 0, 1259, 0], [0, 0, 16425, 0], [0, 0, 0, 6032], [0, 15243, 0, -15243], [-10053, 0, 10053, 0], [0, 0, 0, 24384], [0, 0, -11589, 0], [0, 0, 0, -14562], [0, 0, 22765, 0], [-3766, 0, 0, 0], [0, 26861, 0, 0], [0, -2097, 0, 0], [0, 0, 0, 35334], [42535, 0, -42535, 0], [30676, 0, 0, 0], [0, 0, -37343, 0], [0, -2815, 0, 0], [0, 0, 0, 9276], [18571, 0, -18571, 0], [-29048, 0, 0, 0], [0, 0, 0, 32090], [0, -24537, 0, 0], [0, 0, 20825, 0], [0, -22847, 0, 22847], [0, 42737, 0, -42737], [6162, 0, 0, 0], [0, 0, 0, 7748], [0, 25923, 0, -25923], [0, 31935, 0, 0], [3408, 0, 0, 0], [0, 0, 13909, 0], [0, 0, -24537, 0], [0, 0, 0, 32758], [38561, 0, -38561, 0], [0, -39661, 0, 39661], [0, 0, 0, -46618], [0, 0, 0, -14816], [0, 0, 16875, 0], [0, -15837, 0, 0], [0, -9915, 0, 0], [43681, 0, -43681, 0], [0, -47113, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_350_j_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_350_4_j_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_350_4_j_d(:prec:=4)
    chi := MakeCharacter_350_j();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_350_4_j_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_350_4_j_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_350_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![1, 0, -1, 0, 1]>,<11,R![1225, 35, 1]>],Snew);
    return Vf;
end function;