
// Make newform 637.2.a.h in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_637_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_637_2_a_h();".
// 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_637_2_a_h();".
// 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, -5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-3, -1, 1]];
    Rf_basisdens := [1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_637_a();"
function MakeCharacter_637_a()
    N := 637;
    order := 1;
    char_gens := [248, 197];
    v := [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_637_a_Hecke(Kf)
    return MakeCharacter_637_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+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 := [[-1, 1, 0], [-1, 0, -1], [-2, 1, 0], [0, 0, 0], [-1, -1, 0], [1, 0, 0], [-1, 0, -1], [-2, -2, 1], [1, -3, 1], [-3, 0, 2], [0, 1, 2], [-3, 1, -2], [-2, -4, 4], [3, 2, -2], [-6, 0, 1], [5, -1, -1], [-8, 1, 1], [8, 2, -2], [-4, 2, -4], [1, 3, -2], [-4, 6, 1], [-1, 2, 2], [-8, 3, -2], [4, -4, 3], [2, -4, 1], [-5, -4, 1], [4, -4, -4], [6, -1, 3], [-7, -3, 4], [3, 5, -3], [8, -1, -3], [-6, 4, 2], [7, -3, 4], [3, -2, -1], [3, 3, -6], [9, -1, -4], [-1, -4, 5], [6, 0, -2], [6, 6, -3], [-1, -6, 1], [-9, -4, 0], [-11, 4, -3], [-6, 9, -3], [8, -2, 0], [-14, 0, -2], [3, 10, -5], [-15, 3, 3], [4, -1, 2], [-10, -2, 0], [0, 1, -5], [7, 0, -8], [-14, -4, 4], [0, 5, -4], [-9, 6, 3], [-19, 2, -1], [3, 0, 6], [20, 2, -4], [0, -6, 2], [-3, 0, 0], [-1, -1, 2], [-2, -2, 8], [-10, -10, 5], [-6, 5, -10], [5, -6, 7], [25, 0, -3], [4, -10, 4], [-2, 10, -4], [-13, 3, -9], [-10, 4, -6], [2, 6, 7], [8, -5, 5], [-17, 1, -2], [-13, -6, 3], [-4, -11, 1], [1, 3, 12], [-18, 8, -2], [0, 9, -7], [-2, 0, 5], [4, 2, 0], [2, 4, -11], [-1, 2, -11], [11, 1, 4], [-6, 6, -12], [10, 4, 4], [-2, -6, 0], [-13, -5, -3], [2, 8, 2], [-9, -11, 10], [0, -2, -4], [-7, 5, 8], [-31, 2, 1], [0, -9, -4], [-8, -12, 6], [2, 3, -1], [7, -5, -4], [8, -2, -10], [8, 4, 9], [8, -4, 0], [-16, 2, 2], [-1, 17, -10], [-15, 5, 7], [12, -2, -10], [-18, 10, 2], [7, -4, 12], [-19, -7, 5], [4, 10, -4], [-22, -5, -4], [-2, -3, -8], [9, -2, 8], [10, 4, 4], [5, 20, -11], [4, 12, -4], [-2, -4, -12], [0, 6, -2], [-10, -4, -6], [-7, -2, 6], [-16, 0, 12], [-29, 8, 1], [-32, 2, -8], [9, -7, 7], [-12, -9, -2], [9, -17, 1], [-1, -14, -1], [22, 0, 4], [-28, 12, 3], [-27, -2, 2], [9, -9, -10], [5, 0, 5], [14, 18, -6], [32, -1, 2], [35, -5, 0], [0, -12, 2], [25, -8, -6], [1, -6, -6], [4, 13, 2], [4, -11, 10], [-10, 4, -16], [-2, 13, -4], [-6, -6, 10], [25, -4, -8], [6, -2, 16], [16, -8, -8], [-20, 1, -1], [-18, 0, -6], [-1, 14, -15], [4, -16, 0], [10, -9, 6], [6, 16, -18], [-26, 2, 6], [4, 4, 4], [-21, 13, 4], [-22, 2, -4], [-4, 11, 9], [18, -10, 12], [11, 17, -1], [23, -2, -4], [-10, -8, -6], [-16, -1, -4], [17, 0, -11], [6, 3, 12], [0, 2, 10], [-25, 10, 0], [-11, -5, -2], [-6, 12, -10], [13, 5, -2], [8, -22, -1], [2, 11, -1], [2, -10, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_637_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_637_2_a_h();".
// 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_637_2_a_h(:prec:=3)
    chi := MakeCharacter_637_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_637_2_a_h();".
// 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_637_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_637_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-6, -4, 2, 1]>,<3,R![-14, -2, 4, 1]>,<17,R![-14, -2, 4, 1]>],Snew);
    return Vf;
end function;