
// Make newform 671.2.a.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_671_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_671_2_a_a();".
// 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_671_2_a_a();".
// 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, 3, -1, -5, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [2, 0, -5, 0, 1], [-2, 1, 5, 0, -1], [-3, -3, 5, 1, -1], [3, 3, -9, -1, 2]];
    Rf_basisdens := [1, 1, 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_671_a();"
function MakeCharacter_671_a()
    N := 671;
    order := 1;
    char_gens := [123, 551];
    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_671_a_Hecke(Kf)
    return MakeCharacter_671_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 := [[0, 1, 0, 0, 0], [0, -1, 0, 0, 1], [0, 0, -1, 0, 0], [-1, 1, 2, 0, -1], [1, 0, 0, 0, 0], [-2, 1, 0, 0, -1], [-1, 0, 1, 0, 0], [-3, 2, 1, -1, -2], [0, -1, -2, -1, -1], [-3, 1, 4, -3, -1], [-1, 1, -2, 2, 1], [0, -3, -3, 1, 3], [-1, -1, -1, 2, 2], [2, -3, -4, 2, 3], [-1, -3, -1, 0, 1], [1, -3, -1, 5, 3], [-2, 2, 0, -3, 0], [1, 0, 0, 0, 0], [-1, 4, 0, 0, -4], [1, 1, 2, 2, 2], [-2, -2, 2, 4, 1], [-5, -2, -3, 1, 0], [-1, 4, 4, 3, -1], [-4, 1, 5, 2, 0], [-1, 6, 4, -7, -2], [-2, 0, -1, -3, -4], [5, -8, -10, 0, 4], [-1, -1, -4, 5, -1], [-4, 4, 5, 5, 1], [1, -7, -6, -5, 4], [5, 5, 0, -6, 3], [0, 6, 2, 0, -7], [1, 10, 4, -6, -3], [4, -9, -3, -4, 6], [-3, -9, -4, 6, 9], [-4, 1, 2, -8, -3], [-5, -11, -1, 1, 1], [0, 6, 1, -7, -1], [-3, 1, 8, -3, 0], [8, -3, -7, -4, 0], [-6, -2, 5, -1, 0], [-2, 4, 2, 3, 5], [-5, 4, -1, 9, -4], [-6, 3, 4, 4, 4], [-1, 2, 6, -2, -10], [5, -2, 8, 1, 3], [-11, -6, 2, 0, 4], [-7, -5, -1, 5, 0], [1, -3, 2, 1, 0], [-6, -5, 7, -4, -5], [-8, -11, -5, 0, 0], [4, -1, -2, -7, 4], [-3, 3, -9, -2, -1], [13, -8, -1, 5, 6], [5, 7, -3, 8, -5], [-2, 15, 11, 2, -7], [-1, 0, 1, 2, -2], [6, -11, -13, 3, 11], [-13, 3, 9, -5, -2], [-1, -15, -17, 3, 1], [-11, -6, -9, 4, -4], [2, -9, -10, 3, 5], [-1, -2, 1, 2, -5], [-2, -1, -4, 0, -6], [0, -14, -16, 4, -1], [3, -2, 5, 6, -1], [-10, 13, 13, 1, -6], [-6, -5, 2, -5, -12], [0, -4, 1, 2, 10], [-13, 2, 11, -9, -5], [12, 5, 4, -10, -2], [6, -12, -8, 4, 10], [9, 5, 0, -4, -7], [10, 8, 2, -14, -3], [2, -1, -4, 18, -1], [15, 9, 6, -8, 6], [16, 24, 6, -4, -5], [0, 9, 0, -8, 0], [-7, -6, -2, 12, 7], [-18, -11, -2, 7, 9], [-8, 3, -1, -11, -2], [-5, 16, -1, -3, -9], [-7, 14, 9, -17, -9], [7, 3, -2, -6, 2], [5, -2, 4, -9, 6], [6, 2, 2, 6, -2], [-2, 16, 5, -2, -5], [-9, 10, -1, -3, -5], [8, 16, 1, -2, -1], [-6, -7, -3, 10, 4], [9, -6, -2, -9, -9], [-4, -22, -4, 9, 7], [-2, -5, 0, 7, -8], [-11, 5, 6, -8, -11], [-1, 16, 2, -10, -5], [-3, -5, -7, 18, -2], [18, -15, -16, 13, 6], [12, 8, -3, -2, -3], [11, 15, 13, -14, 4], [-10, -5, -2, -11, 4], [-13, -5, 4, -4, -4], [13, 19, 2, -3, 2], [2, -9, -1, 4, -7], [-6, 21, 27, -7, -12], [7, 4, 5, 4, -2], [2, 8, 2, -11, -4], [13, 8, -17, 3, -5], [-5, 10, 13, -2, -6], [-5, -8, -2, 5, 16], [-8, -5, -6, -2, -5], [12, -5, -9, 2, 8], [-1, -13, -6, 17, -4], [-6, 6, 14, 6, 6], [4, 13, 19, -3, -15], [8, -19, -24, 9, 0], [7, 4, -16, 8, 4], [5, -1, 1, 7, 11], [6, 6, 8, 6, -3], [5, -2, -6, -11, -4], [-11, -10, -11, 15, -10], [13, 16, 7, -2, 7], [-18, 7, 4, 13, -12], [5, 26, 13, -5, -1], [8, 14, 4, -13, -4], [18, -11, -12, 11, 1], [-11, 19, 18, -15, -13], [-17, 18, 14, -10, -29], [15, 11, 2, 3, 0], [-1, 17, 2, 9, 14], [13, 15, 18, 7, -11], [18, 16, 13, -15, 11], [-3, -6, 0, 1, 6], [4, -4, -7, -12, 2], [-10, 22, 24, -11, -13], [17, -7, -8, -10, 19], [-7, 10, 0, 0, -14], [12, -13, 5, -4, 9], [30, -2, -8, 8, 1], [-20, 12, 25, -3, 3], [-15, 16, -8, -18, 4], [-4, 2, 13, 2, 6], [-10, -3, 3, 13, 5], [2, 25, 26, -13, -4], [10, 17, 4, -17, 1], [-7, -1, 2, 8, -1], [14, 3, 13, -8, 2], [-23, 4, -1, 7, -6], [-12, -1, -3, -9, -14], [20, -21, -27, 11, 4], [-4, -20, -4, -15, 7], [10, 10, 0, -11, 8], [-17, -32, -10, -5, 10], [1, -7, 8, 9, 0], [12, -1, -19, 11, 16], [-1, -18, -2, -5, 8], [-22, -17, 1, -5, -7], [-23, 6, 19, 11, 8], [-9, -20, -23, 11, -11], [4, -12, -1, 12, 9], [-1, -8, 10, 10, 1], [24, -22, -34, 11, 9], [-21, -17, -7, 3, 10], [-14, -11, 11, -15, -2], [6, 20, 9, 1, 16], [15, -12, 3, 16, 7], [26, 5, -2, -1, 22], [-11, 22, 4, -13, -5], [30, -5, -15, 7, 6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_671_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_671_2_a_a();".
// 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_671_2_a_a(:prec:=5)
    chi := MakeCharacter_671_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_671_2_a_a();".
// 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_671_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_671_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 2, -4, -3, 2, 1]>],Snew);
    return Vf;
end function;