
// Make newform 5070.2.a.bc in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5070_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_5070_2_a_bc();".
// 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_5070_2_a_bc();".
// 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 := [-2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 2]];
    Rf_basisdens := [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_5070_a();"
function MakeCharacter_5070_a()
    N := 5070;
    order := 1;
    char_gens := [1691, 4057, 1861];
    v := [1, 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_5070_a_Hecke(Kf)
    return MakeCharacter_5070_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, 0], [1, 0], [-1, 0], [0, 1], [0, -2], [0, 0], [-2, 1], [0, -1], [0, -3], [-6, -1], [-4, 0], [6, 2], [2, 2], [4, 2], [8, 0], [2, 4], [8, -2], [6, 0], [0, -2], [0, 2], [6, 3], [8, -2], [-12, 2], [10, 2], [6, -1], [2, 5], [4, -2], [-4, 0], [-6, -1], [-2, 3], [4, 2], [8, -5], [6, 4], [12, -2], [2, -2], [-12, 0], [-6, 4], [0, -4], [8, -6], [2, -4], [16, -3], [-2, -2], [0, 4], [6, -3], [2, 4], [16, -2], [12, -4], [-8, -1], [-4, 0], [10, -5], [14, -3], [-16, 0], [6, -4], [8, 1], [-2, -1], [8, 3], [-6, -3], [4, -4], [26, 0], [-14, 2], [-12, -2], [-6, 8], [16, 2], [-24, 0], [-14, -6], [-14, -4], [0, 3], [-6, 6], [-4, -2], [2, -5], [6, -4], [0, 10], [20, 2], [-14, 4], [16, -3], [24, 2], [2, -3], [-2, -10], [10, 6], [-2, 2], [-8, -1], [10, 5], [16, 0], [-6, -6], [0, 8], [36, -2], [2, -10], [-18, -1], [-14, 0], [-16, 7], [-4, 4], [0, 4], [-8, -3], [-24, -5], [8, -3], [-32, 3], [26, 4], [10, 6], [12, -8], [-6, -3], [12, -4], [-22, 4], [-36, 2], [26, 2], [4, 6], [-2, 9], [20, -8], [6, -2], [40, 0], [18, 4], [12, 6], [6, -10], [-2, 0], [32, 1], [28, 2], [18, -8], [0, -14], [0, 3], [-14, -10], [-16, 3], [-6, 5], [10, 8], [18, -2], [-28, -2], [8, 7], [-14, 5], [10, -1], [-24, -2], [-12, -12], [6, -2], [0, 5], [8, 10], [0, 4], [-14, 12], [-22, -2], [14, 0], [42, 4], [24, 0], [-6, 8], [2, 4], [16, -5], [-6, -10], [-36, -6], [-4, -2], [-42, -4], [0, 0], [-10, -10], [-18, -1], [4, 12], [0, 10], [6, 16], [18, -10], [20, 4], [-32, -3], [-20, 10], [8, 6], [8, 12], [-6, 14], [-6, 8], [-38, -6], [-36, 2], [-42, 5], [16, -3], [24, -3], [14, -2], [8, -2], [8, -18], [50, 4], [6, 8], [10, 4], [-16, -4], [-6, 15], [-16, 2], [-10, 9], [8, 18], [34, 6], [16, 11], [-30, 10], [-4, -4], [-18, -10], [0, -11], [8, -13], [-30, -4], [30, -6], [-8, -10], [-14, 3], [-14, 16], [8, -2], [-34, -10], [8, 12], [-22, 0], [28, -12], [-12, -8], [-6, -14], [4, 6], [6, -5], [46, 2], [10, 0], [6, 6], [24, 11], [-14, 12], [-8, -8], [-10, -4], [10, -8], [-8, 6], [26, -6], [-52, -4], [28, -10], [2, 18], [-28, -4], [26, 2], [-46, 3], [-12, 0], [12, -10], [0, 4], [-42, -8], [12, 0], [10, 6], [-40, -2], [10, 12], [22, -2], [12, -2], [18, 14], [-32, 13], [-36, 6], [6, -10], [14, 1], [-16, 14], [36, -12], [-24, 14], [26, 0], [20, -12], [-20, -4], [-14, 2], [-12, 2], [40, 2], [-26, -6], [2, -16], [40, -11], [-24, -10], [-4, -6], [20, 0], [12, 8], [18, -7], [14, -14], [-40, -4], [-8, -1], [48, -4], [-40, 7], [-8, 11], [-42, 2], [-14, 6], [-16, -4], [-6, 8], [18, 0], [0, -4], [-34, -4], [0, -24], [-14, 0], [46, 3], [-12, 4], [-12, 0], [10, 7], [-38, 4], [22, -2], [-28, -6], [50, -4], [-22, 6], [-48, 10], [26, -14], [-34, 2], [-24, 16], [-10, 1], [16, -8], [-6, 8], [-72, -3], [4, 28], [2, -7], [-10, 14], [-16, 13], [24, -1], [12, 10], [8, 15], [2, 23], [-8, -2], [-8, 10], [2, 18], [10, 12], [68, 2], [-6, -20], [42, -5], [60, 0], [-10, -14], [8, -20], [10, 8], [-22, 9], [0, 22], [50, -6], [16, 3], [-32, -4], [2, 10], [-38, 0], [0, -26], [-4, 22], [-12, 20], [38, 5], [28, 16], [30, 2], [-8, 16], [-22, -4], [-64, -7], [2, 0], [-30, 2], [-28, -12], [0, -12], [74, 0], [8, 22], [-16, 10], [-2, 19], [50, -8], [-28, -18], [30, -9], [-30, 5], [24, -9], [70, -2], [10, -24], [32, -7], [-32, 2], [40, -5], [26, -4], [2, 5], [-46, -2], [32, 4], [20, 24], [32, -5], [20, 20], [26, -7], [22, 0], [-2, 26], [-36, -14], [14, 12], [-18, -13], [-54, 2], [8, -22], [-22, -12], [0, 19], [6, -12], [16, 26], [48, 6], [18, -2], [-48, -7], [62, -3], [-30, -14], [-44, 8], [-66, -6], [14, -7], [-32, 16], [-48, -2], [-34, 5], [24, -16], [30, 4], [42, 8], [24, 19], [-16, 8], [68, -6], [-22, -18], [-22, 20], [24, -13], [42, -16], [-16, 7], [68, 6], [-8, 2], [-38, -5], [8, -10], [-38, -20], [-32, 0], [0, 12], [46, 1], [-70, -6], [-6, -6], [10, -14], [-42, -12], [-32, -9], [-2, 10], [0, -35], [32, 8], [-76, 6], [18, 0], [-24, 14], [32, -17], [-26, 4], [34, 12], [-80, 2], [-4, -24], [80, 6], [-38, -22], [28, 4], [58, -12], [-4, -20], [-22, 4], [-6, 21], [54, -5], [8, 11], [62, -2], [2, -8], [8, -12], [38, -6], [-14, -26], [24, 0], [16, -20], [2, 10], [18, -20], [20, 18], [20, -18], [18, 16], [-62, -5], [40, -20], [20, -26], [-26, 2], [0, -13], [2, 11], [-66, -4], [-32, -16], [48, 13], [14, 7], [10, 0], [12, 16], [50, 10], [-32, 21], [-32, -8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5070_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5070_2_a_bc();".
// 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_5070_2_a_bc(:prec:=2)
    chi := MakeCharacter_5070_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(2999) - 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_5070_2_a_bc();".
// 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_5070_2_a_bc( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5070_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-8, 0, 1]>,<11,R![-32, 0, 1]>,<17,R![-4, 4, 1]>,<31,R![4, 1]>],Snew);
    return Vf;
end function;