
// Make newform 2070.2.a.x in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2070_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_2070_2_a_x();".
// 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_2070_2_a_x();".
// 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 := [-5, -1, 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_2070_a();"
function MakeCharacter_2070_a()
    N := 2070;
    order := 1;
    char_gens := [461, 1657, 1891];
    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_2070_a_Hecke(Kf)
    return MakeCharacter_2070_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], [0, 0], [1, 0], [0, 1], [-1, -1], [4, -1], [1, 1], [4, -1], [-1, 0], [-4, 2], [2, 3], [-4, 0], [5, -1], [4, -4], [8, 2], [-6, 0], [10, -2], [5, -3], [4, -4], [-3, 3], [2, -6], [8, 0], [6, 0], [-8, 4], [1, 5], [-8, -2], [-1, 3], [-8, -2], [2, 3], [-6, 0], [2, -6], [2, -4], [-2, 1], [2, -6], [-10, -1], [-9, 1], [-4, 6], [14, 3], [-4, 8], [17, -1], [0, 6], [-6, 7], [-22, 2], [-10, 6], [-1, -7], [-2, 8], [-10, 0], [0, 4], [12, -6], [6, -8], [-2, -2], [8, -4], [-28, 0], [-28, -1], [-6, -6], [-13, -1], [0, -6], [2, -9], [10, -4], [0, 6], [-6, -8], [-6, 12], [-3, 7], [-12, 0], [-10, -3], [-8, 1], [-10, 6], [-14, -1], [-7, -1], [26, 0], [4, 4], [-2, 4], [-12, 4], [2, -6], [-1, 3], [24, 0], [1, -11], [-3, 7], [-6, -6], [-20, -1], [-28, -4], [2, -9], [-8, 10], [-18, 7], [6, 7], [38, -1], [0, -9], [-10, 0], [18, -6], [-10, 0], [2, -10], [6, 12], [10, 2], [0, 6], [12, 4], [-16, -1], [-24, 6], [-18, 0], [10, -4], [16, 8], [-10, 9], [-8, -8], [8, -4], [18, -12], [5, 9], [34, -4], [29, -1], [14, -10], [15, 9], [-20, -1], [-28, 0], [14, 0], [-3, -15], [0, 1], [10, -10], [-8, 4], [-4, -6], [-14, 4], [-16, -7], [16, 4], [-31, -3], [38, 0], [-14, 16], [6, -3], [32, -6], [-18, 3], [15, 7], [10, 7], [23, -9], [26, 0], [8, 0], [4, -5], [2, 6], [4, 8], [20, 5], [-26, 8], [-12, 12], [-14, 2], [22, 10], [-25, 5], [-20, -10], [-14, 4], [30, 4], [-2, -14], [2, -12], [28, -2], [28, -7], [2, -4], [-16, 6], [26, 2], [-23, 11], [-16, 8], [-26, 5], [-8, 4], [-4, -6], [-8, -2], [-48, 4], [6, 12], [50, 3], [-41, -5], [10, 7], [7, 7], [-22, 6], [-7, 5], [18, -9], [9, -3], [-35, -1], [-22, 12], [2, 18], [7, 13], [16, 4], [-12, -2], [32, -10], [37, -7], [11, -15], [-16, 2], [-2, -4], [-48, 0], [-29, 11], [6, 1], [38, 6], [37, -11], [-4, 15], [12, -6], [-7, -7], [18, 3], [36, 7], [36, 4], [-13, -15], [41, -1], [42, -2], [-8, 16], [20, 0], [40, 4], [10, 16], [-30, -3], [-16, -6], [14, -24], [42, 0], [16, -2], [-1, -13], [2, -21], [-19, 3], [-40, 12], [-8, 16], [2, -4], [-30, -8], [-9, -15], [28, -5], [4, 14], [24, -2], [46, 4], [36, 1], [16, -2], [0, -12], [-28, 18], [42, 10], [-6, 12], [52, -11], [-2, -8], [28, 2], [-2, 8], [7, -23], [-64, 3], [36, 0], [-36, 10], [26, -1], [-38, -11], [-28, 3], [52, -8], [36, 1], [29, -3], [-40, 0], [-3, -9], [-16, 6], [24, -12], [44, -6], [-32, 10], [44, -10], [-21, -3], [-32, -2], [-22, 12], [-6, 4], [-14, 2], [-52, 14], [14, -13], [-24, -14], [-12, -3], [-9, 7], [-13, -1], [-10, -18], [-2, -26], [-24, 21], [-52, 12], [-10, -4], [10, -14], [23, -9], [14, 18], [28, -11], [14, 0], [45, -11], [14, 8], [-16, 24], [-36, -2], [-14, 28], [29, 9], [7, 7], [-10, 20], [2, 6], [24, -9], [-52, 6], [8, 0], [-34, 9], [48, 4], [2, -12], [-26, 2], [67, 1], [0, -26], [38, 0], [20, -7], [-20, -20], [8, 0], [-15, 33], [10, -28], [-30, 13], [-3, -9], [14, 12], [14, -10], [17, -21], [-37, 17], [-14, -8], [-62, -2], [2, 2], [28, -5], [-9, -23], [-41, 19], [0, 10], [29, -1], [38, 8], [11, 21], [2, -3], [4, -2], [-50, 2], [-16, 32], [-4, -27], [-18, 10], [-29, 13], [18, 1], [36, 6], [-25, 9], [-4, -10], [-10, -7], [50, 2], [11, -15], [24, -15], [20, -30], [18, 12], [54, 3], [-26, -16], [29, -13], [-25, -9], [-11, 29], [14, -4], [-28, 6], [-27, 9], [-4, 24], [4, 11], [-38, -1], [-52, 11], [-64, 2], [38, 18], [-59, 7], [40, 8], [-42, 9], [-12, -8], [-42, 9], [-27, -11], [6, 24], [45, -5], [5, 9], [-18, 4], [-15, 21], [-20, -8], [24, -2], [-38, 16], [32, -4], [10, -4], [14, -21], [-38, -8], [20, -10], [40, -22], [38, -24], [10, -2], [-15, -23], [-26, 8], [12, -18], [-14, -14], [16, -23], [-62, 4], [-66, 12], [-42, -2], [-18, -24], [-18, 18], [-59, 13], [29, 3], [34, 20], [59, -19], [8, -24], [-20, 14], [22, -32], [10, -22], [62, -4], [53, -19], [-28, 36], [-8, 17], [-16, 2], [-14, 10], [-19, -3], [-4, 26], [16, -4], [-11, -11], [29, -7], [-46, 6], [14, -4], [8, -36], [-4, 20], [-26, 23], [-52, 15], [-58, 24], [-40, -13], [18, -20], [60, -9], [20, -4], [52, -4], [54, 12], [30, 7], [2, 0], [-20, -20], [-12, 1], [6, 12], [2, -6], [42, -6], [-52, 0], [74, -1], [-8, 16], [29, -9], [16, 2], [0, -3], [-88, 6], [-44, 28], [-6, -2], [-5, 1], [44, -16], [5, -15], [22, -25], [-6, -12], [22, 7], [-4, -6], [1, -23], [8, -37], [45, -3], [44, 6], [56, 2], [-16, 26], [56, -18], [14, 8], [56, 8], [24, -15], [-10, -6], [-1, 5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2070_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2070_2_a_x();".
// 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_2070_2_a_x(:prec:=2)
    chi := MakeCharacter_2070_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_2070_2_a_x();".
// 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_2070_2_a_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2070_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-5, -1, 1]>,<11,R![-3, 3, 1]>,<13,R![7, -7, 1]>,<17,R![-3, -3, 1]>,<29,R![-12, 6, 1]>],Snew);
    return Vf;
end function;