
// Make newform 8030.2.a.k in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8030_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_8030_2_a_k();".
// 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_8030_2_a_k();".
// 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 := [-3, -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_8030_a();"
function MakeCharacter_8030_a()
    N := 8030;
    order := 1;
    char_gens := [1607, 2191, 881];
    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_8030_a_Hecke(Kf)
    return MakeCharacter_8030_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, -1], [-1, 0], [-2, 2], [1, 0], [2, -1], [0, 0], [-1, 1], [2, 0], [2, -2], [-2, -2], [2, 0], [6, 0], [3, -1], [1, -5], [0, 4], [-6, 1], [-8, -3], [0, -3], [-1, 7], [1, 0], [-6, 2], [8, -5], [-6, -3], [-8, 4], [-8, 8], [4, -4], [-7, 5], [10, -4], [-15, 3], [10, -9], [16, -4], [6, 6], [2, -6], [-1, -3], [-16, 1], [6, 2], [2, 6], [-6, 0], [-10, 12], [-1, 1], [-22, 2], [8, 4], [-4, 10], [5, -3], [-10, 8], [-5, 9], [8, -6], [24, 2], [3, -7], [20, -4], [-8, -5], [-22, 0], [-12, 8], [-18, 2], [-14, -4], [-2, -2], [-8, 3], [23, 3], [-6, 3], [-18, 10], [0, 0], [-3, 1], [24, 1], [-11, 7], [-9, 15], [-10, -5], [8, 2], [-6, -6], [7, 5], [12, -8], [8, -4], [-14, 12], [-2, -12], [7, 9], [14, -12], [-18, 0], [-19, 1], [-14, 13], [-11, -11], [4, -12], [20, 7], [-4, 13], [-7, 7], [-6, -14], [20, -12], [-18, 0], [-7, -1], [-15, -9], [10, -6], [0, 0], [28, -6], [28, -8], [-42, 0], [-26, -4], [-3, -1], [14, 12], [-6, -6], [14, 12], [-30, 10], [-28, -2], [6, -2], [20, -4], [10, -16], [-8, -4], [-36, -5], [-4, -12], [-21, -7], [12, 2], [2, -9], [-13, -11], [-22, 3], [-8, -11], [-14, 10], [32, -4], [-34, 0], [24, 0], [-17, 25], [14, -9], [-5, 5], [24, -16], [12, -11], [1, -19], [-20, 14], [-14, 9], [-12, 16], [16, -7], [-6, 12], [-12, 22], [-22, 2], [0, -12], [12, -14], [-24, 8], [11, -1], [7, -1], [25, 1], [-24, -4], [2, -4], [-6, -13], [2, -10], [16, 4], [17, 11], [-29, 1], [-30, -4], [-2, -16], [0, -17], [-44, 6], [1, -13], [-15, 7], [26, 9], [-2, -2], [20, -10], [-30, 8], [46, -4], [50, 4], [-5, 7], [-11, 13], [-24, -14], [-17, -3], [-4, 21], [-28, 7], [20, -11], [-2, -13], [-5, -3], [16, 11], [15, -31], [-6, 20], [-16, 10], [58, -1], [-7, 9], [0, 0], [4, -22], [36, -6], [22, -16], [-16, 6], [-19, 29], [22, -22], [21, -5], [-32, 0], [-14, -4], [46, -2], [-16, 28], [6, -23], [-6, -16], [-20, 14], [-41, -3], [-34, 9], [6, 12], [-27, 1], [-40, 0], [20, 5], [44, 0], [-32, -10], [3, -3], [-36, 0], [4, 0], [-10, 4], [-4, -2], [-12, 15], [50, 3], [-24, -3], [8, -21], [-30, 17], [-6, 20], [22, 14], [34, 4], [33, -7], [-20, 17], [36, 2], [28, -14], [36, 0], [8, 17], [-22, -9], [-12, 12], [-15, 1], [-34, 4], [10, 18], [-2, -15], [-36, 20], [-6, 8], [20, -3], [-2, -22], [-48, -6], [-16, 32], [39, -21], [-24, -18], [18, -16], [-6, -14], [-20, 30], [66, -5], [-8, -10], [-5, 33], [-56, 15], [9, -19], [-5, -9], [9, -13], [-32, -18], [11, -33], [9, -21], [6, -8], [0, -6], [-26, -9], [51, -7], [8, 2], [6, -32], [-56, 13], [14, -31], [-32, 12], [22, 9], [-6, -3], [12, 0], [-30, -1], [40, -20], [14, 17], [-2, -8], [-48, 4], [-10, -16], [-44, 25], [-38, 24], [2, 18], [25, -1], [42, 12], [-12, 34], [-51, -7], [24, 6], [-34, 33], [32, -9], [-12, -14], [-44, 21], [0, 6], [-23, 27], [38, -10], [-18, 24], [12, -4], [12, 13], [9, -17], [-23, 1], [0, 11], [6, 22], [-12, -2], [66, -7], [11, -9], [4, -22], [12, 21], [17, 21], [12, -27], [-14, -24], [-14, 23], [-40, 17], [36, -6], [42, -12], [-37, 7], [-40, -2], [-2, 6], [20, -2], [-30, 20], [-84, 2], [21, -15], [-30, 41], [40, -20], [-66, -4], [69, 1], [-28, 7], [19, 11], [-64, -4], [38, -2], [-4, -2], [-25, 17], [0, 12], [-46, 8], [42, 11], [40, 6], [-10, -2], [58, 6], [60, -8], [-38, 16], [2, 0], [42, -14], [14, -22], [-16, 10], [10, 16], [-48, -9], [13, -21], [-58, -3], [-17, -31], [76, -8], [-30, 11], [-14, 0], [-18, -18], [14, -24], [36, -16], [-82, 2], [16, -48], [52, 0], [-22, 27], [20, -11], [58, -22], [4, 22], [-6, 11], [54, -15], [6, -34], [-16, -17], [6, 4], [5, -7], [-9, -5], [67, 1], [-23, 15], [-28, 10], [44, -29], [44, -26], [26, -23], [44, 20], [27, -3], [-52, 22], [70, 0], [1, -13], [-36, 2], [-3, 7], [-22, 20], [10, 15], [-18, 5], [30, 4], [50, -8], [-26, 19], [-70, -2], [16, -4], [-26, -17], [28, 10], [2, -1], [72, 8], [-14, -34], [-14, -7], [-41, 9], [-18, 16], [11, 3], [24, 8], [0, -28], [85, -5], [-32, 8], [28, 0], [40, -17], [-62, 20], [-38, 1], [30, -6], [-22, 18], [64, -10], [44, -18], [2, 9], [-88, -4], [-31, -5], [-46, 12], [-6, -25], [-68, -4], [-52, -8], [36, -22], [-55, 35], [-6, -12], [-50, -18], [24, 16], [-14, 20], [56, -3], [-5, 5], [22, -6], [-8, 16], [40, -4], [36, 4], [14, -7], [18, 30], [32, 2], [10, -17], [-6, -40], [54, 1], [-12, -25], [47, -3], [38, -36], [0, 9], [12, 16], [8, 0], [-10, 20], [-18, -2], [0, -15], [-18, -10], [60, -28], [18, -43], [57, -25], [-30, -19], [44, -40], [-42, 4], [-51, -13], [12, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8030_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8030_2_a_k();".
// 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_8030_2_a_k(:prec:=2)
    chi := MakeCharacter_8030_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_8030_2_a_k();".
// 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_8030_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8030_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-3, 1, 1]>,<7,R![-12, 2, 1]>],Snew);
    return Vf;
end function;