
// Make newform 8512.2.a.bk in Magma, downloaded from the LMFDB on 12 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8512_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_8512_2_a_bk();".
// 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_8512_2_a_bk();".
// 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 := [12, -9, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-7, 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_8512_a();"
function MakeCharacter_8512_a()
    N := 8512;
    order := 1;
    char_gens := [5055, 6917, 7297, 3137];
    v := [1, 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_8512_a_Hecke(Kf)
    return MakeCharacter_8512_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]];
    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, 0, 0], [0, -1, 0], [-1, 0, -1], [-1, 0, 0], [3, -1, 1], [1, -2, 1], [1, -1, 1], [-1, 0, 0], [5, 0, -1], [1, -1, 1], [-1, -1, 1], [3, 0, 3], [-1, 3, -1], [-2, 0, -2], [-5, 2, 1], [-10, -1, 0], [7, -2, 1], [-9, 0, -1], [-1, 3, 1], [3, 0, -3], [6, 1, 0], [-2, 0, -2], [-4, 3, 4], [-6, 2, -4], [-3, -2, -3], [-11, -2, 1], [3, 2, 1], [-1, 4, -3], [7, 2, -1], [2, -1, -4], [14, -2, -2], [-3, 3, -1], [-6, -4, 0], [9, -2, -1], [3, 2, 3], [-5, 3, 1], [-1, 3, -5], [8, 5, 0], [3, -4, 1], [-6, 2, -4], [-5, 5, 5], [-7, -1, -3], [8, -1, 4], [-7, 3, 1], [6, 3, 0], [-13, -2, 1], [-10, -1, -2], [-7, 4, 3], [1, -3, -5], [6, -4, -4], [-1, -3, 3], [-9, 4, -3], [12, 2, 2], [-3, 1, -9], [2, 1, 0], [12, -3, 0], [-12, 3, 6], [-1, -3, -7], [4, -2, 6], [9, 6, -3], [-8, 1, 0], [17, 0, 1], [11, -5, 1], [-3, -3, 3], [-8, 2, 2], [2, 4, 0], [10, 5, 2], [3, -1, -9], [2, -9, -2], [-8, 7, -10], [-9, 3, 3], [13, -3, 11], [17, -4, 3], [6, -7, 8], [3, -10, 1], [13, 6, -1], [1, 3, 1], [2, 6, 0], [21, 3, -3], [6, -1, 4], [3, -6, 5], [-5, -8, 3], [2, -4, 2], [25, -4, 1], [-8, 8, 0], [-4, -1, 4], [6, 1, 0], [-14, 1, -4], [-6, 5, 4], [17, 2, 3], [0, -3, 0], [5, -5, 11], [-13, 8, -11], [4, 4, -8], [18, 3, -2], [32, 0, 0], [6, 0, 0], [-7, -4, -7], [3, 8, -3], [-14, 4, 0], [14, 7, -2], [-7, 1, -7], [13, 2, -5], [-5, 4, -5], [11, -6, -3], [12, 1, 2], [6, -8, 6], [-11, -2, 1], [-6, -3, -10], [-16, 11, -6], [-19, 2, 7], [-6, 5, -12], [5, 5, 5], [-6, -1, -2], [-19, 2, -9], [27, -3, 7], [-1, 4, 1], [27, 2, 1], [9, 8, 5], [9, -11, 7], [-18, 8, 8], [-26, -9, 0], [-16, 11, 2], [-23, 4, -5], [16, 2, -4], [16, -8, -6], [-7, 0, -3], [20, 0, 4], [-15, 4, 3], [4, 4, 6], [9, -8, -9], [11, -4, -3], [-18, -7, 2], [-14, -4, 8], [-7, 0, -11], [-24, -4, 2], [-16, 10, -10], [36, -4, 8], [-5, -13, 3], [23, 0, 7], [-10, -4, -10], [-32, 0, 2], [-4, -6, -4], [-7, 0, -5], [12, 4, -6], [-4, 4, 12], [10, -3, 4], [-25, 5, -1], [-3, -1, -1], [10, 5, -2], [2, 2, 0], [0, -11, 6], [-15, -4, -1], [19, -4, 9], [-17, 2, 5], [18, 0, 2], [21, 6, -1], [0, -3, -2], [9, 11, 1], [10, -4, 12], [-18, 13, -6], [-1, 5, 3], [21, -5, -13], [13, -6, 3], [8, -10, 2], [19, 10, 1], [12, 2, -8], [-2, 9, -8], [9, 4, 5], [-12, 0, -10], [-10, 3, -10], [-9, 5, -17], [17, 11, -1], [-1, 14, -9], [-2, -10, -6], [8, -4, 10], [-26, 1, -6], [23, -12, 7], [-11, -8, -9], [-15, 9, 17], [-11, 11, 11], [9, 4, 7], [30, 1, -8], [16, -5, 2], [-7, 10, 3], [-6, -4, -8], [-25, -12, -1], [22, -1, 2], [14, -5, 0], [24, -1, 4], [8, -4, -2], [-27, 9, -9], [2, 12, 6], [37, 8, -3], [-2, 12, 6], [38, -1, -4], [1, 10, 5], [-11, 0, 9], [-7, -2, 13], [6, -4, -6], [-24, 8, 2], [34, -7, 18], [-27, -4, 5], [4, 6, -6], [5, 5, -1], [23, -7, 11], [-10, 0, 2], [6, -6, -6], [32, -10, 10], [-26, 3, -6], [-24, -6, -6], [-58, -5, 0], [-34, 6, -6], [41, -6, 7], [-14, -4, -2], [3, -10, 7], [-9, -18, -3], [2, -12, 12], [4, -8, -8], [-15, 8, -3], [31, -6, -1], [-16, -3, 12], [-14, 16, 0], [-3, -4, 7], [-17, 5, 13], [-2, -3, 0], [3, -4, 7], [4, 13, -4], [32, 0, 0], [37, 9, -1], [30, 11, 0], [-42, 10, -2], [-21, 10, 13], [1, 8, -11], [38, -7, -6], [20, -15, 16], [15, 5, 3], [2, 12, -8], [-9, 4, 9], [17, 5, -1], [54, -6, 2], [3, -24, 5], [3, -6, 1], [-20, 4, -18], [5, 17, -7], [17, -13, 7], [42, -2, -2], [6, 3, 2], [6, -4, 10], [39, 0, -11], [-13, 12, -13], [-15, 3, -7], [-19, -5, 11], [11, -15, 19], [28, -16, 14], [23, -4, -7], [45, -4, -7], [-36, 2, 0], [-16, -13, 2], [45, -4, 1], [48, -8, 12], [22, 8, -6], [-5, 12, -13], [-28, 2, -14], [4, 8, 6], [20, 10, 8], [23, 2, 7], [17, -2, 13], [-16, -8, 0], [2, 1, 12], [17, 7, 5], [18, -2, 6], [-12, 1, 10], [-10, -17, 6], [28, -7, 6], [-44, 12, -12], [-33, -7, -3], [18, 20, 0], [14, -22, 4], [13, 20, 3], [-24, -7, 4], [-1, -20, 13], [20, 8, -4], [-38, 2, 12], [4, -6, 16], [16, 0, 8], [27, -16, -5], [-1, -2, 23], [-21, -12, 9], [-47, -2, 1], [-3, -18, 9], [-32, 5, -8], [-19, -13, 13], [-10, 12, -10], [52, -5, 2], [-4, -9, -6], [1, 6, 11], [6, -8, -12], [-45, 3, 9], [-2, 1, -6], [17, 24, -3], [-11, 18, -7], [14, 6, -14], [6, 3, -18], [17, 13, 3], [-53, 6, -5], [31, 20, -3], [-26, 7, -12], [24, -12, -16], [1, 5, -23], [15, 6, -11], [45, -11, -3], [31, 16, -1], [-32, -9, 12], [14, -4, -6], [0, -16, -10], [-20, -25, 4], [15, -8, 1], [-26, 15, 24], [-18, -12, -12], [3, 0, 5], [17, 25, -7], [-23, 7, -11], [30, 5, -6], [42, 8, 0], [-36, 19, -26], [28, -11, 8], [11, 25, 1], [4, -17, 4], [14, 1, 0], [-30, -2, 8], [8, -4, -22], [-75, 1, -5], [9, 20, -1], [-15, -2, 7], [3, -19, -15], [61, -10, 17], [13, 23, 1], [-37, -2, 11], [-20, -11, 0], [-13, 21, -17], [47, 13, 3], [5, 10, 21], [3, -18, 13], [29, -16, 9], [-1, -19, -7], [-10, 20, 4], [4, -6, 12], [11, 12, 9], [-14, -5, -4], [-25, 12, -15], [47, -2, 7], [-2, 1, -8], [31, 7, 1], [11, -3, 7], [-31, 21, -11], [24, -6, 0], [37, -18, -5], [40, 0, 2], [29, -11, 15], [-5, -22, -5], [22, -13, 24], [-21, 14, -11], [8, -10, 8], [-15, -28, 7], [-32, 0, 10], [-45, 6, 3], [13, 23, -9], [4, 4, 14], [-5, -15, 1], [12, 10, 16], [16, -20, -8], [18, 14, 0], [2, 23, -14], [-18, -11, 8], [30, 2, 14], [-73, 6, -11], [-18, 8, -12], [15, -16, -9], [39, -20, 3], [-31, 22, -7], [-14, 26, -16], [35, 16, -7], [39, 2, 7], [18, 4, 10], [-24, 8, 16], [-77, 3, -3], [-10, -9, 8], [25, -8, -9], [-50, 11, -10], [-23, 21, 1], [-18, 7, 4], [1, -8, 7], [-33, 13, 5], [51, -14, 13], [23, 28, -1], [31, -16, 29], [-8, -4, -2], [30, -20, 18], [-12, 23, 10], [25, -6, 17], [-45, -16, 3], [47, -10, -7], [19, -16, -9], [85, -1, 1], [22, -19, -2], [95, -1, 3], [30, -16, 20], [-2, 8, 6], [11, 11, -7], [35, -8, 15], [-9, 24, -9], [23, 19, -7], [36, -13, 20], [-72, 10, 2], [19, 11, -9], [13, 6, -5], [-14, 7, -6], [-1, 4, 15], [2, -15, -6], [31, -8, -17], [-35, -8, 5], [45, -12, 15], [26, 11, -10], [-5, 5, 3], [-47, 2, -15], [-5, -1, 13], [-1, 4, 15], [13, -2, 3], [10, -14, 6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8512_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8512_2_a_bk();".
// 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_8512_2_a_bk(:prec:=3)
    chi := MakeCharacter_8512_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_8512_2_a_bk();".
// 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_8512_2_a_bk( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8512_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-12, -9, 1, 1]>,<5,R![-6, -9, 2, 1]>,<11,R![36, 1, -7, 1]>,<23,R![-96, 75, -16, 1]>],Snew);
    return Vf;
end function;