
// Make newform 8512.2.a.bu 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_bu();".
// 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_bu();".
// 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, 9, -8, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [1, -6, 0, 1], [-4, 0, 1, 0]];
    Rf_basisdens := [1, 1, 2, 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, -1, 0], [-1, 0, 0, 0], [1, -1, 1, 1], [0, -1, 0, -1], [1, 0, 0, -1], [1, 0, 0, 0], [-2, 1, 0, 1], [0, -2, -1, 0], [-5, 2, 0, 1], [-1, 1, -1, -2], [2, -2, -1, 0], [-1, 4, -3, 1], [0, -2, 1, -1], [3, 2, -1, 0], [1, -1, -1, 2], [2, -4, 1, -1], [3, -2, 0, -3], [-6, 2, 3, 1], [-3, 3, -4, 0], [-3, -4, -1, -1], [-3, 3, -6, 0], [0, -3, 3, 0], [-2, 0, -5, 1], [4, 1, 2, 1], [-2, -1, 4, 3], [0, 1, 4, -1], [0, 4, 3, 1], [-1, -7, 2, 0], [-10, 1, -3, -2], [-7, 2, 2, 3], [-13, -2, -1, 1], [2, -9, 2, -1], [6, -3, -2, 1], [5, 0, 0, -1], [3, -1, 3, -1], [7, -2, 11, 0], [-4, -1, -2, 3], [4, 0, 2, 4], [-7, 6, -2, 1], [-9, -2, 2, 3], [-1, -1, 0, 2], [-13, 6, -6, -1], [7, -5, 2, 0], [-9, 5, -1, 2], [-1, 1, -2, -4], [-6, 1, 0, -1], [-1, 4, 0, -3], [-5, -4, -3, 1], [-17, 7, -9, -3], [-12, -3, 0, 1], [-3, 4, -7, -1], [-9, -4, -6, 3], [2, 1, 9, 5], [7, 7, -4, 0], [1, 1, 6, 0], [2, 4, 3, -4], [4, 2, -2, -2], [0, -3, 2, -5], [-5, 5, -2, 4], [-2, 5, -4, -1], [4, 0, 7, 2], [-7, -5, 7, 1], [14, -6, 12, 4], [4, 1, 7, 6], [-1, 1, -6, 2], [5, 2, -6, -5], [-11, 5, -10, -6], [-21, 1, 2, 2], [-3, 6, 0, 1], [9, -2, 14, 1], [-10, -2, -11, 1], [4, -1, 1, -5], [0, 5, -4, 1], [-2, 1, 0, 1], [-13, 2, -8, 1], [-24, 9, -9, 0], [-7, 2, -12, 1], [21, 0, 7, -2], [-8, 7, -2, 3], [-4, -9, 10, -1], [-8, 1, -1, 2], [13, -3, 11, -2], [6, -2, 4, 6], [-22, 5, -7, 1], [-3, 1, 12, 0], [15, -6, -7, -2], [-7, 8, 1, 4], [-10, 1, 0, -1], [5, 5, 8, -2], [-4, -4, -1, -10], [-9, -5, -3, -2], [-4, 8, 0, 0], [2, -1, -7, -9], [-9, 0, -5, -5], [10, -14, 6, 0], [0, 9, -2, 5], [6, 1, 2, -7], [-26, -6, 2, 0], [5, -7, 12, 6], [7, -8, 6, 7], [-10, 8, -13, -3], [-4, 9, 0, 7], [2, -12, 9, -3], [13, -5, 10, 2], [14, 2, -2, -8], [-20, -5, 4, -7], [-23, 12, -5, 0], [21, 9, 2, -4], [0, -13, 0, -5], [11, -5, -6, -10], [8, -6, 5, 0], [15, 1, 4, 0], [10, -1, 10, 7], [1, 14, -14, 5], [-4, -9, 10, -3], [-25, 9, -21, -2], [-4, 11, 4, 3], [27, -4, 12, 5], [-2, 12, 0, 8], [-5, -5, 8, 0], [13, -7, 14, 8], [-8, -11, 6, 5], [4, 6, 10, -2], [6, -8, 10, 2], [0, 7, 8, 3], [12, 12, 0, 0], [-10, 6, -11, -7], [7, -6, 17, -5], [23, -9, 7, 2], [5, -13, 15, 0], [4, -3, 11, 3], [24, 6, -8, 6], [6, 3, -2, -7], [26, 0, -6, 2], [36, -2, 10, 2], [15, 0, 3, 11], [7, 4, -2, 5], [-9, -13, 5, 0], [-6, 9, -7, 0], [22, -2, -4, -2], [16, -12, 8, 6], [8, -1, 0, 1], [16, 4, -8, -8], [6, -10, -12, -6], [0, -11, 9, -1], [-15, 12, -8, -7], [-5, 4, -6, -3], [8, -3, 7, -9], [-34, 5, -19, -2], [3, -1, 8, -4], [-12, 2, -1, -5], [-8, -13, 10, -5], [8, 5, -8, -3], [-25, 12, -7, 1], [4, -5, 12, 1], [7, -13, 12, -8], [-3, -4, -4, 3], [-12, 8, -18, -2], [-1, 2, 17, 6], [13, -12, 16, -3], [29, -8, 12, -1], [2, 2, 21, 1], [-8, -2, 2, -2], [4, -5, 10, 5], [-14, 23, -3, 8], [12, -11, 13, 3], [-18, 5, 12, 7], [6, -6, -8, -14], [-3, 13, -20, 6], [-43, 10, -18, -3], [-17, 4, -6, -7], [-15, 17, 5, 4], [-8, 6, 0, -4], [-10, -6, 12, 6], [-12, 5, 5, 7], [-18, 7, -8, 3], [16, 9, -6, -3], [-18, -4, -7, 10], [15, -4, -6, -7], [28, -12, -11, -7], [37, 4, -9, 4], [17, -4, -11, -8], [-8, -9, -4, 1], [-2, -5, -5, -12], [5, -11, 21, -4], [-28, 9, -1, 1], [5, 5, -8, -6], [19, -10, 27, 2], [10, -18, 8, -6], [2, -16, 13, -2], [34, 4, 4, -4], [-22, -3, -4, -9], [-8, -5, -23, -6], [-12, 7, -15, 1], [-26, 8, -9, 3], [-12, -5, -4, 11], [-10, -13, -6, 3], [-17, 2, -13, 1], [-18, -2, -16, 2], [27, -9, 4, -6], [-18, 9, -4, 3], [16, 1, -11, -2], [-5, -10, 16, -5], [13, -5, -7, -1], [2, 3, -5, -4], [8, -10, -12, 0], [20, -4, 4, -2], [-15, 12, 7, 6], [-17, -10, 1, -7], [-25, 6, 13, 4], [18, -2, 18, -2], [5, 13, -9, -4], [-16, 8, -10, 6], [17, -3, 9, 10], [16, 17, 0, -5], [-17, -12, 1, 1], [-7, -10, -13, 1], [-4, 5, 2, 3], [-4, 0, 11, -3], [-43, 19, -32, -6], [20, 2, 16, -2], [32, 13, 2, -5], [-15, 9, -11, 1], [33, -7, -10, -4], [-25, 19, -11, 6], [-23, -3, -6, 2], [0, 4, 16, 12], [16, 12, -1, 8], [-47, 7, -8, -2], [-16, -11, -13, -4], [4, 2, -3, 5], [22, -3, 16, 11], [-27, 3, -24, -8], [-34, 9, -15, -1], [7, -4, 16, 1], [22, -16, 16, 0], [11, 1, 7, 8], [-19, 8, 2, -5], [-26, 22, -26, 2], [40, -13, 10, 5], [-10, 33, -14, 1], [0, 5, -7, 0], [41, 2, 16, -1], [-5, 7, -3, 9], [-13, 12, 1, 15], [15, 5, 14, -4], [24, 4, 22, -2], [20, -5, -14, -3], [16, -1, 6, -7], [-17, 22, -12, 1], [-11, -10, 22, -5], [9, 0, 0, -7], [2, -18, 12, 2], [17, -19, 5, -6], [-16, -3, -30, -1], [28, -19, -3, -6], [27, 15, -14, 10], [-18, -1, -6, 1], [-5, -14, 9, -1], [2, 14, -2, 12], [50, -20, 37, 7], [40, -6, -6, -2], [36, 2, 14, 0], [48, 1, -3, 2], [-34, 21, -26, -5], [-8, 19, -12, 13], [-12, -4, -8, -8], [-25, 5, 12, 6], [-21, 0, 20, -1], [-6, -16, 16, 2], [-27, 5, -20, -2], [-3, 15, -16, -6], [-4, -9, -1, 13], [25, -4, 1, -7], [-17, 4, -32, -3], [-15, -14, 13, 3], [30, 12, -4, 10], [-26, 1, -8, -15], [15, -1, 6, 4], [8, 15, -14, -7], [-22, -1, 9, 14], [14, -18, -18, -10], [30, 2, -12, 0], [32, 2, -6, 0], [-44, -10, 9, 1], [21, -33, 23, -2], [30, 4, -7, 3], [-2, 3, 2, 1], [-26, -13, 6, 11], [-15, 19, -2, 10], [-1, -22, 6, 3], [-11, 2, 17, 19], [13, 13, -6, -8], [-6, 9, 5, 13], [14, -7, 20, 1], [13, -12, -9, -5], [-37, 11, -35, -3], [-13, -5, 6, -10], [-4, 3, 0, -13], [14, -9, -6, -3], [-40, 18, -20, -4], [-7, 9, -28, -6], [-6, 8, 9, -6], [0, 11, -10, -3], [14, -25, -12, -7], [-1, 13, -6, -4], [8, -16, 20, -4], [19, 10, 28, -1], [11, 9, 27, 2], [-15, -8, 0, -15], [-10, -7, -26, -7], [-5, -1, -6, 2], [46, -14, 38, 8], [-25, -6, -3, 15], [19, 5, 14, 10], [1, -21, -1, -4], [51, -1, 6, 4], [16, -13, 13, 2], [-44, -6, -3, 7], [-26, -14, -1, 10], [47, 0, -8, -9], [-3, 28, -17, 4], [22, 4, 36, 4], [-30, 17, -17, 5], [13, 15, -12, 18], [6, 6, -13, 2], [-41, 8, -17, -10], [19, -19, 0, -14], [8, 6, 24, 4], [-11, 4, 17, -3], [17, -11, 13, 5], [-28, -23, 14, -9], [-5, -3, 9, 0], [-13, -16, 0, -7], [0, 26, -13, 11], [-13, -1, -17, 3], [20, -5, 22, 3], [-7, 20, -9, 2], [23, 5, 3, -5], [-33, 19, -13, -3], [2, -5, 2, 13], [0, -5, -26, 1], [-19, 27, 9, 6], [-33, 12, -22, 7], [46, -19, 37, 8], [-28, -4, 8, -6], [-4, -5, -6, 3], [-10, 19, -15, -13], [23, -7, -9, -14], [-34, 9, -14, -9], [-33, -5, 22, 2], [13, -22, 18, -5], [30, -6, 15, 14], [-35, 7, 21, 7], [4, -5, 15, 10], [28, -3, 34, 1], [-72, 1, -15, 0], [-21, 13, -19, 9], [-4, 25, 0, -1], [-33, -5, 2, -16], [20, -13, 2, -15], [-18, 16, -14, 0], [6, 3, 14, -7], [30, -2, 12, 6], [33, -21, -5, -8], [-4, 0, -23, 4], [16, -8, 0, -12], [22, -2, -3, -16], [61, 6, 3, 7], [-30, -12, -26, 2], [-36, -14, 0, -4], [5, -3, -14, 6], [27, -5, -2, -14], [-36, -3, -29, 4], [4, -9, 14, -5], [24, -8, 10, 2], [42, 9, 6, -9], [29, 11, -9, 12], [-34, 3, 2, 9], [46, -5, 27, 10], [14, -9, 12, -5], [39, -3, -9, 6], [26, -23, 25, -4], [-14, 0, -6, -10], [19, 16, -20, 3], [10, 15, -11, 7], [-20, 33, -22, 7], [-15, -23, 16, 2], [29, -2, -8, 3], [39, 17, 6, -12], [9, 3, 41, 4], [3, 4, -12, -7], [-55, 9, 17, 6], [-2, -3, -2, -23], [16, -11, 28, 7], [30, -9, 33, 18], [-40, 6, 8, 2], [-47, 7, -10, -4], [34, -21, -14, -11], [-52, -2, 9, 1], [-36, -2, -35, 5], [28, -6, 21, 11], [-23, -2, -4, -7], [30, -25, 25, 13], [-15, 2, 16, 13], [-2, -4, -8, -24], [-4, 38, -12, 10], [-11, 2, -4, -3], [-20, -16, 15, 15], [28, -7, -16, -3], [-41, 10, -20, 5], [1, -4, -17, -8], [-21, -18, 5, 5], [23, -1, 11, 3], [28, 1, -10, 11], [29, 9, -4, -2], [14, -11, -14, -13], [-69, 15, -20, -6], [14, 2, 35, -3], [26, 7, -10, -1], [-58, 23, -22, -3], [35, -15, 30, -4], [5, 7, -21, -11], [-44, -13, -4, 3], [11, 0, -10, 13], [-21, 3, 5, 4], [10, -15, 4, -17], [-2, -12, 0, -14]];
    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_bu();".
// 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_bu(:prec:=4)
    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_bu();".
// 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_bu( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8512_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-3, -1, 1]>,<5,R![3, -9, -8, 1, 1]>,<11,R![79, 28, -27, -2, 1]>,<23,R![-36, -54, -9, 6, 1]>],Snew);
    return Vf;
end function;