
// Make newform 3072.2.a.i in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3072_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_3072_2_a_i();".
// 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_3072_2_a_i();".
// 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, -4, -6, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, -4, -1, 1], [3, 2, -1, 0], [-3, 4, 2, -1]];
    Rf_basisdens := [1, 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_3072_a();"
function MakeCharacter_3072_a()
    N := 3072;
    order := 1;
    char_gens := [2047, 2053, 1025];
    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_3072_a_Hecke(Kf)
    return MakeCharacter_3072_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 := [[0, 0, 0, 0], [-1, 0, 0, 0], [-1, 0, 0, -1], [1, 0, 1, 0], [0, 1, -1, -1], [-2, 0, 1, 1], [0, -1, -1, 1], [0, -1, -1, 1], [0, -2, 0, 0], [-3, 2, 0, 1], [3, 0, -1, 0], [-4, -2, -1, 1], [0, 3, -1, 1], [0, 1, 1, 3], [0, 2, 0, 0], [-5, -2, 0, -1], [0, 4, 0, 0], [-4, -2, 1, -1], [4, 2, 2, 2], [-2, 0, -2, 0], [-2, 4, 2, 0], [3, -4, -1, -4], [0, -5, 1, -3], [-2, 6, -2, -2], [0, -6, -2, -2], [-5, 0, 2, 1], [1, 0, 1, 4], [0, 4, 4, 0], [-6, 0, -1, -1], [-2, -4, 0, -4], [3, -4, -1, 0], [-4, -2, 2, 2], [0, 3, -1, -3], [4, -6, -2, -2], [-7, 4, 0, 1], [1, -8, 1, 4], [-8, 4, -3, -1], [0, 1, -3, -1], [-10, -2, 2, -4], [-9, -6, 2, 1], [0, -2, -2, 6], [-6, -2, -3, 1], [-10, -6, -2, 0], [-4, 0, 6, 4], [-9, -2, 0, 3], [7, 4, -1, -4], [4, 10, -2, -2], [-3, 8, 1, -4], [0, -5, 1, -3], [2, -6, 3, -5], [-2, 2, 6, 2], [-14, 10, -2, -4], [-4, 2, -4, 2], [0, 3, -3, 5], [10, -2, -2, -2], [-16, 4, 4, 4], [5, 8, 6, -1], [13, -8, 1, 4], [-2, -2, -5, -1], [10, -2, 2, -2], [-12, -4, 0, 0], [-1, 6, -8, -5], [4, 2, 2, -6], [-4, 12, 0, 4], [-4, -4, -6, 0], [-1, -8, 6, 1], [-12, -6, 6, -2], [10, 0, 2, -4], [16, 9, 3, -1], [8, -2, 7, 5], [-2, 6, 2, 6], [-6, -4, -6, 8], [7, 12, 3, 0], [-4, -4, 5, -5], [0, 3, -1, -7], [-10, -10, 6, 0], [11, -10, -2, 1], [8, -4, 1, 3], [0, 5, 5, -5], [4, -2, 6, 2], [0, -7, 3, 7], [2, -6, -1, 7], [-4, 10, 8, 4], [4, 8, 0, 0], [11, 12, -5, 4], [0, 13, -1, -5], [-16, 1, 1, -5], [-16, -2, 2, 2], [5, 10, -2, 3], [-9, -12, 3, 0], [16, -9, 5, 9], [-6, 4, -2, -12], [19, -4, 3, 0], [-4, 0, -4, 0], [-12, 0, -4, -4], [-18, 12, -2, -8], [19, -2, -4, -5], [0, -17, -1, -3], [8, 15, -1, 1], [2, -10, -1, -9], [0, 7, 7, -3], [9, -8, 4, -3], [16, -13, -3, -3], [-16, 3, -1, -7], [4, -12, 0, 0], [-2, 16, 2, -4], [-20, 10, 2, -2], [-2, 12, -8, -4], [-12, -14, -4, 0], [-2, -4, -2, 0], [-1, 8, -5, 0], [0, -4, -3, -1], [-6, -4, -8, -4], [-12, 0, -8, 0], [15, 16, -1, 0], [8, 11, -1, -3], [-8, 7, -5, -7], [12, 2, -4, -8], [-3, 10, -6, -1], [-20, 4, 0, -4], [16, 0, -1, -3], [-12, 12, -4, -4], [9, 20, 4, -3], [16, 7, -3, 1], [8, 5, -7, 3], [-3, -4, -2, 7], [18, 6, -7, 5], [-10, -8, 10, 4], [-1, -8, 7, -4], [10, -10, -3, 9], [4, -4, 0, 8], [-22, -2, -6, 0], [-3, 4, 1, -12], [-2, 10, -3, -11], [8, -11, 5, -1], [12, -8, 2, 12], [7, -24, -2, 1], [-8, 9, 5, 3], [3, 8, 12, 7], [-8, 1, 5, 11], [8, -3, 9, -1], [-7, -4, 6, -5], [-25, 12, 7, 8], [16, -6, -2, 2], [18, 16, 7, -1], [-14, 4, -2, 12], [-16, -12, -7, -5], [-8, -7, -3, 7], [32, -9, 3, 1], [-6, 10, 10, 8], [-8, 22, -7, -5], [2, 2, 10, 10], [-8, -11, 5, 7], [8, 4, -8, 0], [32, -13, -1, -3], [2, -6, -2, -12], [19, 0, 3, 8], [8, -1, -1, -3], [10, 0, 0, 16], [-11, -14, 2, 7], [8, -2, 6, -10], [-8, -9, -1, -3], [-23, 8, 1, 0], [-8, -11, -1, -5], [-16, 3, 3, 9], [-12, 0, -8, 4], [5, -4, -7, -12], [4, -6, -1, -15], [-12, -10, -2, -6], [-5, 2, -2, 9], [-20, -16, 4, -8], [-12, 12, 7, 17], [10, -6, -2, -4], [22, -12, -4, 4], [-3, -8, -7, 4], [26, 18, -2, 2], [-28, -8, 4, 4], [-11, 20, -4, -7], [15, 8, -9, 4], [16, -6, -1, 13], [-5, -12, -1, -4], [24, 15, 5, -7], [-18, 4, 11, 7], [10, -6, -2, -14], [-14, 2, 2, 0], [11, 10, -14, -3], [-6, -2, -5, 11], [-4, -20, 0, -8], [-4, 34, 0, -6], [24, 8, 8, -8], [-6, 4, 0, -20], [24, -16, -4, -12], [-28, -16, 4, 4], [3, 10, -6, -3], [-4, 2, -2, -2], [18, -16, -4, -16], [-2, 4, -10, 0], [-42, 6, -5, -1], [-14, -10, 2, 6], [8, -4, 0, -16], [13, 16, 12, 1], [1, 4, -3, 8], [-2, -4, 7, 3], [10, -4, 6, 8], [-8, -3, 3, 19], [5, -18, 4, -11], [21, -4, 1, 4], [-32, 7, 1, 9], [-18, 20, 8, 4], [8, -1, -9, -11], [-12, -12, 6, -8], [-23, 4, 4, -3], [-25, 8, -1, -12], [-32, 0, -4, -4], [30, -8, 6, 8], [-16, 8, 4, 8], [-25, -8, -5, 0], [-8, -3, 9, -9], [38, 14, -2, 0], [-35, 4, 4, -7], [-42, 0, -5, 7], [3, -12, 3, 8], [-8, -35, 1, 3], [11, 8, 7, -20], [-32, -1, -3, 9], [-4, -4, 17, 7], [6, 0, 8, -8], [2, -4, 2, 8], [-15, -4, 1, 8], [20, -8, 4, 8], [-10, 24, -1, 3], [8, -33, -1, -11], [-17, 20, 3, 8], [18, 14, 18, 6], [-16, -1, 7, -19], [50, 0, -2, 4], [-8, -18, 10, -6], [-51, 8, 0, 5], [16, -17, -3, 17], [-8, 10, -8, -8], [-32, 3, 5, -3], [-20, 22, -2, -10], [-13, -4, -5, -4], [-8, -4, -9, 13], [26, 0, 4, 16], [10, 12, 10, -8], [21, 8, 9, -12], [0, 4, -4, 0], [-4, -10, 6, 6], [20, -4, -4, -16], [-10, -8, -3, -11], [-2, 8, 12, 16], [24, 10, -8, 8], [6, 20, -8, -4], [-17, 18, 0, 3], [16, 22, 10, -2], [-18, 4, -11, -11], [28, 0, 0, -16], [-19, 10, -16, -7], [-4, 8, 0, 16], [-15, 8, -11, -8], [-8, 0, -4, 4], [2, 4, -11, -15], [-48, 0, 3, -7], [8, 3, -1, 1], [4, -6, 10, -14], [11, -24, -8, -5], [-10, -8, -4, 16], [0, -15, -3, -13], [-35, -20, 4, -7], [-20, -18, 15, -7], [28, -12, -4, 4], [26, -12, 0, -4], [-15, -8, 5, 8], [6, -8, -8, 8], [11, -12, -5, 0], [-24, -7, -1, 7], [40, 12, 1, 3], [14, 8, 2, -20], [-8, 22, 10, 6], [-6, -12, -14, -16], [-21, 12, 3, 20], [16, -22, -8, -8], [-16, 24, 1, -5], [-40, -7, -3, 3], [24, 20, -16, 8], [20, 10, -4, 18], [-13, -8, 0, 11], [1, 16, -7, 8], [-10, -22, -10, -6], [-11, -10, 2, 15], [8, -10, -18, -2], [30, 8, -8, 0], [48, -2, -2, 6], [-24, 18, 1, 11], [37, 6, -10, 11], [-9, -16, -5, -4], [-21, -8, -10, -11], [8, -21, 5, -3], [8, 13, -23, -9], [-36, -28, -10, 8], [-25, 2, -2, 5], [37, 16, -15, 4], [44, 24, 0, 0], [8, -11, -3, -1], [28, 2, 4, 2], [0, -9, -7, -7], [6, 10, 9, -15], [-14, -24, 10, 0], [-32, 20, -1, 5], [2, 26, 6, 4], [39, 20, 6, 1], [32, -13, 7, 5], [-8, 3, 7, 17], [24, -8, 8, 16], [2, 16, -4, 8], [28, 18, -14, 14], [10, 36, -2, -4], [18, -12, 14, 0], [-26, -2, 6, 14], [-4, 4, 24, 8], [12, 4, -10, -8], [9, 30, 0, -3], [11, -16, -9, 16], [-16, -31, -3, -1], [20, 22, -2, -14], [-8, -41, 3, -3], [-8, 5, -11, 7], [44, 4, 8, -12], [-15, 14, 14, -1], [-26, -12, -7, -7], [9, -36, -8, -7], [-23, 4, 5, 4], [16, 4, -4, 8], [-4, 22, -6, 10], [16, 15, 9, -15], [48, -8, 3, 9], [-50, 20, 4, 12], [10, 0, -14, 20], [7, -24, -13, -12], [-20, -8, -1, -15], [0, 1, 1, -25], [13, -2, 4, 21], [-23, -4, 1, 0], [-13, -2, 0, -17], [8, 47, 1, 1], [22, -4, 3, 7], [-20, 2, -14, 10], [-26, -20, -2, 8], [5, 14, 0, 1], [64, 5, -7, -1], [12, -10, 6, 2], [-7, 18, -4, -15], [21, 12, 1, 4], [-12, 36, 3, 5], [18, -22, 10, 18], [14, 6, -2, 16], [-28, -22, 2, -2], [24, -9, -13, -15], [16, -24, -4, 4], [-8, 22, 15, 5], [-34, -30, 18, -6], [-2, 18, -6, -20], [-16, -17, 1, 9], [-12, -16, -16, 8], [-14, -16, -2, -4], [7, -8, 6, 1], [13, 16, -3, 8], [0, -8, -12, 0], [40, 11, 13, 5], [0, 11, 7, 1], [20, -20, 16, 4], [35, -2, 4, 11], [-17, -8, 7, -4], [-6, -6, -7, 13], [-8, -5, 29, 13], [-18, -34, 6, 0], [2, 20, 0, -4], [-8, 3, 23, 17], [-4, -26, 12, 6], [27, -10, -6, -3], [40, -11, 13, 15], [17, 40, -7, 12], [-14, 36, 4, 4], [-32, 11, -1, 13], [-14, -2, 22, 4], [29, -28, -7, -12], [18, -10, -5, 11], [20, 18, 14, 6], [28, -14, -4, -8], [-6, -4, 8, 4], [-9, -10, -2, 5], [4, 2, -10, -18], [-8, 3, 3, 13], [18, -6, -26, -12], [-48, -10, -6, -6], [41, -16, -3, -8], [32, 25, 1, -1], [48, -5, -5, -3], [25, -14, -2, 7], [48, 8, -1, -11], [8, 15, 19, -3], [-45, 4, -1, 12], [-16, -29, -9, 9], [53, 8, -10, -1], [37, 24, 13, -12], [10, 0, -11, -7], [-8, -11, 5, 11], [-32, 19, -13, -19], [-40, 7, 1, -7], [-14, 40, -2, 12], [-25, -4, -16, 15], [-24, -13, 21, -3], [0, -15, 17, -5], [-22, -12, -18, -8], [23, -28, -10, -15], [10, -10, -10, 4], [5, 16, 5, 16], [-32, -21, -5, 1], [10, 8, 18, 8], [43, -8, 20, 15], [20, -8, 7, 17], [16, 8, 12, 20], [32, -13, -7, -11], [4, -12, 2, 16], [51, -4, 10, 1], [-16, 5, 7, 19], [6, -10, 6, 14], [-8, 31, -17, -11], [-8, -34, 8, -8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3072_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3072_2_a_i();".
// 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_3072_2_a_i(:prec:=4)
    chi := MakeCharacter_3072_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_3072_2_a_i();".
// 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_3072_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3072_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![8, -16, -4, 4, 1]>,<7,R![4, 24, -8, -4, 1]>,<11,R![32, -32, -24, 0, 1]>,<19,R![16, -64, -32, 0, 1]>],Snew);
    return Vf;
end function;