
// Make newform 1040.2.a.h in Magma, downloaded from the LMFDB on 22 December 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1040_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_1040_2_a_h();".
// 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_1040_2_a_h();".
// 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, 0, 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_1040_a();"
function MakeCharacter_1040_a()
    N := 1040;
    order := 1;
    char_gens := [911, 261, 417, 561];
    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_1040_a_Hecke(Kf)
    return MakeCharacter_1040_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], [-1, 1], [-1, 0], [-2, 0], [3, -1], [1, 0], [0, 2], [1, -3], [-3, -1], [-6, -2], [-5, 3], [-4, 0], [0, -2], [-5, -3], [-6, 0], [0, -6], [3, 7], [2, 6], [4, 6], [-3, 1], [-4, 0], [-2, -6], [6, 0], [-6, 4], [2, 0], [6, 4], [-5, -3], [-9, 5], [2, 0], [12, 2], [-11, 3], [0, 0], [-6, -4], [-2, 6], [6, 8], [7, 3], [-10, 0], [4, -6], [-6, -4], [12, 2], [12, -4], [-10, -6], [12, 4], [-10, 0], [-6, 4], [-20, 0], [-8, 0], [-2, 0], [0, -2], [-4, -6], [-6, 0], [9, -3], [8, 6], [18, -2], [6, -8], [-3, -1], [-6, 8], [-5, 15], [-16, 6], [12, -6], [-17, 9], [-12, -4], [-2, -12], [6, -6], [-4, 6], [24, 0], [13, 9], [-16, 6], [-3, -5], [-4, 6], [0, 16], [-3, -3], [-17, -3], [-10, 0], [7, 15], [-6, 4], [6, 0], [8, -12], [-30, 4], [-28, -6], [6, 2], [-10, 12], [3, -13], [14, -12], [-32, 0], [-9, -15], [24, 2], [-10, -12], [0, 2], [-8, -6], [-33, -3], [-27, 5], [16, -6], [6, 2], [13, -15], [3, 13], [36, -2], [-18, 6], [19, -3], [-16, 6], [7, -3], [-12, -8], [3, -11], [-6, -2], [-26, 6], [-4, 0], [12, 2], [6, -12], [18, 6], [2, -24], [1, 15], [26, 0], [6, -16], [19, 15], [-11, -3], [6, -4], [-14, -12], [-3, 11], [-6, -8], [18, -2], [-10, -12], [-4, 6], [0, -6], [-36, -2], [1, -27], [42, 0], [20, -6], [12, -8], [-29, 9], [38, 0], [13, -3], [30, 8], [-26, 6], [8, 6], [-6, 8], [14, -12], [0, -4], [10, 24], [-6, 20], [-6, -2], [-23, -3], [-6, -20], [31, 9], [-18, 0], [-10, -6], [-9, -5], [8, 0], [6, 24], [-14, -18], [18, -12], [2, 0], [-30, 10], [-29, -3], [15, -19], [7, -27], [36, 0], [22, 18], [48, 2], [14, -12], [0, -18], [-6, 20], [-6, -4], [40, -6], [-12, 4], [12, 16], [24, -6], [-8, 0], [44, 6], [-34, 12], [-36, 6], [15, 15], [8, 6], [-6, -14], [-16, -24], [-20, -12], [18, -16], [25, -15], [-24, -6], [-23, -21], [14, 0], [-14, 0], [-6, -6], [32, -18], [18, -12], [24, -10], [-18, 14], [2, 36], [22, 0], [8, -18], [-21, 7], [-4, -30], [-30, 4], [-20, 12], [42, 4], [-39, 7], [-42, 4], [-4, 18], [2, 12], [36, -12], [3, 21], [-12, 14], [16, -12], [26, 0], [2, 30], [-51, 5], [-24, -2], [-11, 9], [-57, 1], [-6, 16], [-14, -18], [14, -12], [-30, 6], [19, 21], [6, 4], [-9, -17], [-28, 6], [-41, -3], [6, -16], [-12, 2], [6, 24], [26, -18], [-41, 3], [-30, 12], [-38, 0], [15, 5], [14, -6], [30, -8], [6, -2], [43, -15], [9, -15], [-10, 12], [-2, -6], [1, -3], [-30, 12], [-17, -15], [36, -18], [-28, -6], [-24, 16], [24, 16], [-36, -4], [-6, 8], [34, 18], [-29, 21], [-4, -6], [-18, -16], [36, 12], [-50, -12], [-33, -13], [7, 15], [-9, -7], [-40, -12], [-30, -8], [-6, -12], [26, 0], [-30, 0], [27, -17], [-46, 18], [-20, 6], [-6, -24], [32, 12], [25, 27], [-27, 11], [44, 6], [32, 18], [-12, -28], [10, -30], [0, -34], [6, 12], [64, 6], [0, -22], [14, -12], [4, -6], [38, 24], [4, 0], [-28, -18], [-26, -24], [0, -2], [-28, -30], [8, 18], [6, -2], [-21, 29], [1, 9], [9, -9], [-16, -18], [34, 0], [30, 30], [2, 0], [30, 12], [-23, 9], [18, 14], [-6, -2], [-9, 13], [6, 28], [-9, 19], [-70, 0], [6, -24], [16, -12], [-42, -8], [-18, 38], [22, 24], [-10, 36], [-42, -12], [-32, 12], [15, -39], [28, 24], [8, 12], [-9, -3], [-10, -12], [51, -1], [38, 12], [21, 7], [12, 10], [30, 6], [1, 3], [54, 16], [-22, 30], [21, 1], [-63, -3], [-40, 12], [-18, 40], [46, 18], [-4, 0], [18, 14], [4, 30], [36, -24], [-10, -6], [67, -9], [34, -12], [-51, -13], [30, 28], [-46, 24], [30, -12], [34, 6], [0, 22], [-17, 15], [-72, -6], [14, 0], [-30, -32], [8, 6], [1, 15], [2, -36], [-36, 2], [30, -24], [70, -6], [-30, -28], [-12, -12], [50, -24], [34, 12], [-21, -1], [-30, -16], [49, -3], [14, -24], [-18, -32], [-29, 21], [14, 0], [-12, -6], [-57, -5], [21, 1], [30, 24], [18, 0], [68, -12], [54, 10], [-21, 17], [15, -1], [73, 15], [20, 6], [6, 20], [-8, -30], [62, 12], [12, -44], [-32, 12], [-72, 6], [54, -18], [28, -24], [-28, 18], [-51, -15], [6, 10], [-34, 36], [6, 14], [20, -30], [48, 18], [0, 20], [-2, -24], [-24, -12], [37, 21], [48, 26], [43, 27], [-28, 18], [-2, 24], [21, -5], [56, -6], [12, -32], [-9, -45], [19, 9], [15, -29], [62, 0], [-11, -39], [18, -6], [-32, -12], [54, -8], [-58, 24], [-84, -2], [-38, 24], [-60, -12], [-30, -16], [-20, 36], [-52, 12], [54, 8], [40, -18], [-33, -13], [-34, 12], [-24, 10], [-39, -5], [-26, -6], [-10, 24], [-42, 0], [9, -11], [-35, -21], [-24, -8], [39, 1], [78, -14], [14, 24], [-12, -10], [-6, 6], [-34, 12], [-24, 8], [-33, 33], [-12, -30], [13, -27], [78, 10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1040_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1040_2_a_h();".
// 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_1040_2_a_h(:prec:=2)
    chi := MakeCharacter_1040_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_1040_2_a_h();".
// 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_1040_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1040_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 2, 1]>,<7,R![2, 1]>,<11,R![6, -6, 1]>],Snew);
    return Vf;
end function;