
// Make newform 3549.2.a.v in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3549_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_3549_2_a_v();".
// 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_3549_2_a_v();".
// 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, 7, -7, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-4, 0, 1, 0], [1, -6, 0, 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_3549_a();"
function MakeCharacter_3549_a()
    N := 3549;
    order := 1;
    char_gens := [1184, 1522, 3382];
    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_3549_a_Hecke(Kf)
    return MakeCharacter_3549_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, -1, 0, 0], [-1, 0, 0, 0], [-1, 0, -1, 0], [1, 0, 0, 0], [-1, 1, 0, -1], [0, 0, 0, 0], [-2, -2, 0, 0], [-2, 1, -1, 1], [0, 3, 1, -1], [-2, 1, 1, 1], [0, -1, -1, -1], [2, -2, 0, 2], [-7, -1, 0, -1], [-4, 1, 1, 1], [-1, 2, -1, 2], [-2, 1, 1, -3], [-7, 1, 0, 1], [-2, 0, 0, 2], [-6, 4, 2, 0], [-1, 1, 0, -1], [8, -1, -1, 1], [0, -1, -1, 1], [-1, 2, -1, 2], [-1, 2, 1, 2], [10, 1, -1, -1], [2, -4, -2, -2], [-6, 0, 2, -2], [4, 4, 2, 0], [2, -4, 2, 0], [2, -1, -1, -1], [4, 6, -2, 0], [8, -2, 2, 2], [1, -3, 4, 1], [-2, 0, 2, -2], [1, 1, 4, 1], [-8, 4, 0, -4], [-2, -4, 0, 0], [-6, 0, 2, -4], [3, 4, 1, 4], [-10, 0, -2, 2], [-8, -1, -3, 3], [-6, 2, 2, 2], [0, 0, 2, 0], [2, 0, 2, 0], [-17, -1, 0, -1], [4, -2, 2, -2], [8, 3, 3, -3], [2, -1, 1, 3], [-3, 5, 4, -3], [-4, -4, -4, 2], [-2, 1, 1, -3], [3, 3, 2, -3], [12, -5, -1, -3], [0, -4, 4, -4], [6, 0, -2, 6], [-8, 1, -1, -3], [-6, -2, -4, 4], [-2, -2, 4, 2], [-2, 1, -3, -3], [-15, 3, 2, -1], [-8, 0, -4, 0], [9, -2, 3, -2], [0, -3, 1, -3], [0, 6, 2, 2], [-2, -4, 4, -2], [-9, 1, -6, 1], [8, -12, -4, 4], [4, -11, -5, 3], [-8, -4, -2, -4], [4, 7, -1, -3], [1, 3, 0, 3], [-9, -9, 2, 1], [6, -2, 4, 0], [8, -4, 2, -2], [-12, 0, 0, 0], [-11, -7, -4, 1], [-6, -4, -8, 0], [-2, -3, -5, -1], [3, 1, 2, -3], [2, 13, 3, -5], [4, -8, -4, 0], [-16, -2, -6, 6], [-7, -5, 4, -3], [14, -6, -6, -4], [-6, 0, 2, -2], [-4, -1, -3, 3], [-5, 1, -6, -3], [18, 8, -2, 0], [-5, -3, 0, 1], [-18, 0, -6, 4], [-4, -6, 6, -2], [13, -2, -3, 6], [8, -8, 4, 0], [16, -4, -6, 0], [0, -2, 6, -2], [-4, 4, 8, 4], [11, 0, 3, 0], [14, -8, 2, 6], [-10, 0, -6, 6], [4, -8, 0, 0], [8, -5, -5, -1], [-3, -7, -8, 1], [8, 6, -2, -2], [-2, 1, 1, 1], [0, 11, -1, 1], [-6, 6, -4, -8], [-17, 2, 3, -2], [-21, 12, -1, -8], [4, 11, 1, -1], [10, -6, -2, -4], [2, 4, 6, 2], [0, 6, -2, -6], [-15, 9, 4, -3], [-22, 6, 0, 6], [2, -10, 0, 2], [6, -5, 7, -1], [8, 8, 0, 4], [24, -6, -2, 6], [18, 4, -4, 4], [-8, -7, -5, 1], [8, -13, -1, 1], [14, 9, -3, 5], [-26, 6, -4, -8], [-21, -7, 0, 3], [-10, -5, -3, 7], [6, 3, -9, -1], [10, 6, -4, 2], [28, -6, -2, -2], [8, -2, 6, -2], [-4, -13, 7, 1], [-18, 6, 0, 2], [-9, 7, -6, 1], [-8, -3, -3, -3], [-4, -15, -5, 7], [1, 6, -5, 2], [-8, 1, -11, 7], [13, -5, 0, 7], [2, 9, 7, -11], [2, 2, 4, 4], [-18, -1, 3, 3], [0, -4, 0, 4], [-23, 15, 2, -1], [8, -8, 0, 8], [7, -1, 2, 1], [-2, 18, 2, -8], [-15, 9, -4, -3], [-18, 7, -3, -3], [10, 8, -6, 6], [-16, 2, -2, 10], [1, -9, 4, -3], [32, -2, 6, 2], [-26, -2, 0, 0], [-4, 2, -6, -8], [12, -2, -2, 2], [-8, -1, -9, -3], [-4, 5, -9, -3], [20, 6, 2, 0], [1, -6, -5, 10], [-2, 8, 4, -2], [-7, -4, 9, 0], [-15, 5, -6, 3], [-14, 15, 7, -1], [10, 2, 4, -2], [24, -4, 0, 4], [-1, 3, -8, 7], [-23, 2, 5, -2], [-12, 0, 8, -4], [-14, 6, 2, -14], [4, -8, -8, 0], [6, 8, 6, -2], [5, -11, -6, -9], [22, 10, 4, -4], [-12, 9, -9, -7], [12, 0, 0, 12], [-2, -10, -4, 0], [-14, 10, 0, -4], [-6, 2, -8, -2], [-7, -9, 2, 7], [-10, 6, -4, 8], [-18, -10, 2, -4], [2, 0, 14, -4], [28, 4, -4, 4], [28, 0, 2, -6], [-5, -12, -1, 8], [3, 19, 2, -3], [-10, 0, -2, 2], [-26, 7, 3, -5], [-24, 3, -5, -1], [4, 10, 2, 6], [-25, -12, -7, 8], [14, 10, -2, -2], [-13, -9, -2, 1], [-16, -7, 5, -11], [-31, 2, 7, 2], [0, 5, 7, -15], [22, 0, -6, 2], [14, -4, 2, 8], [-8, 1, 3, -1], [11, -14, -7, -10], [0, 6, 2, 14], [31, -11, -2, -3], [2, 4, -2, 2], [-12, -9, 3, 5], [6, -24, -4, 4], [13, 0, 11, 0], [6, 12, 2, -10], [-14, -3, 7, 1], [-8, 7, -11, -1], [27, 5, 2, 9], [8, 6, 10, 6], [-16, -9, -3, 5], [-18, 24, 10, -2], [4, 2, 6, 0], [-17, -9, -4, 3], [9, -20, -5, -4], [10, -1, 9, 9], [24, 7, -5, -3], [6, 0, 10, 2], [-13, 13, 4, -1], [21, -9, -2, -9], [14, -3, -7, 9], [-2, 10, 0, 14], [-9, -5, -12, 7], [20, -14, -6, -6], [-28, 8, 0, -4], [26, -13, 3, 7], [38, 10, 0, -4], [4, -15, 3, -7], [8, 2, -6, -10], [1, 1, -2, -1], [-6, 4, -8, 4], [10, -10, 4, 8], [-20, -2, 2, 6], [-6, -12, 4, -16], [-2, 0, 2, 6], [11, -22, -9, -2], [-24, 0, -4, -2], [-11, 3, 2, 3], [-24, -7, -1, 9], [20, 4, 0, 12], [-13, -17, -2, 1], [-28, -10, -10, 6], [-20, -2, -6, 2], [-6, 8, 2, 0], [-5, 6, -11, -6], [-24, -10, -6, 2], [-22, -1, 5, -1], [-3, -18, 9, 2], [-30, 10, 4, -2], [-4, -23, 3, 5], [-6, 0, -2, -16], [5, -17, 4, -5], [21, 3, -8, 5], [-18, 12, -4, -6], [-14, 4, -10, -10], [19, -1, 6, 9], [-8, 15, 5, 1], [22, 6, 8, 2], [2, -6, 4, 8], [8, -13, -1, -11], [0, -5, -5, 9], [-16, -11, 3, -7], [-10, 1, 3, -5], [2, 6, -14, -6], [11, -11, 8, -7], [12, 12, -12, 0], [7, 7, 0, -1], [5, -4, 13, -12], [14, -6, 20, 6], [26, 4, 12, -4], [-22, 8, 0, -2], [28, 24, -4, 0], [-4, 0, -16, 6], [20, -5, 7, -9], [6, -26, -6, 8], [-2, -22, -8, 14], [-35, -11, -10, 11], [-14, 0, -6, -8], [6, 8, 6, 8], [32, 6, -2, -6], [24, -6, 2, -2], [14, -18, 0, -14], [-4, 0, -4, 16], [-34, 9, -9, -13], [-38, 6, 0, -2], [-28, 7, 9, 7], [16, 7, -7, 13], [-7, -3, 0, -19], [-28, 16, 0, 4], [-26, 28, 10, -2], [-34, 9, 9, 9], [-44, 12, -4, -8], [31, 9, 6, -3], [-19, -14, -3, -2], [34, -19, 1, 1], [-10, 4, 10, -2], [6, -8, -2, -14], [6, 0, 10, 10], [12, 2, -2, 2], [18, 1, 15, -3], [-2, 2, -14, 2], [27, -5, -4, 19], [24, 17, 5, -7], [40, 7, 9, -1], [8, 3, -13, -1], [-2, -12, 14, 0], [16, -12, -2, -12], [2, -2, 14, -2], [-11, -1, -12, -7], [32, -5, 5, -3], [12, -12, 8, 8], [-49, -13, 4, 3], [-38, 8, -8, -16], [0, 9, -7, -9], [-41, 19, 2, -15], [10, 12, 4, 6], [-15, -22, -3, 2], [53, 5, 10, -1], [18, 11, -7, 5], [-22, 3, -5, -5], [2, -4, -14, -2], [26, 2, -4, -2], [-46, 6, 0, -12], [0, -16, -12, 8], [43, -9, 0, 3], [-50, 4, -4, 2], [-18, -20, -6, 0], [-36, 5, -11, -11], [-4, 4, 10, -8], [30, -4, 16, 8], [10, 2, 16, -6], [14, 21, -11, 5], [-24, 8, 8, 0], [19, -6, 3, -10], [-10, 0, 14, -4], [31, -8, -11, 4], [-24, -14, 2, -10], [-43, -5, 0, 3], [-6, 26, -4, 0], [6, 6, -12, 8], [0, 0, 8, -16], [30, 0, 12, 12], [-23, 23, 4, 3], [8, 16, 4, -16], [31, 17, -6, 1], [-8, 4, -6, 12], [10, -10, 2, -4], [12, 10, -14, 2], [19, -8, 5, -20], [-6, -30, -4, 0], [-16, -5, -9, -13], [12, -26, -2, 6], [-7, 17, 16, -7], [-28, 22, 10, -6], [56, 4, 10, -10], [-26, -10, -8, 0], [23, -2, -1, -6], [-43, 8, 11, 8], [-26, 0, -8, -4], [1, 9, 18, 11], [12, -10, 10, 6], [2, 0, -6, 18], [-16, -17, 9, -9], [9, -1, -4, -3], [-6, -14, -8, -4], [22, 7, 7, -13], [19, -4, -1, 20], [-2, 2, 0, -10], [-10, 23, 11, -5], [24, 4, -6, -8], [42, 2, 8, -4], [43, -1, 2, -11], [14, 5, -3, -3], [-16, 14, -10, -6], [22, -5, 19, 7], [39, 7, -4, 11], [-44, 20, 6, -4], [18, -18, -12, -16], [-34, -4, -10, -6], [18, -6, -10, 6], [17, 10, 7, -6], [25, -5, -6, -1], [-12, -18, -14, 6], [-5, 7, -4, 11], [12, 9, -3, 13], [29, 4, 7, 0], [0, 4, -4, 4], [30, 0, 8, 6], [58, -12, -2, 8], [4, 10, -6, -2], [4, 18, -6, -2], [9, -8, -3, -4], [-29, -21, 10, -11], [32, 0, -4, 8], [-49, 13, -8, -17], [-20, -15, -9, 7], [36, 12, -8, -4], [-14, -10, -4, 0], [-16, -11, -15, -1], [-33, 23, 12, 7], [30, 7, 5, 5], [10, 9, -11, -7], [-6, 32, 6, -4], [-1, 2, -11, -2], [39, -28, -9, 8], [4, -22, -10, -6], [0, 2, -10, 18], [-13, -25, 0, 11], [-14, -1, -11, 19], [-33, 10, -9, -10], [-42, 22, 6, -2], [-6, -1, -17, 7], [-32, -19, -5, 1], [-48, 8, -8, -12], [38, -13, 3, 7], [-6, 6, -4, 0], [-13, 13, -12, -9], [-8, 4, 4, -20], [-45, -7, -16, 17], [-4, -24, -8, 4], [-26, -7, -7, 5], [12, 20, 8, -2], [27, -13, -2, -15], [4, -10, -2, 6], [32, 16, 0, -2], [17, -6, 3, -2], [0, 7, -7, -1], [-13, 5, 2, 1], [-4, -31, 5, -3], [8, 0, 4, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3549_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3549_2_a_v();".
// 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_3549_2_a_v(:prec:=4)
    chi := MakeCharacter_3549_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_3549_2_a_v();".
// 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_3549_2_a_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3549_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![2, -7, -7, 1, 1]>,<5,R![52, -28, -14, 3, 1]>],Snew);
    return Vf;
end function;