
// Make newform 3249.2.a.y in Magma, downloaded from the LMFDB on 07 November 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_3249_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_3249_2_a_y();".
// 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_3249_2_a_y();".
// 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, -5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-4, 0, 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_3249_a();"
function MakeCharacter_3249_a()
    N := 3249;
    order := 1;
    char_gens := [362, 2890];
    v := [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_3249_a_Hecke( : Kf := Kf)
    return MakeCharacter_3249_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, 1, 0], [0, 0, 0], [-1, 0, -1], [0, 0, 1], [-1, 2, -1], [-1, 2, 0], [0, 0, 0], [0, 0, 0], [-5, 2, 1], [-2, 0, -2], [4, 2, -1], [-1, 0, 0], [2, 0, 2], [-2, 2, -1], [6, 0, 0], [3, 0, 3], [1, -2, 1], [3, 2, -2], [4, -4, -1], [4, 4, -2], [7, 0, 2], [4, -2, -1], [0, 4, 0], [5, 0, -1], [-2, 6, 2], [-6, -4, 0], [-6, 4, -5], [2, 0, -4], [-8, 2, -4], [5, -4, -1], [10, 0, 2], [4, 0, -2], [-8, 4, -2], [-12, 2, -5], [-3, 0, 3], [6, -4, -2], [9, -4, -2], [6, 4, 1], [-5, -2, -5], [-10, 4, 2], [3, 2, -3], [6, -2, -2], [9, -6, 3], [-9, 2, -2], [-17, -4, 1], [-10, -2, -3], [-2, 2, -7], [-2, -6, 5], [-5, -2, 1], [-1, 2, 0], [-6, -8, 0], [5, 2, -1], [7, 4, 2], [-4, -4, 2], [-3, 8, -3], [-8, 4, -2], [1, 4, -5], [0, -12, 4], [6, -10, -2], [15, -4, 3], [8, 4, 0], [-2, 8, 4], [4, -2, 2], [-1, -6, -1], [8, 6, 0], [13, 4, 1], [-14, -2, -1], [11, -6, 0], [-17, -2, 1], [9, 2, 4], [-19, 4, -7], [4, -4, 4], [-2, -10, -1], [10, 2, 2], [12, -6, 1], [-5, -6, 7], [-9, -12, 3], [-3, -2, -8], [7, 0, 1], [16, 6, -4], [-3, 2, -3], [0, -2, 4], [-10, 8, -4], [13, -6, 8], [6, 8, 1], [10, 4, -2], [10, -4, 10], [19, -2, -4], [-11, 4, -5], [2, 0, 9], [18, -8, 6], [0, 8, 6], [-2, -4, 2], [0, 4, -6], [0, -4, -11], [-14, 4, -2], [18, -4, 0], [-27, 8, 3], [-6, 4, 7], [11, -12, -6], [-10, -2, -9], [-12, 0, 6], [10, -12, 4], [14, -4, 14], [-6, 4, -5], [-12, 6, -8], [-7, 10, 5], [-1, -4, 5], [29, 2, 5], [9, -10, 4], [6, -6, 1], [-6, 6, -2], [-9, 8, 9], [-2, -4, -1], [8, 4, -3], [-28, -8, 2], [-36, 4, 1], [6, 0, 6], [-18, 8, 0], [-23, 2, -11], [2, 8, 0], [-11, 2, 8], [2, 16, -4], [-23, 10, 7], [-36, 4, 4], [17, 8, 5], [9, 10, -2], [11, -18, 5], [-8, 14, 11], [-14, 6, 2], [28, -8, 11], [17, 14, -7], [-10, -8, 3], [1, 14, -10], [23, -8, -7], [-17, 4, 8], [24, -12, 6], [2, 0, -9], [-26, 4, -8], [-29, 0, -5], [-14, 8, -10], [-6, 12, 6], [10, 0, -10], [30, -4, 6], [-23, 14, 2], [-11, 10, -5], [-13, 16, -12], [6, 0, 0], [-4, -12, 3], [6, 8, 0], [-5, -16, 8], [1, -4, 1], [-10, 14, 3], [-5, -6, 7], [8, -8, -12], [14, -8, -10], [-2, -16, 5], [33, -8, -3], [-17, 14, -10], [-20, 12, -8], [4, 8, -2], [-13, -8, 5], [12, -10, 1], [10, -20, 4], [-12, -12, -6], [23, -6, 11], [-2, -6, 11], [45, 4, -2], [-30, 0, -8], [-2, -8, -2], [11, 6, -1], [7, -6, -10], [14, -12, -4], [-9, 2, 4], [42, -8, 7], [22, -4, -8], [-28, 10, 9], [30, -8, 0], [0, 12, -11], [-11, -8, 2], [-32, 2, 2], [8, -4, 8], [9, 16, -8], [-33, 0, 3], [-10, 20, -16], [-6, 8, 0], [-12, 6, -8], [16, 8, -1], [24, -14, 4], [23, -14, -13], [-15, -4, 4], [22, -4, -2], [0, -22, 1], [12, -8, -6], [7, -2, 1], [-34, -12, 2], [-2, -10, -10], [23, 4, 6], [-26, 4, -2], [4, -12, 10], [25, -16, -5], [8, -18, 18], [-3, -6, 10], [19, -10, 20], [-25, 2, -19], [-14, -8, -2], [18, 20, -2], [18, 4, -6], [4, 12, 4], [-26, 12, 11], [9, 14, -8], [-10, 20, -16], [-14, 14, -13], [-15, -6, -3], [25, -2, 7], [-15, 4, 10], [-18, 4, -2], [3, -24, 15], [6, 12, 6], [5, -4, -1], [37, 6, 2], [20, -8, 15], [-5, 28, -5], [28, 4, -7], [18, 16, -6], [5, -18, 0], [14, 8, -4], [-27, -18, 9], [22, 8, 2], [18, 12, -6], [25, 4, -4], [-20, 4, -1], [48, 2, -2], [22, -8, -14], [14, 28, -6], [25, 2, 1], [-9, -2, -8], [-24, -8, 0], [2, -28, 14], [23, -10, -7], [-9, 6, -15], [22, -8, 2], [-4, -14, -9], [-1, 24, 0], [9, -4, -9], [4, -16, 16], [-24, 8, -8], [-15, -10, -15], [-18, -4, 7], [29, -6, 5], [17, 0, -12], [19, 12, 7], [44, 0, 8], [2, -10, 6], [-31, -4, -13], [26, 8, 14], [-23, -6, -10], [6, 8, 19], [-28, 8, -4], [3, -6, 10], [-18, 4, 7], [11, 10, -13], [25, -22, 2], [-5, 16, -4], [-23, 4, -5], [-4, -28, 3], [-16, 4, -16], [-26, -12, 10], [-22, 4, -9], [-50, 8, 10], [-1, -10, -18], [20, -2, 6], [-14, 0, 8], [12, 0, 4], [9, -6, -14], [-10, -2, -3], [-24, 24, 18], [-33, 18, -14], [33, -12, -14], [-26, 4, 16], [19, -14, 1], [-10, 28, -18], [11, -22, -7], [35, 4, 0], [40, 10, -10], [6, -12, 6], [33, 4, -8], [11, 32, -13], [-44, 2, -7], [22, -4, 4], [0, 28, -18], [-13, -22, -7], [0, 4, -6], [-42, 12, 0], [-27, 28, -2], [19, -8, 1], [38, -8, 18], [39, -8, 21], [45, -6, 9], [26, 0, 21], [18, 10, -2], [-54, -12, 0], [16, 0, 11], [40, 0, -2], [18, -18, 13], [11, 8, 6], [-36, 0, -6], [-1, -14, 12], [5, 26, -1], [-11, 12, 2], [-38, 8, -8], [-35, -24, 7], [6, -28, 12], [16, -6, -1], [-19, 14, -13], [-36, 22, -8], [29, 26, -7], [18, 4, 12], [17, 12, 12], [21, 20, 3], [44, 6, 3], [-53, -6, 2], [-6, 28, -18], [34, 18, -7], [17, -16, -19], [-1, 8, -6], [-14, -16, 11], [26, -26, 9], [43, -14, -5], [-24, -16, -12], [19, -12, 26], [-26, -8, 10], [42, -8, 10], [1, -26, 7], [-6, 36, -2], [-19, -2, -1], [11, -12, 12], [0, 28, -6], [-25, 10, -12], [54, -8, -5], [30, -10, -14], [9, -20, 9], [-61, 8, 5], [10, 8, 26], [-3, -8, 15], [15, 2, 9], [-2, 22, 14], [16, -24, 20], [-25, 14, -7], [25, -4, -11], [-20, 2, -25], [-53, 8, -10], [2, 8, 14], [28, -8, 20], [-18, 8, 4], [-40, 20, 2], [17, 2, 29], [-19, 6, 11], [25, -20, 7], [-54, 0, -6], [32, 6, -12], [31, -8, 19], [13, -2, 25], [-21, 18, -9], [0, 14, -2], [32, -18, 24], [17, -20, 11], [-10, -12, -9], [-23, 28, -22], [-9, 2, -3], [-12, 26, 7], [-56, 8, 4], [8, -4, 26], [30, -6, 7], [-15, -4, 4], [29, 6, 11], [-37, -6, 11], [-50, 6, 14], [3, -36, 3], [21, -2, 10], [-3, 0, 9], [-64, 8, 2], [32, -26, 18], [46, -8, 16], [-16, 6, 9], [8, 4, 26], [58, 6, 11], [18, -38, -2], [16, -4, -16], [50, -4, 8], [-13, 4, 6], [24, 12, -18], [-4, -24, -10], [2, -18, -3], [31, 6, 13], [46, -14, 14], [62, 6, 9], [-29, -20, -5], [-4, 8, -3], [-64, 4, -4], [-21, 12, -8], [-17, 36, 7], [-48, -2, -14], [-92, 4, -2], [22, -20, -20], [-30, 16, 1], [61, -2, 2], [47, 8, 11], [-4, -4, 12], [1, -10, -23], [-29, 6, -22], [-55, 16, -7], [-45, 6, -27], [50, -18, -9], [-36, -10, 4], [-41, 28, -11], [-57, 2, 9], [2, -8, 18], [29, -12, 23], [3, 14, -15], [-21, 0, -9], [-33, -14, -2], [-11, 26, 1], [-45, 6, 15], [-31, 20, -18], [-47, 28, 7], [-57, -22, 3], [24, -24, 6], [4, 18, -1], [-36, 0, -6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3249_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3249_2_a_y();".
// 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_3249_2_a_y(:prec:=3)
    chi := MakeCharacter_3249_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_3249_2_a_y();".
// 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_3249_2_a_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3249_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![3, -5, -1, 1]>,<5,R![-12, -8, 2, 1]>,<13,R![3, -21, 1, 1]>],Snew);
    return Vf;
end function;