
// Make newform 2940.2.a.m in Magma, downloaded from the LMFDB on 15 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2940_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_2940_2_a_m();".
// 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_2940_2_a_m();".
// 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 := [-7, 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_2940_a();"
function MakeCharacter_2940_a()
    N := 2940;
    order := 1;
    char_gens := [1471, 1961, 1177, 1081];
    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_2940_a_Hecke( : Kf := Kf)
    return MakeCharacter_2940_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, 0], [-1, 0], [0, 0], [1, 1], [0, -1], [-1, -1], [-3, 2], [-1, -1], [5, -1], [-1, -2], [-2, -1], [3, 3], [2, 3], [-6, 0], [-2, -2], [-3, 3], [-8, 0], [-2, -1], [11, -1], [0, 5], [-3, -2], [3, -3], [1, -5], [-8, 0], [-3, -3], [-12, -1], [-9, 3], [-13, 0], [6, 0], [-10, -3], [-16, 2], [-7, 5], [-5, 0], [4, 4], [6, 4], [-4, -2], [-16, 0], [-17, 1], [6, 6], [-10, -4], [-1, -2], [-2, -8], [-16, -3], [3, 3], [2, -8], [-2, -4], [-2, 6], [-17, 1], [7, 0], [16, -2], [6, 0], [18, 2], [3, -3], [13, 7], [-2, 10], [-12, 6], [-18, -4], [-18, 1], [24, 0], [12, -1], [0, 0], [-20, 3], [-11, 1], [8, -5], [5, -7], [1, 8], [-20, 5], [-8, -8], [10, 0], [7, -11], [3, 9], [6, 5], [8, 3], [-21, -2], [22, -2], [-19, -7], [14, 1], [14, 2], [15, 8], [-16, 2], [-9, 4], [24, -6], [-6, -1], [0, 2], [-6, -6], [-2, 4], [-12, 1], [17, -1], [-8, -7], [6, -12], [-4, -4], [0, 1], [18, 6], [1, -10], [4, 10], [12, -6], [18, 6], [-18, 5], [5, -12], [-38, 2], [-18, 0], [26, -4], [-17, -5], [39, -2], [24, -7], [-9, 3], [33, 3], [18, 6], [27, -4], [30, 5], [-20, 8], [4, -2], [19, 0], [-4, 12], [17, 11], [-2, -9], [-15, 3], [31, 1], [-24, 0], [25, -6], [-8, -1], [25, 1], [-3, -3], [-13, 4], [27, -3], [-12, 4], [20, 2], [10, -9], [6, 5], [29, 0], [-21, -3], [3, -8], [-30, -8], [-17, -11], [9, 2], [7, 13], [10, 6], [3, 3], [-2, -8], [-12, 2], [-7, 17], [-8, -10], [-4, -10], [-25, -2], [-19, -1], [-30, 5], [11, -1], [0, -10], [-8, 10], [14, 0], [20, 8], [2, 9], [17, -1], [-18, 1], [13, 7], [21, 4], [-17, 7], [6, 17], [3, -3], [-43, -1], [-26, 10], [6, 13], [-32, -8], [-33, -3], [15, 15], [39, -2], [-30, 5], [-9, -14], [21, -15], [-10, 14], [6, 14], [5, -13], [-12, -11], [-11, 12], [-12, -12], [3, -14], [0, -6], [20, -8], [13, 12], [22, -1], [9, 3], [12, 1], [-14, 10], [18, -12], [-42, -6], [-16, 18], [-10, -5], [-25, 12], [-15, -3], [-14, -3], [-29, 1], [16, 8], [-60, 0], [-13, 11], [39, 3], [-1, 24], [32, -12], [40, -2], [15, -9], [0, 18], [21, 2], [16, -6], [0, -16], [3, 15], [-16, -10], [-36, -4], [-6, 0], [26, 2], [4, -18], [-28, 3], [-51, 3], [54, -2], [-44, 4], [-7, -1], [27, -4], [0, -8], [-24, 0], [-2, 10], [-6, 12], [29, -6], [5, -2], [-27, -15], [-24, 16], [-16, 14], [-8, -4], [0, -12], [-12, -6], [-18, 11], [-35, 1], [18, 13], [12, 8], [27, -8], [35, -13], [14, 1], [-7, -1], [5, -14], [39, 9], [20, -4], [-1, -13], [-27, -9], [-3, -10], [-34, 7], [2, -18], [-54, 6], [-18, -18], [44, 1], [-5, 19], [44, -6], [-20, -20], [24, 1], [-52, 8], [22, 4], [53, -8], [-14, 10], [23, 5], [-3, -26], [-54, -7], [-25, -19], [-14, -12], [-30, 13], [6, -18], [-53, 0], [-50, 3], [24, 18], [21, -4], [-13, 11], [6, 12], [56, 6], [33, 3], [15, 20], [24, 10], [28, 9], [35, 6], [-12, -22], [-30, 2], [11, 5], [-25, -12], [27, -2], [-8, -20], [10, -14], [-49, 0], [-6, -12], [-37, -8], [-26, 6], [-37, -1], [2, 9], [59, -7], [-17, 12], [11, -1], [-10, -10], [-26, -2], [39, 9], [-29, 1], [16, -7], [6, -24], [35, 16], [1, 1], [43, 7], [-14, -9], [30, 20], [8, -22], [4, 14], [-56, 4], [25, 14], [-30, 7], [-42, -12], [-19, -14], [54, 6], [28, -13], [21, -15], [-7, 29], [-27, -3], [-10, -6], [10, -2], [-23, 6], [57, -3], [-33, -3], [22, 9], [-33, 3], [-21, -10], [6, -5], [-40, 2], [20, -15], [10, 16], [71, -2], [69, -2], [0, -7], [-36, -12], [-52, 2], [3, 22], [77, -1], [37, 0], [39, -3], [-25, -12], [-4, 2], [24, -8], [57, 9], [-39, -4], [34, -3], [16, -1], [-7, -7], [-38, 10], [-45, -8], [3, 9], [2, -4], [11, 10], [40, 8], [33, -3], [-28, 2], [62, -8], [54, 4], [-48, -6], [-8, 0], [37, 20], [-26, 4], [42, 12], [-42, 6], [0, 12], [-53, 13], [-16, 12], [21, -3], [36, 18], [-18, 24], [46, -9], [-20, 20], [41, 17], [-40, -3], [-20, 8], [-66, -6], [39, -4], [87, -3], [15, -9], [5, -14], [24, 25], [39, -3], [-12, 12], [20, 10], [7, -11], [-50, 15], [8, 20], [-35, -23], [32, -24], [28, -8], [12, 2], [18, -18], [-35, -4], [-42, 20], [68, -3], [-23, 7], [-27, -8], [27, -3], [3, -15], [4, -3], [-61, 11], [8, -24], [-24, -28], [48, 6], [-15, -26], [-6, 24], [-14, -18], [-33, -21], [-16, 6], [-31, -1], [-23, -11], [35, 10], [36, -2], [14, -22], [-8, 21], [44, -16], [6, 8], [78, 6], [45, 21], [20, -24], [6, -17], [-5, 19], [77, -7], [0, -7], [40, -20], [18, 18], [10, -2], [-20, 27], [-33, 3], [4, -26], [12, -10], [23, -31], [-58, 8], [-12, 24], [24, 20], [41, 11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2940_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2940_2_a_m();".
// 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_2940_2_a_m(:prec:=2)
    chi := MakeCharacter_2940_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_2940_2_a_m();".
// 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_2940_2_a_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2940_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-6, -2, 1]>,<13,R![-7, 0, 1]>,<17,R![-6, 2, 1]>,<31,R![-27, 2, 1]>],Snew);
    return Vf;
end function;