
// Make newform 6240.2.a.cb in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6240_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_6240_2_a_cb();".
// 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_6240_2_a_cb();".
// 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 := [-1, -4, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [-3, 1, 1], [-3, -1, 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_6240_a();"
function MakeCharacter_6240_a()
    N := 6240;
    order := 1;
    char_gens := [1951, 2341, 2081, 2497, 5761];
    v := [1, 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_6240_a_Hecke(Kf)
    return MakeCharacter_6240_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], [1, 0, 0], [-1, 0, 0], [2, 0, 1], [0, 1, 0], [1, 0, 0], [0, -2, 1], [2, 1, 1], [2, 0, 3], [0, 1, -3], [4, 0, 0], [2, -1, 0], [-2, -1, 0], [0, 2, 0], [0, 0, -4], [2, 3, 0], [0, -2, 2], [-2, -1, -2], [0, 0, 0], [4, -1, 2], [0, 1, -5], [8, 1, -2], [-8, 0, -4], [-2, -1, 2], [8, 0, 1], [-4, -1, -1], [8, 0, -4], [-4, 1, 2], [8, -3, 1], [4, 3, -3], [8, 4, 4], [6, -1, 5], [2, 2, -6], [0, -3, -4], [6, 1, -4], [4, -4, 4], [6, 2, 6], [-4, -3, 0], [-4, 2, -2], [2, -2, 2], [-6, -1, -1], [-6, -3, 4], [0, 2, -4], [12, -4, 3], [10, -2, 2], [0, 4, -4], [4, 0, -4], [6, -1, -3], [-4, 6, -4], [8, -3, -3], [0, 0, 5], [-8, -1, -2], [2, 6, 0], [-22, 1, -1], [-4, -5, -7], [-10, 5, -5], [4, -5, -1], [12, -6, 4], [14, 4, 2], [6, 4, 4], [0, 0, 0], [18, 0, 2], [-8, -5, -6], [0, -2, -4], [6, -6, 2], [18, 4, 0], [-6, 9, -3], [6, 2, -10], [-8, -5, 12], [16, 3, 5], [2, 4, -6], [-4, 0, -4], [0, -2, 4], [-2, 6, 6], [6, 5, -5], [4, -4, -2], [12, 3, 11], [10, -1, -8], [6, -6, 2], [-2, 4, -6], [22, 5, 9], [-4, -1, -9], [-8, 0, 8], [-18, -2, -10], [4, -5, 2], [-8, 7, 0], [6, -7, 6], [-8, 0, 5], [-6, -3, 6], [-2, 2, -1], [-12, 1, 6], [-8, -3, -6], [18, 2, 1], [-2, -7, 5], [-2, -1, 7], [6, 9, -1], [10, -5, 10], [-10, 2, -10], [20, -2, 8], [-20, 5, -1], [-12, 0, -8], [26, 2, -4], [-24, -5, -8], [-2, 2, -2], [-16, -5, -4], [28, 2, -9], [-20, -6, -8], [6, 2, 8], [-16, -8, 0], [-6, -5, 0], [16, -2, 4], [-22, 1, 0], [2, -2, 10], [-2, -3, 11], [0, 6, -18], [-14, 0, -4], [8, -1, -6], [6, 4, 1], [-18, -4, -16], [6, 7, -7], [-36, -3, 7], [18, 2, 0], [14, 1, 6], [-24, -6, -8], [18, 9, 13], [-28, -5, -5], [8, -3, 13], [-12, 2, 6], [-4, -4, -6], [-14, 3, -4], [-2, 3, -13], [12, 0, 10], [-4, 3, 2], [-18, -10, -8], [-2, -6, 10], [2, -8, -4], [10, 8, -6], [-4, 2, 6], [-14, 1, -4], [2, 0, -8], [-22, 1, -3], [-18, 3, 4], [24, 0, -4], [-32, 4, 0], [6, 12, -8], [24, 7, 2], [-14, 5, -8], [4, 2, 15], [12, -1, -2], [-12, -4, 18], [14, -8, 12], [-34, 0, -2], [20, 0, 0], [-22, -10, -13], [-16, -6, 20], [-4, 4, -10], [4, -7, 14], [-10, 7, -2], [10, -10, -4], [6, 9, 8], [-32, -4, -4], [-24, -2, -3], [10, 5, -1], [10, 3, -17], [-18, 8, -4], [4, 2, 14], [-8, -5, -14], [-18, -4, -18], [26, 6, 8], [-14, -9, 20], [-20, 0, 0], [-16, -1, -3], [-4, 8, -14], [20, -6, 3], [-16, -1, -10], [-10, -14, -2], [-14, 7, -3], [-18, -11, -12], [20, -10, 2], [18, -2, 10], [2, -12, 5], [22, 1, -3], [-18, -6, -8], [22, 2, -4], [-4, 0, -2], [-24, -7, -15], [-50, -4, -8], [8, 13, -10], [-2, 6, -10], [0, -7, 6], [2, 8, -8], [-12, -14, -4], [-28, -3, -2], [-10, 7, 4], [-8, -10, 10], [4, -4, 3], [6, -6, -10], [6, 4, 18], [6, -6, 4], [-18, -14, -3], [42, 4, -4], [-20, -11, -2], [-34, 7, -10], [42, 1, 12], [-32, 11, -4], [-2, -5, 26], [-28, 6, -8], [8, 2, -2], [-42, 1, 4], [-20, -5, 14], [-2, 4, 6], [8, -7, 17], [-4, 6, 6], [-16, 16, -4], [40, -8, 8], [-14, -4, 16], [-4, 8, -14], [-18, 12, -2], [-12, 12, -6], [10, -8, -8], [-38, 7, -4], [8, 12, -26], [6, -6, 22], [18, 9, 7], [8, -5, 4], [-30, 9, 4], [-24, -12, -19], [4, 2, 10], [36, 4, -2], [-8, -3, 0], [22, 6, -10], [52, -1, 6], [-36, 6, -4], [14, 16, 6], [0, -10, -16], [-4, -6, 22], [-10, -2, -18], [10, -4, -4], [10, 11, 3], [-28, -12, 6], [-8, -16, 8], [-20, -1, -10], [20, 10, -22], [-12, 1, 15], [-34, -8, -20], [0, 2, 0], [-2, 19, -7], [44, 3, 10], [2, 3, 9], [-26, 9, -9], [2, -6, 6], [-58, -4, 0], [16, 8, -24], [2, 15, 2], [-14, -7, 0], [12, 9, 6], [-18, 8, 4], [-28, -11, 0], [18, -6, -14], [0, 4, -19], [-52, 4, -6], [-4, -1, -2], [0, -19, 1], [-18, -12, 16], [-2, 14, 8], [-16, -9, -12], [34, 3, 10], [38, 7, 10], [16, -11, -6], [-18, 4, 12], [-22, -2, -2], [-12, -9, 12], [-4, -14, 15], [-4, -12, 20], [2, -6, -4], [-14, -14, 9], [12, 16, -4], [20, 3, 11], [-10, -6, -6], [-22, 11, -5], [-74, -2, -3], [48, 2, 2], [-10, 17, -9], [-40, -3, 9], [8, -8, -4], [-4, 6, -18], [46, -10, 6], [18, -12, 2], [-24, -2, 22], [-22, 10, 0], [-24, 3, 17], [12, -4, -20], [-42, -10, 8], [12, 12, 12], [6, -4, -4], [-4, 13, 3], [-24, 12, 0], [62, -5, 2], [-10, -11, 1], [-28, 5, 4], [-26, 16, -10], [34, 6, 2], [-72, -3, -2], [-16, 1, -8], [-12, -4, -4], [28, -4, 15], [-12, -17, -2], [10, 3, 4], [-32, 4, 4], [-58, -2, 4], [-38, 8, -1], [-14, -12, 0], [14, 16, -26], [20, 6, -28], [-8, -22, 4], [34, 11, -6], [32, -17, 4], [12, 7, -18], [16, 13, -25], [26, 2, -24], [0, 14, -2], [40, 13, 11], [-4, 5, -5], [54, 10, 7], [38, -10, 12], [-22, -13, -8], [-18, -3, -5], [8, -13, 6], [14, -18, -7], [26, 14, -10], [4, -3, -21], [30, 3, 2], [12, -8, 20], [-12, 14, -8], [18, -13, 5], [-36, -4, 8], [-4, -5, -9], [10, 16, 2], [-10, 0, 6], [-40, -6, -20], [22, -5, 6], [28, -2, 19], [22, 13, -32], [-56, 0, 0], [-46, -14, -10], [70, 1, 5], [54, -5, 6], [-32, -1, -14], [12, 0, 28], [6, 3, -14], [38, -15, 11], [-4, -12, -5], [-2, -5, 20], [4, -24, 6], [-6, -9, -16], [32, 17, 23], [-40, -5, -2], [0, 14, -14], [16, -7, 11], [16, -6, -8], [46, 4, 12], [26, 8, 16], [6, 5, 3], [-4, -24, 8], [0, 22, -12], [-18, 20, -2], [34, 4, 10], [-26, -18, -21], [34, -1, 2], [38, 3, 21], [0, -2, -6], [-44, 4, 10], [72, -5, 5], [-48, 7, -2], [-10, -12, 4], [-4, -11, -10], [40, 8, 8], [36, -2, 19], [-10, 20, -2], [6, 6, 2], [-2, -10, 6], [50, 10, 18], [54, 11, 17], [46, -16, 0], [38, 1, -1], [16, 16, -4], [-16, 0, 26], [-58, 4, -8], [32, -4, 24], [38, -14, 9], [26, 0, -4], [2, -2, -20], [-16, -11, -8], [44, -18, 4], [20, 7, 2], [6, -14, -2], [-52, -2, -20], [-6, 14, -16], [-4, -5, -2], [18, 0, 24], [-28, -7, -17], [-12, 6, -17], [54, -4, 7], [-10, 0, -28], [58, 9, -2], [-28, 3, 14], [-6, -10, 30], [14, -16, 16], [52, 11, 8], [20, -23, -2], [-2, 22, -2], [10, 9, -20], [40, -5, -4], [-48, -7, -4], [26, -2, -16], [8, -3, -27], [16, -3, -14], [-16, -16, 28], [6, -6, 32], [14, 0, 5], [8, -11, 17], [22, 5, 10], [-8, 10, -4], [74, -3, -1], [16, 17, 15], [-22, -10, -12], [-20, 1, -22], [46, 9, -18], [-58, -11, 15], [0, 6, -24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6240_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6240_2_a_cb();".
// 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_6240_2_a_cb(:prec:=3)
    chi := MakeCharacter_6240_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_6240_2_a_cb();".
// 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_6240_2_a_cb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6240_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, 2, -5, 1]>,<11,R![-16, -12, 1, 1]>,<17,R![148, -50, -1, 1]>,<19,R![56, -16, -4, 1]>],Snew);
    return Vf;
end function;