
// Make newform 6720.2.a.cq in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6720_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_6720_2_a_cq();".
// 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_6720_2_a_cq();".
// 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;
    Rf_num := [[1, 0], [0, 2]];
    Rf_basisdens := [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_6720_a();"
function MakeCharacter_6720_a()
    N := 6720;
    order := 1;
    char_gens := [1471, 3781, 4481, 5377, 1921];
    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_6720_a_Hecke(Kf)
    return MakeCharacter_6720_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], [-1, 0], [1, 0], [-1, 0], [-2, 1], [0, 1], [2, 0], [-2, 1], [0, 2], [2, 0], [-2, 1], [-2, 0], [2, 2], [-4, 0], [4, -2], [4, 1], [0, -4], [-2, -2], [0, -2], [6, -3], [4, 1], [4, 0], [-8, 2], [-6, 2], [16, -1], [-2, 4], [8, 0], [-12, 2], [2, 0], [0, -3], [12, 2], [-4, -2], [-4, -1], [-2, 1], [14, -2], [8, -2], [-4, -1], [-12, 0], [8, 0], [18, -2], [6, 1], [-6, 0], [2, 3], [2, 4], [16, -1], [-10, -3], [-4, 0], [20, -2], [0, -2], [-2, 6], [-16, 1], [-2, 5], [6, 6], [-16, -4], [-10, -2], [-4, 4], [6, 0], [10, -1], [-2, -4], [-10, 6], [-4, 4], [18, -2], [16, -2], [-8, 0], [0, -5], [4, 5], [-4, 0], [10, 0], [0, 4], [-10, 2], [18, 4], [-14, -1], [12, 6], [26, 2], [24, -2], [-8, 4], [-2, -6], [-8, 1], [-6, 0], [2, 0], [12, 6], [2, -4], [-2, -7], [32, -1], [-10, 5], [-24, 0], [10, -4], [6, 6], [-18, -4], [28, -2], [24, -2], [0, -4], [-4, -6], [-10, 1], [16, -2], [16, 0], [6, 8], [-6, -10], [8, -6], [-14, -4], [32, -2], [-20, -3], [-12, -4], [-6, -4], [-16, 6], [16, -5], [-4, -4], [18, 8], [-30, -1], [22, -2], [28, -2], [-2, 0], [-24, -3], [22, -3], [12, 8], [-18, 6], [24, -6], [0, 4], [24, -1], [34, -1], [14, 2], [22, -2], [10, 6], [-8, -4], [18, 7], [-6, 4], [10, -8], [-4, -6], [4, -2], [0, -3], [8, 6], [12, -4], [4, -12], [26, 2], [-38, -2], [6, 10], [2, -2], [8, 6], [-30, -2], [-18, 6], [-10, 9], [2, 8], [12, 6], [4, -2], [10, 4], [-12, 6], [8, -3], [-18, -2], [34, 3], [-40, -2], [-34, 0], [26, -6], [8, 10], [4, 2], [-8, -6], [2, 7], [-28, 0], [18, -10], [4, 9], [-2, 0], [-16, 0], [-32, 5], [-8, 0], [24, 4], [-24, -3], [20, -6], [4, 16], [-24, -7], [-22, 0], [10, -2], [30, -3], [-22, -8], [-42, -3], [26, -4], [-6, 3], [18, 6], [4, -12], [-34, -6], [4, -6], [2, -12], [0, 8], [-32, 4], [26, 10], [-30, 4], [-60, 0], [14, -4], [-42, 0], [0, 6], [-38, -4], [-36, 2], [-8, -1], [-20, -6], [16, -14], [6, -16], [0, -12], [34, 4], [-22, 8], [26, 2], [-18, 2], [16, -8], [-30, 4], [-2, 1], [-4, -5], [14, -6], [16, -4], [-22, 14], [34, -5], [-28, 6], [-34, -2], [-6, -13], [-18, -6], [-10, 4], [8, -12], [32, -6], [28, -2], [-50, 2], [-4, -6], [-14, 2], [-12, 4], [0, -13], [-22, -8], [14, 5], [-10, 6], [24, 4], [-12, 4], [18, 4], [6, 2], [-38, 3], [24, 12], [30, 1], [2, -18], [-18, -11], [40, -2], [-2, -10], [8, -2], [36, 6], [-38, 0], [48, -5], [6, 9], [36, -2], [-28, 10], [22, -15], [-16, 16], [-6, 16], [46, 6], [28, 6], [-4, -14], [-60, 2], [-52, -4], [0, 18], [6, -12], [-6, -10], [-40, -6], [-6, -20], [-30, -6], [-10, 9], [26, 0], [-12, -16], [-30, -2], [-24, 15], [28, -14], [8, 8], [-10, -6], [28, 11], [-14, -4], [-6, -1], [-30, -4], [34, 2], [16, 6], [-48, 7], [2, 12], [-52, 8], [16, 15], [12, 0], [8, -17], [44, 2], [-12, -14], [26, -4], [-38, 8], [56, 0], [-12, -2], [60, 0], [68, -2], [-54, -4], [24, -2], [-14, 11], [6, 2], [12, -15], [14, -15], [26, 2], [18, -12], [-24, -2], [0, -15], [-36, -6], [10, 2], [-18, -4], [58, -1], [38, -12], [48, 0], [56, 2], [-4, 1], [-24, -17], [-16, -6], [0, -20], [8, 14], [34, 8], [-12, 10], [22, -18], [14, -7], [74, 2], [12, -2], [-14, -4], [74, 0], [32, -14], [24, 6], [-58, -6], [44, 2], [-38, 7], [-8, 7], [10, 8], [58, 3], [30, 6], [-50, 4], [16, 4], [36, -5], [-50, -6], [-32, -6], [4, 16], [0, -22], [20, 1], [-70, 0], [32, 3], [42, -5], [-12, -12], [-4, 8], [12, 8], [74, 0], [62, 6], [14, -10], [12, -2], [10, 18], [24, -15], [-18, 8], [-32, -2], [32, 15], [10, -21], [22, -14], [-40, 14], [36, -6], [50, 6], [62, -3], [-2, -14], [-54, 8], [-32, -12], [18, 16], [-30, 8], [28, 14], [-4, 14], [16, -15], [-12, -4], [-22, 10], [-62, 6], [-84, 0], [18, -13], [32, -10], [-34, 10], [-42, 8], [-8, -20], [42, -16], [54, -3], [2, -13], [-36, -8], [-42, -6], [-22, -1], [-50, -8], [-36, -2], [18, -21], [24, -1], [66, 6], [-4, 17], [30, -8], [0, 21], [-28, 18], [-12, 3], [-10, 13], [0, 8], [-20, 16], [32, 9], [-16, -10], [-20, -2], [34, -4], [6, -24], [26, 11], [0, 6], [-42, 5], [10, 0], [-6, 19], [58, -10], [-32, 18], [-6, 16], [18, 12], [-72, -3], [-16, 0], [22, 6], [-50, 12], [-6, -1], [-34, -8], [2, -16], [60, 8], [0, 16], [28, 9], [36, 9], [0, 24], [-12, -20], [50, 16], [-10, 16], [6, 21], [-20, -10], [-38, 0], [-60, -10], [46, 14], [-8, 9], [12, 8], [-16, 0], [-48, -13], [-6, -10], [4, -10], [2, -24], [-14, 11], [36, 14], [-6, -4], [-14, -5], [-28, 0], [-8, -20], [-44, 7], [42, -14], [26, 16], [-6, -4], [80, 6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6720_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6720_2_a_cq();".
// 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_6720_2_a_cq(:prec:=2)
    chi := MakeCharacter_6720_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(3067) - 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_6720_2_a_cq();".
// 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_6720_2_a_cq( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6720_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-8, 4, 1]>,<13,R![-12, 0, 1]>,<17,R![-2, 1]>,<19,R![-8, 4, 1]>,<23,R![-48, 0, 1]>,<29,R![-2, 1]>,<31,R![-8, 4, 1]>],Snew);
    return Vf;
end function;