
// Make newform 6720.2.a.bf in Magma, downloaded from the LMFDB on 20 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_bf();".
// 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_bf();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
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], [-1], [1], [1], [4], [2], [-2], [-4], [0], [-10], [0], [-6], [-10], [-8], [4], [-6], [0], [-10], [-8], [8], [-2], [-4], [-12], [6], [6], [6], [8], [12], [-6], [-18], [0], [-16], [-18], [12], [14], [20], [10], [-8], [12], [6], [-20], [-2], [16], [-14], [-14], [0], [12], [-8], [4], [-10], [-2], [16], [10], [24], [-2], [8], [14], [8], [-22], [10], [-28], [14], [-4], [-8], [22], [2], [-20], [2], [-36], [14], [-18], [-32], [-8], [-38], [20], [20], [14], [-30], [2], [-22], [32], [26], [24], [-34], [-24], [-4], [-6], [34], [30], [16], [20], [-24], [8], [12], [-20], [12], [-34], [30], [44], [2], [-8], [10], [-4], [26], [-4], [22], [-12], [-34], [-8], [18], [-32], [26], [-2], [4], [-20], [10], [-44], [-12], [-46], [36], [-18], [26], [30], [12], [4], [38], [-46], [-24], [-16], [-6], [12], [24], [-44], [-38], [-26], [2], [-18], [28], [54], [-22], [-28], [-18], [-32], [12], [14], [-8], [-6], [-10], [20], [24], [50], [-42], [-32], [36], [-16], [-8], [-20], [14], [-42], [46], [-28], [14], [16], [8], [-18], [-28], [-20], [-46], [2], [14], [-20], [54], [48], [-22], [-56], [30], [12], [-2], [24], [46], [-32], [40], [-22], [54], [8], [22], [34], [-4], [26], [56], [-34], [52], [-44], [-34], [-36], [6], [-14], [-62], [-18], [44], [62], [32], [-54], [34], [-32], [6], [48], [-4], [50], [-20], [42], [22], [-16], [12], [-48], [58], [-56], [22], [0], [-46], [50], [16], [-30], [8], [68], [42], [-26], [0], [56], [-60], [-14], [4], [-20], [18], [-12], [-28], [-62], [-54], [44], [0], [-76], [60], [32], [26], [14], [24], [32], [-8], [-4], [72], [2], [70], [-8], [-46], [54], [-60], [34], [-60], [-18], [-26], [-40], [-36], [46], [-30], [30], [60], [-50], [62], [-32], [-22], [-26], [-16], [6], [44], [14], [-56], [-4], [-6], [-22], [48], [-4], [-60], [-12], [-10], [68], [-24], [-22], [2], [16], [-2], [30], [36], [78], [-80], [-14], [-50], [16], [-42], [-48], [-28], [-66], [18], [-76], [28], [36], [2], [-60], [54], [24], [10], [36], [-82], [34], [-40], [-72], [10], [-16], [48], [-82], [-38], [-84], [34], [46], [-56], [-42], [-6], [20], [76], [48], [18], [-46], [-46], [-8], [36], [76], [68], [82], [46], [-70], [-16], [58], [-58], [70], [52], [-86], [-20], [70], [32], [-40], [-34], [-52], [-30], [-58], [-24], [50], [6], [32], [-72], [22], [-24], [10], [-26], [-8], [-84], [4], [34], [-2], [16], [-54], [12], [-12], [24], [-90], [-40], [-62], [80], [-32], [-74], [38], [-42], [-82], [30], [64], [6], [76], [-44], [-20], [-62], [88], [-12], [50], [38], [-92], [-36], [-88], [-22], [-32], [-2], [60], [-58], [46], [-90], [16], [-42], [6], [-40], [-46], [-86], [100], [-56], [-34], [18], [36], [-28], [90], [46], [-24], [64], [-34], [28], [38], [-14], [32], [-72], [22], [6], [-44], [-78], [-92], [8], [42], [60], [-60], [-108], [-62], [78], [90], [106], [64]];
    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_bf();".
// 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_bf(:prec:=1)
    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_bf();".
// 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_bf( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6720_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-4, 1]>,<13,R![-2, 1]>,<17,R![2, 1]>,<19,R![4, 1]>,<23,R![0, 1]>,<29,R![10, 1]>,<31,R![0, 1]>],Snew);
    return Vf;
end function;