
// Make newform 6720.2.a.bu in Magma, downloaded from the LMFDB on 23 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_bu();".
// 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_bu();".
// 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], [2], [4], [-2], [-6], [4], [10], [-2], [2], [-10], [4], [8], [-4], [4], [2], [4], [-6], [0], [4], [-4], [-6], [4], [6], [-8], [20], [14], [12], [-8], [0], [16], [-22], [6], [16], [0], [-20], [-8], [6], [18], [6], [-18], [-22], [24], [-10], [20], [12], [20], [2], [-12], [-6], [14], [-4], [2], [-8], [30], [-14], [10], [30], [4], [-2], [-8], [0], [28], [28], [-12], [-6], [-24], [34], [6], [-2], [28], [-18], [-8], [-8], [6], [-20], [10], [10], [0], [-6], [10], [28], [6], [24], [-30], [-2], [26], [24], [-12], [-24], [-24], [2], [16], [16], [-26], [-10], [-32], [18], [-20], [-36], [4], [18], [-40], [28], [-12], [6], [30], [14], [-36], [-38], [-20], [-46], [-44], [38], [-40], [24], [-16], [30], [42], [6], [6], [24], [-34], [18], [10], [-36], [44], [-44], [16], [-40], [-12], [-34], [42], [-18], [-42], [32], [46], [-2], [-38], [-46], [16], [36], [6], [52], [-36], [18], [-50], [12], [-22], [10], [20], [-40], [20], [-34], [-44], [-30], [8], [34], [-16], [-28], [-40], [-12], [-28], [-24], [52], [28], [-2], [-26], [-30], [-42], [-6], [-54], [26], [-10], [-28], [-18], [20], [62], [8], [-12], [14], [46], [-24], [-42], [50], [-24], [-50], [12], [-52], [-36], [64], [2], [8], [6], [-18], [-26], [-30], [-16], [-30], [-50], [-16], [14], [60], [-10], [-46], [-36], [-18], [22], [22], [-2], [-8], [-36], [44], [-26], [-40], [-30], [-48], [8], [-38], [62], [30], [48], [-52], [38], [-46], [14], [-48], [50], [54], [-6], [16], [14], [-64], [56], [-6], [-48], [58], [68], [-12], [18], [0], [46], [-22], [4], [-52], [-24], [-60], [-36], [2], [34], [52], [26], [14], [10], [-10], [-28], [62], [-60], [72], [-16], [2], [-16], [-74], [-34], [10], [6], [-36], [-24], [30], [-20], [-44], [28], [-68], [-68], [36], [2], [10], [-76], [-56], [20], [8], [62], [-16], [-66], [-10], [28], [46], [-30], [26], [-12], [-12], [-40], [14], [-22], [82], [-30], [44], [88], [64], [-48], [-16], [-56], [80], [-62], [-68], [-22], [50], [-18], [-48], [-30], [-62], [44], [4], [-34], [-8], [78], [-52], [82], [-18], [-42], [-62], [-8], [48], [-74], [-56], [-44], [28], [28], [86], [24], [-6], [4], [44], [4], [34], [-6], [-26], [-28], [-74], [-12], [70], [-72], [64], [-66], [50], [36], [-84], [-2], [-14], [38], [18], [0], [-74], [-42], [12], [-24], [36], [48], [-10], [-18], [-24], [-18], [64], [46], [58], [-64], [58], [82], [-74], [56], [-50], [-30], [2], [80], [-82], [52], [94], [72], [-2], [-76], [-64], [40], [34], [-24], [4], [4], [-20], [-16], [86], [-70], [-42], [16], [18], [-38], [10], [-30], [40], [-62], [30], [12], [24], [2], [26], [-14], [34], [-54], [44], [20], [-48], [28], [-64], [68], [2], [78], [-30], [12], [30], [0], [-26], [76], [-24], [12], [-84], [-2], [-92], [34], [-42], [36], [-6], [54], [108], [64], [-72], [-18], [10], [62], [84]];
    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_bu();".
// 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_bu(: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_bu();".
// 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_bu( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6720_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-2, 1]>,<13,R![-4, 1]>,<17,R![2, 1]>,<19,R![6, 1]>,<23,R![-4, 1]>,<29,R![-10, 1]>,<31,R![2, 1]>],Snew);
    return Vf;
end function;