
// Make newform 6720.2.a.br in Magma, downloaded from the LMFDB on 14 May 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_br();".
// 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_br();".
// 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], [0], [-2], [-2], [4], [-8], [2], [-4], [6], [-6], [-4], [0], [-6], [0], [-6], [4], [-8], [10], [-12], [4], [-6], [-6], [-6], [0], [4], [-6], [-2], [8], [0], [-2], [12], [-6], [-12], [-10], [-4], [-24], [18], [0], [-6], [8], [10], [10], [4], [20], [8], [-20], [-14], [-2], [-24], [-14], [0], [-18], [16], [18], [28], [-18], [-30], [-12], [-14], [20], [0], [10], [18], [20], [-22], [-12], [-22], [-26], [-16], [0], [-34], [20], [24], [10], [14], [-30], [10], [8], [-22], [24], [34], [4], [-12], [-30], [10], [-14], [16], [4], [-32], [0], [-16], [12], [24], [18], [18], [-4], [10], [-28], [-22], [-20], [-38], [-28], [-38], [-12], [14], [-16], [42], [32], [22], [-2], [-20], [44], [-6], [4], [8], [42], [8], [-46], [-22], [-6], [-36], [20], [34], [-6], [8], [48], [38], [44], [8], [20], [-2], [18], [18], [50], [4], [-46], [18], [-28], [10], [16], [44], [-14], [-32], [-42], [22], [20], [40], [54], [-46], [-20], [56], [12], [48], [-20], [-22], [2], [-14], [-12], [46], [-48], [0], [-34], [0], [-20], [14], [-14], [58], [-40], [-14], [-24], [34], [-4], [26], [-4], [-22], [-8], [2], [24], [56], [14], [6], [-16], [-22], [-2], [-36], [26], [-40], [2], [44], [-12], [-30], [-52], [6], [2], [14], [-2], [0], [50], [-12], [38], [34], [8], [-22], [20], [-20], [-14], [28], [-70], [-54], [32], [12], [-56], [26], [-32], [42], [-40], [26], [58], [36], [26], [16], [-28], [-14], [-42], [-24], [-8], [-24], [46], [60], [52], [50], [28], [-8], [34], [-46], [-16], [56], [-52], [36], [-16], [34], [54], [24], [-16], [40], [20], [8], [70], [-22], [24], [74], [42], [-56], [-46], [-28], [-30], [42], [-40], [20], [-30], [46], [38], [76], [50], [18], [-12], [34], [-14], [76], [10], [-28], [-14], [-48], [-12], [-46], [-70], [16], [-56], [-68], [-8], [-22], [20], [-40], [34], [-46], [60], [58], [-14], [28], [46], [0], [-82], [2], [20], [-30], [40], [44], [10], [10], [44], [60], [-36], [-78], [60], [74], [24], [14], [64], [-54], [-30], [36], [24], [-38], [56], [-48], [82], [2], [-4], [-78], [50], [72], [6], [-30], [-4], [84], [48], [2], [34], [-38], [36], [-20], [-68], [60], [-6], [-10], [-38], [40], [-34], [22], [58], [-20], [-46], [-40], [-30], [-28], [0], [2], [-36], [-86], [18], [-56], [-70], [14], [32], [48], [-42], [-48], [-18], [-30], [80], [0], [-52], [90], [-38], [-72], [-70], [-72], [4], [0], [-54], [-68], [-10], [-96], [-48], [-86], [10], [-94], [-30], [-2], [56], [22], [-68], [40], [-44], [-2], [-28], [-72], [66], [-38], [-72], [68], [24], [-54], [-20], [-102], [12], [-86], [50], [70], [-8], [78], [-6], [-44], [6], [18], [4], [-40], [-22], [18], [12], [20], [-46], [-78], [-16], [-56], [-58], [48], [-46], [94], [64], [-96], [42], [10], [84], [58], [28], [0], [42], [64], [4], [-72], [-98], [-46], [10], [10], [-76]];
    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_br();".
// 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_br(: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_br();".
// 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_br( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6720_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![0, 1]>,<13,R![2, 1]>,<17,R![2, 1]>,<19,R![-4, 1]>,<23,R![8, 1]>,<29,R![-2, 1]>,<31,R![4, 1]>],Snew);
    return Vf;
end function;