
// Make newform 6720.2.a.cg 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_cg();".
// 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_cg();".
// 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], [-6], [0], [-8], [-10], [-8], [-2], [-2], [-8], [4], [-10], [-4], [6], [0], [-12], [-6], [-8], [4], [14], [2], [6], [-16], [4], [18], [6], [16], [-4], [6], [16], [6], [0], [10], [-16], [-12], [-18], [4], [-18], [12], [10], [22], [24], [4], [-24], [-20], [22], [-18], [-12], [-14], [20], [-14], [0], [-2], [8], [22], [-22], [-28], [30], [-20], [0], [-6], [30], [-20], [-6], [-20], [-18], [18], [-20], [16], [30], [28], [-28], [-10], [-14], [34], [26], [-28], [26], [-36], [34], [32], [-4], [26], [-6], [-18], [8], [-12], [-8], [8], [-20], [-28], [-12], [-2], [-18], [28], [10], [32], [-18], [-4], [-6], [20], [-14], [-12], [-38], [20], [10], [-40], [14], [-2], [-8], [-8], [-6], [4], [28], [14], [12], [30], [10], [22], [-28], [0], [-42], [-14], [8], [-16], [-14], [4], [-8], [-16], [-2], [-42], [10], [-18], [-4], [-18], [42], [0], [-18], [0], [28], [-34], [8], [-22], [-38], [-24], [-8], [-2], [-2], [-48], [20], [-8], [-20], [-24], [-18], [42], [-18], [-52], [-50], [-8], [36], [6], [28], [16], [10], [2], [-42], [36], [-10], [44], [34], [56], [-58], [4], [38], [-16], [-34], [-24], [52], [-18], [2], [-24], [54], [38], [36], [-38], [24], [18], [-12], [52], [-34], [28], [-46], [2], [-34], [-46], [-20], [46], [-64], [-14], [58], [-28], [-42], [-8], [-44], [-30], [-8], [58], [-10], [8], [12], [8], [-62], [16], [54], [-48], [62], [-14], [56], [34], [-40], [60], [-46], [26], [-4], [8], [28], [46], [-16], [64], [34], [4], [28], [-46], [-42], [-68], [-40], [-36], [40], [-56], [10], [-54], [0], [-8], [-20], [-36], [-24], [-34], [6], [-48], [-62], [6], [4], [58], [68], [-2], [-22], [72], [36], [46], [-6], [-14], [-64], [30], [-50], [64], [38], [-58], [24], [-70], [24], [18], [-24], [-12], [58], [26], [36], [4], [80], [-12], [-10], [-20], [36], [2], [-42], [-32], [-2], [14], [12], [-66], [4], [62], [-66], [40], [-66], [36], [-36], [10], [-18], [-64], [-36], [68], [2], [20], [22], [-60], [46], [36], [-2], [-30], [-16], [0], [-46], [20], [-12], [-14], [26], [0], [2], [78], [64], [6], [2], [-36], [-52], [-24], [6], [66], [-34], [0], [36], [36], [-68], [-70], [-6], [82], [-32], [-2], [30], [54], [88], [46], [-12], [38], [-32], [0], [-50], [8], [26], [6], [-16], [-30], [-6], [24], [-36], [-42], [-16], [46], [38], [24], [4], [12], [58], [-58], [48], [-54], [12], [-24], [-24], [30], [32], [-50], [12], [-44], [66], [70], [-70], [30], [-66], [24], [-18], [8], [-84], [56], [-6], [-8], [-68], [34], [54], [36], [92], [-28], [-86], [64], [6], [60], [-90], [14], [54], [-24], [18], [-42], [-24], [62], [-54], [-84], [-12], [18], [22], [-36], [-20], [-78], [-34], [52], [-24], [82], [-92], [62], [-38], [-96], [84], [-86], [86], [4], [18], [32], [-24], [-86], [60], [20], [84], [-70], [-42], [90], [-70], [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_cg();".
// 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_cg(: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_cg();".
// 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_cg( : 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![6, 1]>,<19,R![0, 1]>,<23,R![8, 1]>,<29,R![10, 1]>,<31,R![8, 1]>],Snew);
    return Vf;
end function;