
// Make newform 6720.2.a.by in Magma, downloaded from the LMFDB on 16 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_by();".
// 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_by();".
// 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], [0], [6], [0], [6], [-10], [0], [-12], [6], [4], [-2], [-8], [-4], [10], [0], [-12], [-10], [-14], [-10], [0], [4], [2], [-18], [-16], [-12], [14], [0], [-10], [8], [-22], [-24], [-12], [14], [-4], [-10], [20], [26], [6], [0], [-4], [8], [28], [14], [-10], [-4], [18], [-20], [18], [-24], [14], [-32], [-2], [-22], [4], [-2], [12], [16], [-22], [-18], [20], [10], [-20], [22], [18], [-12], [-32], [6], [-28], [-12], [38], [2], [18], [26], [12], [10], [20], [18], [-40], [12], [-6], [26], [-18], [8], [-12], [24], [8], [36], [-20], [20], [-18], [6], [-4], [42], [40], [-18], [-4], [10], [12], [2], [4], [-22], [12], [-22], [8], [22], [-26], [-8], [48], [-38], [-44], [-36], [-18], [-44], [22], [26], [-42], [-28], [16], [6], [50], [8], [32], [-14], [-20], [-48], [-8], [22], [-34], [26], [14], [-20], [-50], [-38], [16], [-2], [32], [-20], [-10], [40], [26], [10], [-8], [-16], [22], [-10], [-56], [52], [-48], [20], [16], [54], [-38], [14], [28], [-58], [40], [-4], [30], [-36], [-40], [58], [-62], [-42], [60], [14], [-12], [18], [16], [30], [60], [54], [0], [-26], [40], [12], [6], [18], [-64], [22], [-2], [4], [-38], [24], [-30], [36], [-20], [-34], [-20], [-30], [2], [38], [-46], [44], [30], [8], [2], [10], [12], [-42], [-64], [20], [-14], [24], [10], [22], [-24], [12], [-24], [18], [64], [-50], [8], [-66], [50], [64], [18], [24], [44], [-30], [-38], [-60], [-8], [52], [54], [48], [24], [18], [-44], [-4], [50], [22], [20], [72], [12], [-40], [-40], [-6], [-6], [16], [-40], [36], [52], [0], [-58], [-50], [-8], [-46], [6], [60], [-22], [20], [30], [-6], [-24], [36], [-42], [26], [-30], [16], [14], [22], [24], [-26], [-66], [-32], [10], [-32], [-62], [-56], [36], [42], [26], [44], [68], [40], [-28], [-2], [60], [60], [2], [-42], [72], [54], [-66], [-52], [6], [-20], [22], [30], [64], [-34], [12], [12], [-38], [46], [-40], [-4], [12], [-62], [4], [-66], [-36], [-10], [84], [78], [50], [40], [-72], [34], [44], [-4], [2], [42], [-32], [-46], [-2], [-64], [14], [2], [36], [44], [-16], [-58], [50], [-2], [-40], [4], [60], [-84], [74], [-86], [-30], [-32], [-58], [-42], [-58], [0], [30], [-36], [94], [-8], [-32], [-18], [-88], [58], [-10], [-64], [-46], [-70], [-24], [-12], [14], [-40], [86], [-66], [96], [-52], [12], [-70], [70], [-16], [42], [-76], [88], [0], [-18], [-8], [-74], [-60], [-20], [-78], [-66], [42], [30], [54], [-8], [38], [56], [-20], [-16], [-38], [-16], [-4], [2], [22], [-36], [-52], [92], [74], [56], [46], [52], [-74], [-42], [-50], [-40], [-14], [102], [16], [-58], [42], [-20], [-4], [34], [-10], [12], [-60], [-62], [78], [-36], [72], [2], [-60], [-50], [42], [88], [-36], [-102], [-106], [-92], [-30], [80], [72], [10], [-12], [12], [-44], [-38], [78], [-38], [-70], [72]];
    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_by();".
// 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_by(: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_by();".
// 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_by( : 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![0, 1]>,<29,R![-6, 1]>,<31,R![0, 1]>],Snew);
    return Vf;
end function;