
// Make newform 6720.2.a.bw 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_bw();".
// 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_bw();".
// 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], [2], [-8], [2], [2], [4], [0], [10], [-12], [-6], [12], [0], [-6], [8], [4], [2], [-14], [10], [-8], [-4], [2], [-6], [-16], [12], [-14], [12], [10], [0], [18], [-20], [24], [-14], [-20], [-14], [8], [-14], [26], [16], [28], [16], [-12], [-14], [18], [24], [18], [4], [18], [16], [-30], [-8], [18], [-6], [-4], [-6], [20], [24], [-22], [2], [4], [2], [12], [-22], [18], [0], [16], [-14], [20], [-16], [26], [-14], [2], [26], [28], [-22], [8], [2], [16], [-4], [34], [10], [-14], [-32], [20], [0], [-24], [36], [-20], [-24], [-14], [-30], [-20], [-30], [-4], [-30], [-12], [10], [36], [-14], [28], [18], [0], [-38], [-32], [34], [-46], [28], [48], [-30], [-44], [24], [2], [-36], [-30], [-14], [-38], [-36], [-44], [34], [26], [16], [8], [-14], [-36], [-32], [8], [-30], [-30], [18], [-6], [36], [34], [26], [-20], [42], [-8], [-36], [-38], [-40], [10], [26], [28], [-24], [-22], [-22], [-4], [8], [52], [24], [0], [-54], [-38], [18], [52], [-46], [-8], [-12], [42], [-24], [-8], [26], [50], [-22], [-12], [-38], [-32], [-6], [-8], [18], [4], [58], [-40], [26], [16], [12], [50], [-22], [-40], [42], [-38], [4], [42], [0], [-46], [-20], [-4], [-30], [4], [-22], [50], [-54], [18], [-8], [50], [40], [-54], [-14], [4], [2], [-40], [20], [26], [60], [34], [42], [-8], [-36], [-40], [-38], [-48], [10], [48], [18], [-6], [-32], [18], [-64], [4], [-38], [-54], [40], [-8], [4], [-6], [-28], [8], [-6], [-36], [48], [-46], [-6], [-44], [8], [-12], [-68], [-40], [-62], [-46], [24], [48], [44], [-44], [-8], [-22], [42], [-48], [-6], [18], [44], [26], [12], [26], [-22], [16], [-12], [2], [-46], [34], [-28], [-46], [2], [68], [42], [42], [-52], [-54], [24], [2], [24], [-36], [50], [74], [44], [-64], [0], [24], [-30], [28], [-56], [50], [-6], [-32], [-6], [18], [-76], [-30], [68], [-22], [34], [8], [-86], [-12], [-76], [-6], [-30], [24], [4], [-12], [2], [76], [26], [-32], [34], [-32], [-86], [-46], [44], [0], [-38], [-36], [72], [18], [66], [36], [-54], [-78], [-16], [-46], [-78], [60], [28], [-72], [-54], [66], [18], [-88], [4], [84], [-20], [-78], [2], [-38], [-64], [82], [-46], [74], [48], [-30], [44], [50], [-12], [48], [-86], [20], [-54], [-46], [-80], [58], [42], [-48], [20], [-22], [-64], [34], [34], [40], [68], [-28], [74], [-14], [-56], [-70], [12], [76], [24], [-54], [-16], [42], [60], [-40], [-46], [-22], [74], [-30], [66], [88], [58], [-60], [-72], [-40], [58], [4], [0], [-30], [-6], [-28], [68], [0], [-6], [72], [-46], [-44], [74], [-86], [-6], [16], [-22], [74], [80], [26], [-14], [84], [28], [98], [-70], [-36], [44], [58], [-62], [-24], [8], [-78], [72], [-30], [26], [-72], [20], [58], [-14], [4], [106], [28], [40], [26], [92], [-76], [48], [50], [-70], [42], [-22], [20]];
    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_bw();".
// 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_bw(: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_bw();".
// 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_bw( : 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![-2, 1]>,<31,R![8, 1]>],Snew);
    return Vf;
end function;