
// Make newform 6566.2.a.b in Magma, downloaded from the LMFDB on 15 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6566_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_6566_2_a_b();".
// 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_6566_2_a_b();".
// 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_6566_a();"
function MakeCharacter_6566_a()
    N := 6566;
    order := 1;
    char_gens := [4021, 1275];
    v := [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_6566_a_Hecke(Kf)
    return MakeCharacter_6566_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]];
    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 := [[-1], [-2], [4], [0], [4], [4], [4], [0], [0], [10], [-8], [2], [10], [-4], [2], [-6], [4], [8], [-1], [0], [4], [-4], [8], [0], [-14], [-12], [-14], [8], [-2], [10], [16], [0], [-2], [-10], [22], [16], [-22], [8], [6], [-18], [12], [-10], [-12], [2], [-26], [-22], [-8], [-14], [-20], [-8], [-26], [20], [-8], [-30], [24], [-16], [-6], [-8], [-26], [22], [-24], [18], [4], [-28], [-18], [-10], [4], [-10], [-12], [10], [14], [24], [0], [34], [-28], [16], [2], [2], [-18], [14], [16], [38], [32], [-6], [34], [-20], [30], [-2], [18], [36], [-20], [30], [32], [8], [12], [20], [-6], [14], [0], [-2], [28], [-42], [-30], [-30], [28], [-2], [18], [-26], [-16], [16], [-22], [26], [22], [-4], [-24], [22], [-48], [-32], [14], [36], [4], [-34], [24], [36], [-36], [-30], [-6], [-6], [-4], [48], [-4], [-24], [24], [-22], [-12], [26], [54], [18], [6], [2], [14], [46], [40], [-44], [-34], [10], [2], [-18], [-20], [-48], [-22], [0], [-36], [-54], [-24], [-48], [24], [10], [14], [0], [-4], [42], [-8], [12], [-30], [60], [60], [-18], [-34], [-24], [-36], [-14], [32], [62], [-44], [58], [44], [-6], [-28], [-38], [16], [48], [-38], [-12], [0], [-22], [-18], [2], [-26], [-12], [56], [-52], [-20], [60], [-4], [-16], [-18], [-62], [-14], [-22], [54], [-26], [-12], [58], [14], [22], [30], [-24], [-54], [2], [58], [-32], [20], [-58], [24], [-2], [24], [-60], [12], [26], [58], [74], [-30], [16], [2], [-22], [12], [60], [58], [44], [-50], [42], [24], [-6], [-8], [-48], [24], [-26], [36], [-74], [8], [-50], [-4], [-18], [-54], [-36], [-22], [-6], [36], [12], [38], [28], [12], [-4], [12], [4], [42], [4], [14], [68], [40], [-52], [68], [-52], [-60], [40], [14], [50], [-44], [-26], [-36], [20], [22], [64], [12], [-20], [4], [-10], [-6], [-62], [18], [-32], [-56], [0], [-46], [-48], [38], [-18], [60], [-14], [-2], [-14], [-62], [48], [-6], [-58], [-48], [-16], [-24], [-26], [6], [2], [16], [68], [16], [-34], [-12], [82], [-64], [38], [-28], [26], [-54], [36], [-40], [18], [32], [40], [-44], [6], [-16], [26], [-90], [-92], [-42], [-28], [52], [28], [-64], [-2], [-42], [18], [-76], [30], [88], [-48], [-94], [24], [-30], [90], [70], [6], [44], [-32], [50], [-36], [4], [-52], [2], [64], [-64], [-54], [54], [44], [-90], [30], [42], [-30], [26], [-76], [62], [-48], [-56], [-20], [80], [6], [-98], [-72], [-62], [84], [80], [24], [66], [60], [-54], [8], [52], [-68], [78], [-8], [-28], [-30], [-24], [2], [-38], [32], [-52], [-46], [84], [-96], [90], [-88], [64], [10], [-40], [74], [2], [-64], [28], [54], [40], [22], [-28], [-30], [-64], [24], [10], [14], [-56], [6], [24], [42], [64], [48], [-26], [-90], [40], [22], [-72], [-46], [6], [-22], [68], [-14], [-58], [-34], [76], [-10], [16], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6566_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6566_2_a_b();".
// 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_6566_2_a_b(:prec:=1)
    chi := MakeCharacter_6566_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(2999) - 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_6566_2_a_b();".
// 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_6566_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6566_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<5,R![-4, 1]>,<11,R![-4, 1]>,<13,R![-4, 1]>],Snew);
    return Vf;
end function;