
// Make newform 1008.2.a.b in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1008_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_1008_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_1008_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_1008_a();"
function MakeCharacter_1008_a()
    N := 1008;
    order := 1;
    char_gens := [127, 757, 785, 577];
    v := [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_1008_a_Hecke(Kf)
    return MakeCharacter_1008_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], [0], [-2], [-1], [0], [6], [2], [-4], [-4], [10], [8], [6], [2], [4], [8], [10], [12], [-2], [-12], [-12], [-14], [8], [12], [2], [10], [6], [0], [0], [-2], [-18], [-8], [-4], [-2], [12], [-22], [8], [-18], [20], [0], [-2], [-16], [14], [-20], [18], [-6], [-8], [-20], [0], [20], [14], [-2], [20], [10], [-20], [-6], [12], [6], [-8], [22], [-18], [4], [6], [-4], [0], [-22], [18], [4], [2], [24], [14], [-30], [-12], [16], [-10], [-12], [0], [2], [-2], [6], [18], [20], [-26], [12], [-14], [24], [8], [-34], [10], [-18], [40], [36], [-8], [8], [-24], [-4], [40], [-42], [-6], [-20], [-2], [20], [2], [4], [30], [-20], [18], [28], [18], [-44], [10], [-16], [-26], [30], [12], [16], [-10], [-4], [-48], [10], [8], [-10], [-14], [6], [-24], [-44], [-22], [22], [0], [16], [-26], [20], [20], [-32], [-26], [18], [34], [-10], [-28], [-18], [6], [-36], [-6], [16], [0], [38], [0], [-26], [18], [20], [20], [14], [2], [-4], [-48], [-44], [-20], [8], [26], [-54], [-42], [-56], [-58], [48], [4], [6], [-24], [0], [-26], [-30], [38], [-32], [-18], [-60], [10], [-48], [10], [-20], [-30], [-16], [-50], [56], [44], [22], [-22], [36], [-42], [62], [52], [42], [-48], [-22], [-48], [28], [30], [-8], [-6], [-62], [14], [-22], [64], [34], [0], [14], [-38], [-60], [30], [16], [24], [70], [-28], [18], [30], [-40], [36], [32], [-6], [-48], [-14], [60], [-22], [-42], [-64], [46], [0], [-68], [-10], [66], [20], [-64], [24], [-2], [-20], [32], [-10], [-28], [-48], [26], [-6], [-8], [-72], [-64], [-20], [-8], [46], [-6], [-32], [56], [20], [-36], [20], [62], [-62], [-44], [18], [14], [24], [38], [52], [54], [42], [-56], [0], [-66], [-2], [-30], [4], [2], [-70], [12], [-30], [46], [-52], [2], [0], [42], [-32], [80], [-2], [-70], [-60], [0], [64], [72], [6], [76], [12], [-78], [-22], [0], [-6], [-30], [-20], [-50], [-20], [14], [14], [-48], [46], [60], [-44], [74], [42], [-24], [-72], [12], [-46], [72], [-26], [4], [38], [-16], [-14], [38], [-4], [44], [-14], [-20], [-44], [-86], [6], [28], [-22], [78], [32], [-26], [-70], [68], [-76], [-36], [18], [-18], [-6], [-40], [52], [-20], [-4], [-2], [-70], [2], [8], [54], [-18], [-66], [-56], [58], [72], [46], [-28], [-32], [22], [-20], [58], [-30], [0], [-42], [42], [-48], [-28], [-18], [36], [38], [66], [12], [0], [-60], [-70], [54], [0], [10], [-64], [52], [12], [26], [80], [30], [-20], [-12], [50], [-62], [-14], [62], [-26], [-40], [-18], [4], [72], [64], [62], [76], [-8], [-94], [62], [0], [12], [52], [-22], [-64], [-22], [92], [-30], [-18], [-2], [-72], [18], [46], [-56], [-82], [-34], [-60], [84], [-30], [-78], [56], [92], [-54], [-42], [4], [-16], [18], [16], [90], [-10], [12], [36], [42], [-18], [-32], [30], [-20], [16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1008_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1008_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_1008_2_a_b(:prec:=1)
    chi := MakeCharacter_1008_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_1008_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_1008_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1008_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<11,R![0, 1]>,<13,R![-6, 1]>],Snew);
    return Vf;
end function;