
// Make newform 8001.2.a.d in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8001_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_8001_2_a_d();".
// 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_8001_2_a_d();".
// 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_8001_a();"
function MakeCharacter_8001_a()
    N := 8001;
    order := 1;
    char_gens := [3557, 1144, 7750];
    v := [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_8001_a_Hecke(Kf)
    return MakeCharacter_8001_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 := [[1], [0], [-4], [-1], [0], [2], [-6], [2], [0], [10], [6], [10], [-10], [-4], [6], [6], [-12], [-6], [-12], [0], [14], [-12], [0], [8], [-4], [0], [-14], [0], [10], [2], [1], [-18], [6], [-4], [6], [8], [10], [8], [-8], [-20], [-8], [16], [0], [-14], [6], [22], [0], [20], [-6], [16], [-10], [16], [24], [6], [-12], [4], [-18], [2], [6], [14], [-20], [-12], [16], [0], [-4], [-26], [-20], [30], [-28], [-16], [18], [0], [2], [-34], [-12], [-6], [6], [-34], [22], [-36], [18], [-18], [-24], [14], [-28], [4], [-10], [10], [-20], [-16], [-24], [-38], [-8], [36], [-20], [-6], [-6], [-14], [-42], [46], [28], [2], [28], [42], [4], [14], [34], [16], [8], [-12], [22], [34], [14], [-40], [-32], [-26], [38], [-48], [-10], [-36], [10], [30], [-22], [12], [-24], [42], [-10], [-6], [-4], [-14], [16], [-16], [-8], [-34], [-12], [-44], [-18], [30], [42], [-2], [-38], [-10], [-12], [36], [-32], [40], [40], [-12], [4], [-40], [2], [24], [56], [-44], [-16], [12], [0], [12], [-44], [2], [12], [-26], [-16], [10], [2], [6], [24], [-40], [-34], [-38], [12], [32], [56], [-46], [-40], [-20], [-8], [-54], [-46], [-20], [28], [-60], [-6], [54], [40], [32], [54], [-26], [-22], [-42], [-36], [20], [-20], [-54], [16], [30], [-50], [62], [66], [-12], [22], [62], [-42], [0], [28], [44], [-46], [4], [-58], [46], [-66], [22], [40], [10], [42], [60], [-24], [-16], [32], [-62], [-30], [-26], [18], [4], [-2], [42], [-42], [56], [42], [-44], [34], [54], [52], [-54], [-52], [-60], [-20], [6], [-12], [-32], [-12], [-8], [42], [-34], [-72], [18], [-60], [30], [44], [-40], [-14], [20], [-48], [-36], [-68], [-36], [-62], [54], [22], [28], [8], [8], [-70], [18], [28], [-52], [-18], [50], [-76], [6], [68], [-4], [8], [-64], [-68], [-40], [-44], [-50], [-46], [52], [32], [-64], [-34], [8], [-48], [0], [-62], [6], [44], [-48], [78], [-62], [6], [-46], [-2], [-66], [-28], [26], [-22], [46], [-10], [42], [-80], [36], [52], [-50], [-52], [-82], [-64], [-6], [-54], [42], [-22], [52], [8], [-32], [-40], [-20], [-46], [6], [26], [-18], [80], [-80], [-6], [50], [52], [-30], [72], [18], [-38], [42], [-44], [-74], [-4], [20], [-46], [24], [6], [-34], [78], [-46], [72], [-48], [82], [36], [4], [12], [52], [18], [-40], [58], [78], [76], [74], [10], [90], [80], [-78], [56], [30], [32], [-52], [-36], [32], [38], [18], [-16], [26], [12], [-56], [-32], [86], [54], [46], [-60], [72], [72], [50], [48], [-38], [26], [-72], [2], [-84], [-20], [-64], [-8], [-12], [-68], [-38], [-54], [16], [-60], [20], [86], [-2], [6], [48], [-50], [-50], [90], [-68], [56], [-70], [10], [-2], [-86], [46], [-90], [-52], [-38], [36], [-52], [94], [-84], [-104], [56], [82], [36], [-42], [28], [-8], [-66], [-14], [-46], [28], [-10], [-34], [-24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8001_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8001_2_a_d();".
// 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_8001_2_a_d(:prec:=1)
    chi := MakeCharacter_8001_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_8001_2_a_d();".
// 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_8001_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8001_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>,<5,R![4, 1]>],Snew);
    return Vf;
end function;