
// Make newform 9016.2.a.d in Magma, downloaded from the LMFDB on 15 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9016_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_9016_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_9016_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_9016_a();"
function MakeCharacter_9016_a()
    N := 9016;
    order := 1;
    char_gens := [2255, 4509, 1473, 1569];
    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_9016_a_Hecke(Kf)
    return MakeCharacter_9016_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 := [[0], [-2], [-2], [0], [2], [-4], [2], [0], [1], [-2], [0], [-4], [-6], [2], [-4], [0], [-2], [10], [2], [-8], [6], [0], [0], [-10], [-10], [16], [-16], [-6], [0], [-6], [-16], [-18], [-2], [-14], [0], [8], [18], [4], [-12], [8], [16], [6], [0], [-2], [-6], [8], [-4], [4], [20], [14], [-6], [8], [-2], [-12], [6], [4], [24], [8], [-22], [18], [4], [-26], [-14], [12], [22], [18], [-32], [2], [4], [8], [-18], [12], [24], [16], [14], [0], [8], [28], [6], [10], [-28], [40], [-12], [2], [32], [-32], [14], [22], [-8], [-8], [8], [24], [-32], [16], [8], [16], [0], [-6], [0], [-6], [24], [36], [-24], [38], [-2], [-18], [34], [-10], [40], [22], [4], [16], [42], [-28], [28], [-18], [-40], [-8], [42], [46], [-22], [-2], [-38], [-4], [-2], [24], [4], [8], [-24], [14], [44], [24], [-24], [48], [-22], [34], [18], [-12], [22], [38], [-10], [30], [-32], [42], [52], [-24], [-44], [10], [-50], [-48], [-50], [18], [36], [-36], [-10], [-12], [40], [34], [38], [6], [-44], [-2], [-8], [36], [-18], [16], [-48], [-44], [-10], [-28], [-14], [32], [60], [18], [28], [38], [-32], [-38], [-56], [14], [16], [36], [6], [-42], [28], [-10], [-18], [36], [38], [28], [-34], [52], [-2], [-36], [-62], [-62], [-38], [-4], [-18], [4], [36], [-12], [-4], [-2], [36], [-12], [-32], [-24], [26], [-10], [42], [48], [-24], [12], [16], [-42], [40], [-30], [44], [-6], [-14], [-32], [-6], [0], [18], [-54], [-10], [24], [-32], [60], [62], [-56], [-68], [6], [-36], [16], [62], [4], [-4], [16], [-6], [-22], [20], [-26], [30], [-8], [32], [40], [26], [4], [12], [74], [-4], [-38], [24], [56], [-40], [28], [-12], [-14], [-56], [-10], [4], [10], [18], [0], [-28], [-46], [38], [-18], [56], [34], [18], [-72], [-66], [-24], [-32], [22], [18], [-12], [-84], [-68], [32], [-62], [-54], [24], [46], [36], [44], [54], [56], [-8], [-6], [28], [2], [74], [-8], [-36], [-2], [-54], [42], [48], [40], [24], [-50], [-22], [36], [2], [64], [-74], [36], [-44], [-58], [-16], [-16], [-42], [30], [-24], [-70], [-34], [16], [70], [-80], [-24], [-66], [66], [-14], [62], [-8], [0], [34], [-46], [76], [-14], [70], [82], [-4], [-34], [70], [-56], [-2], [94], [12], [-32], [-32], [-76], [4], [44], [56], [-90], [54], [-10], [18], [-80], [20], [22], [88], [88], [58], [24], [92], [62], [-32], [-90], [-2], [42], [20], [-20], [-10], [48], [-34], [88], [20], [-8], [-66], [30], [-44], [34], [-42], [18], [30], [-74], [-56], [-74], [28], [-44], [56], [28], [-46], [-40], [-22], [76], [100], [-38], [-68], [38], [16], [78], [46], [58], [40], [-2], [-32], [78], [28], [-72], [44], [-6], [-56], [6], [22], [-78], [6], [2], [42], [-24], [-48], [56], [-90], [-24], [72], [102], [72], [66], [-14], [24], [18], [-6], [-30], [-60]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9016_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9016_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_9016_2_a_d(:prec:=1)
    chi := MakeCharacter_9016_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_9016_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_9016_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9016_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<5,R![2, 1]>,<11,R![-2, 1]>,<13,R![4, 1]>],Snew);
    return Vf;
end function;