
// Make newform 4026.2.a.b in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4026_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_4026_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_4026_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_4026_a();"
function MakeCharacter_4026_a()
    N := 4026;
    order := 1;
    char_gens := [1343, 1465, 3235];
    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_4026_a_Hecke(Kf)
    return MakeCharacter_4026_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], [1], [3], [-4], [-1], [2], [3], [-4], [-9], [-6], [2], [-1], [-6], [-7], [-6], [6], [6], [1], [-4], [3], [-7], [-4], [-3], [-15], [11], [-12], [5], [3], [-4], [18], [17], [-21], [-6], [-13], [18], [-10], [-22], [-22], [6], [0], [15], [-7], [12], [14], [18], [-25], [8], [20], [18], [14], [-18], [0], [-13], [0], [6], [-6], [18], [11], [-22], [6], [-16], [9], [11], [0], [-4], [-9], [-13], [-16], [-9], [35], [6], [3], [-28], [-19], [-34], [24], [6], [-19], [-3], [32], [-18], [-13], [-18], [8], [-37], [39], [-6], [-16], [-21], [5], [-18], [-12], [32], [-24], [-13], [30], [18], [-42], [-7], [5], [8], [-24], [-36], [12], [14], [2], [42], [9], [45], [35], [35], [-34], [-3], [26], [-22], [15], [23], [-3], [48], [45], [-22], [-28], [-18], [-15], [-16], [0], [17], [0], [-28], [26], [5], [51], [11], [2], [-42], [-4], [-9], [32], [42], [-18], [-7], [-30], [-22], [-15], [2], [42], [-10], [-12], [32], [-36], [41], [-42], [-19], [27], [17], [-30], [29], [-18], [41], [18], [12], [-9], [-31], [-3], [-18], [24], [-16], [17], [-34], [36], [54], [-4], [-45], [5], [-10], [0], [50], [-30], [-34], [35], [20], [-18], [-22], [0], [18], [48], [2], [56], [-46], [-24], [-64], [36], [-10], [-57], [54], [6], [-7], [5], [36], [24], [57], [-34], [-22], [62], [21], [57], [-22], [60], [-3], [-61], [29], [9], [-1], [-54], [69], [8], [-40], [-12], [-18], [-30], [-46], [41], [15], [41], [36], [-61], [-54], [-30], [-22], [-15], [17], [8], [14], [45], [14], [-27], [-58], [-48], [60], [42], [-42], [56], [-40], [38], [57], [0], [35], [-60], [-28], [42], [14], [-18], [-27], [26], [-3], [0], [38], [26], [-48], [-28], [-1], [45], [-58], [26], [54], [8], [-18], [48], [-64], [63], [50], [-16], [-10], [-40], [-4], [29], [-30], [44], [68], [36], [-21], [29], [-24], [53], [23], [30], [-16], [45], [-37], [-6], [-60], [-81], [-54], [48], [44], [-33], [-43], [72], [12], [-55], [14], [78], [-25], [6], [-4], [53], [-12], [-40], [-12], [71], [15], [81], [-21], [-58], [-72], [-10], [-12], [-9], [-25], [-87], [-64], [-46], [-48], [44], [51], [-28], [-79], [-73], [3], [12], [-82], [-57], [-70], [-81], [20], [-24], [38], [24], [92], [2], [-10], [-27], [-90], [-82], [-63], [-66], [-79], [-19], [-9], [-57], [11], [26], [36], [35], [14], [60], [78], [18], [-75], [-51], [8], [-12], [24], [-60], [-28], [20], [90], [44], [35], [54], [56], [66], [-57], [-16], [80], [-78], [-63], [-4], [-36], [-82], [30], [-30], [14], [42], [-46], [93], [-55], [-85], [50], [12], [62], [39], [-15], [-4], [-36], [32], [-82], [0], [-82], [87], [20], [3], [-100], [-9], [42], [-1], [-88], [-90], [-40], [-60], [2], [-6], [-6], [44], [32], [-72], [30], [-1], [-30], [-24], [39], [41], [18], [-84], [35], [102], [-36], [-18], [-79], [96]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4026_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4026_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_4026_2_a_b(:prec:=1)
    chi := MakeCharacter_4026_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_4026_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_4026_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4026_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-3, 1]>,<7,R![4, 1]>,<13,R![-2, 1]>],Snew);
    return Vf;
end function;