
// Make newform 4026.2.a.j in Magma, downloaded from the LMFDB on 25 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_j();".
// 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_j();".
// 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], [2], [0], [1], [-2], [6], [-2], [8], [6], [-10], [2], [-4], [-4], [8], [8], [4], [-1], [10], [-8], [-2], [-12], [-12], [0], [14], [14], [4], [16], [-2], [2], [-14], [0], [-6], [16], [-24], [-8], [10], [-4], [-8], [18], [-4], [-26], [4], [14], [0], [-20], [0], [-2], [4], [2], [-18], [-24], [-2], [0], [14], [12], [6], [-14], [2], [2], [-26], [24], [12], [28], [-22], [6], [-18], [-22], [-16], [6], [-6], [-8], [28], [-10], [-4], [16], [0], [26], [8], [-6], [12], [-34], [-12], [34], [-10], [-28], [6], [2], [-32], [24], [-28], [8], [-8], [-12], [-22], [0], [24], [-12], [-32], [-6], [-40], [-22], [-28], [24], [-10], [-6], [-24], [2], [0], [26], [-22], [26], [36], [8], [-18], [24], [22], [32], [8], [-4], [-46], [-14], [30], [28], [-40], [-2], [-42], [0], [8], [46], [16], [-8], [0], [34], [-30], [-46], [42], [-12], [-30], [56], [40], [2], [34], [-52], [-38], [-8], [-30], [-56], [24], [-48], [14], [22], [-6], [-8], [-10], [32], [10], [-30], [-38], [-6], [24], [-34], [-34], [20], [22], [-44], [16], [-58], [-18], [4], [20], [2], [0], [-26], [58], [-26], [-58], [-56], [-36], [50], [-38], [-4], [34], [40], [32], [0], [14], [-4], [50], [0], [54], [20], [-4], [58], [-12], [66], [34], [-26], [0], [36], [-60], [-36], [38], [14], [44], [42], [-70], [-20], [54], [50], [18], [58], [-8], [44], [56], [10], [58], [-24], [-64], [-12], [-10], [-10], [-44], [52], [28], [62], [-48], [64], [-16], [12], [54], [-22], [-8], [-18], [-44], [56], [-34], [-6], [-16], [-8], [12], [-20], [-10], [-22], [-14], [-48], [-44], [-60], [-20], [-4], [14], [-16], [40], [-26], [24], [-4], [-2], [2], [80], [62], [-66], [-56], [-2], [26], [56], [16], [-14], [-22], [26], [36], [58], [-48], [-10], [48], [26], [56], [-4], [74], [54], [64], [24], [-4], [-16], [42], [-52], [32], [70], [0], [52], [18], [72], [60], [16], [4], [50], [-24], [-20], [60], [-20], [28], [74], [12], [-2], [48], [-26], [58], [12], [78], [20], [26], [0], [74], [-14], [-60], [-20], [46], [-36], [-40], [-62], [-10], [-26], [34], [66], [74], [-54], [-2], [-42], [-46], [32], [-54], [14], [-30], [4], [60], [-32], [-32], [-26], [8], [14], [40], [-6], [36], [-36], [22], [46], [60], [-74], [-18], [-56], [18], [-40], [-10], [8], [58], [14], [28], [64], [28], [-54], [72], [-62], [8], [-36], [76], [-56], [-62], [54], [-44], [26], [40], [-78], [84], [24], [28], [66], [-36], [-48], [-10], [56], [-62], [72], [-28], [56], [-4], [42], [84], [-48], [14], [14], [-72], [98], [78], [36], [82], [-32], [-10], [-32], [18], [-80], [-28], [-78], [-28], [34], [72], [-54], [42], [-18], [72], [16], [-4], [98], [54], [-72], [-82], [-38], [-96], [60], [-8], [36], [28], [54], [58], [-48], [68], [-106], [56], [36], [-64], [-14], [80]];
    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_j();".
// 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_j(: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_j();".
// 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_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4026_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<7,R![0, 1]>,<13,R![2, 1]>],Snew);
    return Vf;
end function;