
// Make newform 4026.2.a.e 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_e();".
// 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_e();".
// 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], [4], [4], [-1], [6], [-6], [0], [-4], [-6], [-6], [0], [12], [10], [-6], [-14], [12], [-1], [4], [8], [-6], [4], [6], [-6], [2], [6], [4], [6], [-14], [-2], [0], [2], [2], [2], [6], [0], [-20], [20], [8], [6], [-24], [-12], [-8], [-24], [-10], [-28], [-18], [-14], [8], [-30], [-10], [-24], [10], [-28], [26], [12], [-28], [16], [30], [2], [24], [2], [10], [0], [22], [-12], [20], [0], [2], [22], [-30], [16], [-12], [-34], [-4], [0], [-10], [-8], [6], [32], [12], [20], [20], [10], [0], [12], [30], [-4], [-34], [4], [-20], [28], [-20], [-14], [-16], [-20], [2], [10], [22], [-6], [-38], [-22], [-14], [-24], [-12], [14], [12], [18], [12], [-2], [-48], [-22], [34], [20], [34], [-18], [4], [0], [-46], [50], [32], [0], [6], [8], [-44], [-30], [-4], [-42], [20], [-26], [30], [-8], [28], [-34], [30], [-36], [-24], [-2], [36], [-16], [50], [22], [-50], [10], [-34], [-42], [-6], [-56], [20], [46], [-10], [14], [16], [-16], [-4], [58], [-16], [34], [26], [-10], [4], [-14], [24], [-40], [18], [-24], [-40], [-10], [-56], [14], [44], [-22], [24], [-22], [22], [-42], [28], [42], [-52], [22], [26], [28], [62], [36], [-30], [34], [2], [-4], [20], [48], [14], [-6], [-60], [-28], [-24], [50], [18], [-44], [-28], [40], [-18], [44], [32], [-20], [8], [-48], [-30], [-24], [-54], [-56], [-2], [36], [16], [20], [-24], [-58], [40], [56], [-46], [-18], [-26], [64], [46], [48], [8], [46], [54], [54], [-32], [6], [32], [-32], [16], [34], [4], [-24], [-34], [22], [-36], [6], [60], [-70], [6], [-16], [18], [-64], [-60], [-24], [58], [-24], [22], [52], [52], [26], [-54], [-12], [68], [28], [-42], [-12], [8], [74], [70], [2], [-18], [28], [-44], [30], [-16], [-6], [40], [76], [58], [20], [8], [0], [-12], [-30], [-36], [-42], [-56], [80], [16], [46], [10], [18], [34], [-30], [-56], [-14], [-2], [48], [76], [60], [-50], [-18], [-20], [54], [-10], [-42], [-62], [14], [56], [-4], [16], [30], [-80], [-26], [52], [-26], [64], [-40], [58], [52], [-64], [-6], [40], [48], [-46], [-6], [4], [-62], [-78], [58], [-6], [-10], [36], [20], [-8], [-6], [-34], [4], [16], [54], [26], [-52], [-2], [44], [42], [16], [-22], [-42], [42], [-54], [20], [44], [20], [52], [-88], [-88], [26], [-2], [26], [32], [42], [-72], [66], [44], [-42], [36], [58], [52], [0], [66], [-92], [30], [-42], [20], [82], [-60], [-64], [96], [30], [32], [-22], [12], [8], [-64], [0], [-4], [-42], [74], [-16], [-48], [-72], [64], [-24], [-68], [56], [-86], [46], [8], [-56], [-96], [66], [68], [72], [-18], [68], [-30], [82], [-82], [-32], [34], [22], [8], [46], [16], [36], [-80], [-38], [38], [36], [12], [-76], [-2], [-40], [-20], [2], [-68], [72], [-22], [42], [-24], [-58], [-38], [-36], [44], [84], [-48]];
    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_e();".
// 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_e(: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_e();".
// 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_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4026_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-4, 1]>,<7,R![-4, 1]>,<13,R![-6, 1]>],Snew);
    return Vf;
end function;