
// Make newform 4026.2.a.d 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_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_4026_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_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], [0], [1], [2], [2], [4], [8], [2], [-2], [0], [-8], [-10], [-6], [-6], [4], [1], [8], [4], [-14], [8], [6], [6], [2], [-10], [-12], [-10], [-18], [-2], [16], [-22], [-6], [-2], [-6], [12], [20], [4], [-16], [14], [16], [4], [-4], [-4], [-6], [12], [10], [6], [8], [2], [6], [0], [18], [12], [-30], [-20], [-20], [8], [-14], [-22], [-4], [-2], [-2], [4], [30], [4], [8], [12], [10], [10], [-6], [0], [12], [26], [4], [-12], [-26], [-16], [-30], [-28], [-12], [12], [-12], [10], [-8], [-20], [30], [-8], [26], [36], [28], [20], [20], [42], [-12], [4], [18], [22], [2], [-10], [38], [-30], [-22], [-4], [-16], [-2], [-28], [2], [-24], [46], [-32], [6], [-42], [4], [38], [-6], [-40], [4], [-38], [-22], [8], [-28], [6], [8], [-36], [2], [44], [30], [28], [-30], [42], [-24], [12], [-18], [-26], [-24], [-32], [2], [-28], [36], [38], [-18], [-54], [-54], [54], [-2], [22], [28], [-44], [-2], [-22], [54], [20], [48], [-48], [34], [-56], [34], [-14], [30], [-20], [58], [16], [48], [-6], [-36], [-16], [10], [-4], [-58], [-36], [-30], [0], [-30], [34], [-26], [8], [-54], [0], [10], [-10], [-36], [46], [56], [50], [-30], [-58], [44], [16], [48], [-26], [-54], [-20], [-12], [16], [58], [58], [20], [8], [16], [42], [-64], [32], [-32], [-24], [-48], [-10], [-24], [42], [-36], [46], [28], [56], [-60], [32], [22], [56], [-20], [42], [14], [-38], [8], [18], [-48], [0], [10], [10], [-26], [-44], [46], [-8], [-4], [28], [10], [-12], [-8], [-50], [-58], [-36], [30], [-12], [22], [26], [-40], [-22], [0], [-4], [0], [30], [64], [2], [-24], [-48], [-46], [30], [60], [-4], [8], [-18], [-32], [24], [34], [-14], [-42], [-22], [-52], [-52], [6], [4], [-34], [64], [-28], [-70], [40], [-28], [-64], [-68], [-50], [-16], [54], [-64], [-32], [-72], [74], [-2], [-30], [66], [-18], [40], [42], [-34], [0], [48], [-12], [-14], [-30], [20], [-26], [22], [26], [-6], [18], [40], [76], [28], [-82], [-24], [-50], [24], [50], [-64], [72], [-22], [60], [16], [50], [-24], [48], [66], [-54], [-8], [-46], [26], [14], [-30], [54], [16], [-24], [-48], [74], [50], [-28], [-44], [62], [46], [-76], [-2], [-16], [-14], [-4], [18], [-54], [-50], [62], [12], [-44], [4], [40], [88], [-48], [78], [86], [-78], [-80], [70], [20], [-46], [-20], [46], [48], [54], [24], [-76], [-6], [44], [22], [78], [56], [2], [36], [44], [8], [66], [-4], [-70], [76], [-72], [-68], [36], [-56], [6], [6], [-100], [-68], [-4], [4], [0], [52], [-20], [50], [62], [-48], [-40], [100], [-22], [32], [52], [54], [76], [54], [-34], [50], [-40], [38], [-26], [-16], [-98], [-44], [4], [72], [-6], [-34], [12], [88], [-40], [38], [-24], [-84], [-66], [-36], [-80], [-90], [42], [-72], [70], [82], [20], [48], [-80], [-24]];
    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_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_4026_2_a_d(: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_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_4026_2_a_d( : 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![0, 1]>,<13,R![-2, 1]>],Snew);
    return Vf;
end function;