
// Make newform 4026.2.a.g in Magma, downloaded from the LMFDB on 24 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_g();".
// 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_g();".
// 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], [4], [1], [2], [6], [-6], [4], [-2], [10], [6], [-8], [-8], [0], [0], [-12], [1], [2], [-12], [-10], [0], [8], [12], [-2], [-10], [-4], [-4], [-6], [2], [-14], [-12], [18], [-12], [0], [-4], [-10], [12], [24], [18], [-4], [10], [8], [18], [16], [-4], [-12], [18], [12], [-6], [-18], [8], [22], [-16], [6], [-4], [-10], [-30], [-10], [-14], [10], [24], [0], [-16], [10], [14], [30], [30], [12], [2], [18], [24], [-28], [-14], [-36], [36], [-16], [6], [-4], [-10], [-12], [-38], [4], [-38], [-26], [-36], [38], [22], [-8], [-24], [12], [-24], [16], [-8], [-14], [8], [0], [16], [-20], [-2], [-12], [2], [0], [-20], [10], [34], [32], [-22], [44], [-22], [26], [-2], [40], [8], [-46], [-20], [-50], [-12], [-24], [8], [38], [-34], [6], [4], [-8], [-42], [-30], [24], [-32], [26], [28], [-40], [-40], [42], [-14], [-26], [26], [-16], [-22], [28], [-52], [10], [-34], [40], [2], [8], [46], [-4], [40], [-32], [26], [30], [42], [-8], [-34], [-48], [10], [34], [18], [-14], [-24], [6], [-42], [12], [46], [-24], [-40], [-62], [-14], [-44], [20], [58], [-48], [14], [38], [6], [-6], [40], [-24], [14], [46], [12], [26], [36], [-64], [-16], [-30], [-4], [-26], [8], [6], [-48], [20], [42], [-44], [-6], [-22], [26], [12], [36], [60], [-24], [-14], [50], [-44], [18], [6], [28], [30], [34], [-62], [42], [32], [-12], [-16], [26], [2], [12], [48], [20], [-22], [-10], [16], [44], [44], [-22], [-36], [-64], [-4], [72], [-46], [54], [28], [-18], [-4], [24], [14], [-54], [-56], [8], [60], [-56], [50], [46], [-30], [24], [-28], [-20], [-8], [36], [2], [-20], [-12], [70], [-40], [-36], [-14], [6], [24], [-22], [-18], [60], [-54], [-42], [12], [-8], [-38], [82], [14], [-36], [22], [16], [70], [-4], [14], [-32], [-60], [6], [-70], [52], [8], [-36], [0], [38], [-56], [80], [-26], [40], [4], [34], [8], [-4], [76], [12], [22], [32], [-52], [-12], [-16], [80], [-14], [-60], [-10], [16], [6], [58], [-76], [-34], [-24], [-26], [-8], [18], [-30], [60], [-44], [38], [-12], [8], [-78], [6], [42], [-62], [-30], [6], [-22], [14], [30], [-46], [64], [2], [-38], [2], [-48], [72], [-52], [-72], [30], [84], [70], [44], [-30], [64], [60], [10], [-90], [52], [42], [-82], [-16], [50], [92], [-10], [32], [-14], [-86], [-16], [64], [-12], [-30], [36], [22], [-92], [96], [-80], [32], [74], [14], [64], [10], [40], [26], [4], [-72], [-80], [-30], [60], [56], [-14], [12], [22], [-24], [16], [60], [-48], [54], [40], [8], [18], [50], [-96], [34], [54], [-84], [90], [-88], [-14], [52], [-54], [48], [12], [38], [80], [10], [-84], [-38], [-22], [-18], [76], [40], [36], [-94], [14], [8], [-62], [22], [24], [-84], [56], [16], [84], [6], [-74], [48], [-20], [54], [-32], [44], [60], [2], [-8]];
    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_g();".
// 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_g(: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_g();".
// 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_g( : 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![-4, 1]>,<13,R![-2, 1]>],Snew);
    return Vf;
end function;