
// Make newform 4026.2.a.a 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_a();".
// 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_a();".
// 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], [1], [-2], [-1], [-1], [0], [0], [-4], [5], [7], [-6], [-1], [4], [0], [-12], [9], [1], [8], [6], [8], [0], [4], [15], [-11], [-11], [-4], [13], [-15], [10], [-8], [9], [-6], [-3], [24], [20], [-1], [15], [12], [-15], [-2], [-17], [-2], [3], [26], [14], [15], [-12], [-12], [-12], [0], [16], [4], [1], [6], [22], [25], [12], [26], [20], [-20], [0], [-27], [-20], [16], [13], [12], [3], [15], [20], [6], [9], [2], [4], [15], [30], [28], [-3], [25], [11], [12], [-17], [-34], [16], [-5], [6], [16], [34], [38], [18], [21], [0], [-32], [20], [-38], [22], [-36], [-10], [19], [40], [3], [-23], [0], [-7], [28], [28], [19], [-14], [4], [-10], [-49], [-11], [-43], [-19], [8], [26], [-14], [42], [24], [1], [5], [-49], [18], [-34], [37], [-27], [-21], [-48], [28], [-22], [1], [37], [-46], [-36], [-40], [25], [-29], [29], [-2], [5], [-28], [-10], [-55], [-28], [32], [-5], [1], [-11], [-53], [-37], [-16], [0], [0], [-16], [8], [21], [-23], [-28], [54], [41], [-24], [-6], [-33], [-48], [-28], [18], [2], [16], [19], [44], [-26], [-2], [-3], [-34], [56], [0], [48], [-52], [0], [-22], [-23], [24], [8], [-42], [19], [-20], [-18], [-49], [58], [-29], [36], [-9], [65], [45], [-22], [30], [20], [-1], [-42], [-18], [30], [42], [13], [5], [20], [15], [-27], [50], [-24], [-14], [-50], [-54], [26], [51], [13], [56], [-37], [-54], [-39], [58], [29], [71], [35], [26], [22], [44], [63], [-56], [-22], [12], [-19], [-24], [-2], [-30], [53], [-51], [2], [33], [-43], [-47], [21], [-20], [59], [-2], [-60], [48], [-40], [-34], [28], [-36], [-28], [35], [-36], [46], [-20], [60], [-29], [-24], [54], [-55], [-15], [21], [-42], [-15], [-6], [19], [-79], [14], [14], [0], [-14], [13], [13], [20], [35], [-44], [-33], [-30], [-10], [77], [-11], [-72], [72], [68], [49], [25], [-32], [-46], [4], [26], [52], [4], [11], [66], [79], [-8], [20], [66], [52], [20], [-26], [-30], [56], [68], [54], [-14], [-56], [-28], [48], [64], [8], [-42], [-78], [-15], [2], [80], [24], [28], [-75], [-54], [0], [-32], [-54], [-63], [52], [68], [24], [68], [32], [-26], [-71], [-2], [6], [51], [-7], [-45], [-84], [-9], [70], [2], [60], [-61], [-48], [-41], [-30], [32], [42], [50], [88], [-41], [-64], [32], [10], [-80], [58], [-14], [-13], [-84], [-72], [30], [58], [-22], [36], [11], [11], [-10], [-47], [-20], [-32], [-55], [98], [32], [78], [-8], [-36], [-75], [-44], [7], [90], [-38], [-30], [-25], [72], [-63], [0], [50], [-82], [-69], [-46], [81], [85], [-54], [54], [-18], [88], [49], [-2], [-42], [-81], [-6], [41], [7], [-23], [-6], [-61], [65], [24], [2], [-29], [60], [-4], [58], [91], [58], [-14], [6], [-102], [-64], [-49], [84], [-6], [-20], [21], [-52], [15], [-82], [43], [-45], [62], [-49]];
    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_a();".
// 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_a(: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_a();".
// 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_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4026_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-1, 1]>,<7,R![2, 1]>,<13,R![1, 1]>],Snew);
    return Vf;
end function;