
// Make newform 4012.2.a.a in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4012_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_4012_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_4012_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_4012_a();"
function MakeCharacter_4012_a()
    N := 4012;
    order := 1;
    char_gens := [2007, 3777, 3129];
    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_4012_a_Hecke(Kf)
    return MakeCharacter_4012_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 := [[0], [-3], [-3], [-1], [-2], [-2], [-1], [-1], [0], [-9], [0], [-10], [-3], [-8], [0], [-1], [-1], [-8], [-6], [-8], [-2], [-11], [14], [-10], [8], [-12], [6], [9], [-14], [-6], [-9], [-4], [11], [4], [8], [-12], [10], [-8], [11], [0], [6], [-21], [18], [-7], [-6], [11], [-2], [0], [-16], [-4], [18], [-3], [11], [1], [-7], [-7], [-14], [15], [25], [-1], [6], [-19], [29], [11], [8], [2], [-25], [-4], [22], [30], [2], [-5], [-26], [-14], [11], [24], [6], [-28], [-34], [-14], [-18], [12], [-32], [19], [-20], [-8], [11], [-38], [2], [-28], [-26], [16], [-25], [33], [1], [-24], [-10], [-10], [1], [8], [-32], [-9], [28], [2], [-4], [-15], [6], [-23], [31], [40], [-13], [2], [27], [31], [-36], [14], [13], [-31], [41], [44], [-45], [26], [-30], [-12], [40], [-12], [-21], [36], [16], [-46], [-2], [20], [2], [1], [-39], [36], [24], [-28], [12], [30], [10], [-20], [4], [4], [-47], [48], [43], [-6], [20], [22], [-49], [-46], [-15], [-30], [-15], [-41], [10], [-58], [20], [-26], [-7], [22], [30], [-27], [-6], [26], [12], [-13], [36], [-22], [4], [32], [7], [-52], [-13], [51], [-57], [-6], [-56], [59], [-17], [-43], [4], [23], [16], [-54], [28], [24], [20], [6], [-52], [-24], [0], [30], [5], [10], [-18], [26], [46], [-20], [-14], [-43], [46], [-4], [-39], [0], [20], [28], [-54], [22], [22], [35], [21], [49], [-29], [34], [-8], [2], [12], [6], [-20], [-36], [4], [39], [30], [-4], [9], [-24], [2], [-3], [-12], [-54], [12], [-10], [16], [-43], [28], [-24], [-64], [-72], [69], [-20], [53], [-27], [-17], [48], [-10], [-38], [-67], [27], [-43], [-8], [-60], [55], [37], [-36], [47], [-42], [54], [-34], [-18], [27], [49], [-46], [23], [-50], [25], [30], [49], [9], [-22], [-76], [-32], [-15], [-33], [-26], [28], [75], [-12], [45], [0], [-14], [18], [14], [30], [50], [-36], [-11], [-56], [59], [68], [12], [-54], [48], [79], [-84], [-72], [-11], [-72], [-62], [57], [-72], [-68], [-26], [-27], [44], [-31], [84], [-70], [6], [-35], [-27], [39], [-52], [-75], [-28], [42], [15], [2], [34], [-52], [54], [3], [87], [-9], [-38], [-86], [20], [70], [-36], [2], [-82], [20], [21], [-44], [3], [-43], [-28], [32], [7], [10], [52], [-72], [-10], [60], [-48], [-22], [-39], [18], [66], [-28], [-37], [-47], [-70], [-26], [-56], [-96], [57], [-13], [81], [38], [-59], [-13], [32], [20], [-14], [6], [-19], [82], [-52], [-20], [22], [-82], [-22], [25], [24], [-4], [-56], [-86], [33], [-77], [-28], [-8], [48], [19], [24], [65], [62], [20], [-62], [8], [0], [52], [-98], [2], [-8], [-79], [75], [55], [61], [-50], [-18], [-53], [78], [69], [-46], [-62], [-45], [74], [28], [-59], [-38], [56], [99], [-13], [-45], [-34], [-32], [-100], [-57], [-68], [-102], [-48], [35], [-27], [-51], [-28], [26], [-39], [64]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4012_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4012_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_4012_2_a_a(:prec:=1)
    chi := MakeCharacter_4012_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_4012_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_4012_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4012_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![3, 1]>,<5,R![3, 1]>],Snew);
    return Vf;
end function;