
// Make newform 6048.2.a.w in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6048_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_6048_2_a_w();".
// 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_6048_2_a_w();".
// 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_6048_a();"
function MakeCharacter_6048_a()
    N := 6048;
    order := 1;
    char_gens := [4159, 3781, 3809, 2593];
    v := [1, 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_6048_a_Hecke(Kf)
    return MakeCharacter_6048_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], [0], [3], [1], [-2], [-4], [-5], [0], [4], [-4], [-8], [1], [-7], [5], [-1], [-2], [11], [-14], [-4], [-12], [0], [13], [-11], [-6], [-8], [6], [-8], [16], [-17], [-6], [19], [-12], [16], [-14], [6], [-5], [-12], [-1], [-5], [6], [6], [6], [6], [-1], [12], [2], [-8], [-24], [28], [22], [0], [6], [-16], [13], [14], [-20], [27], [10], [-7], [0], [14], [5], [20], [27], [10], [8], [-15], [-33], [-12], [-16], [-21], [10], [-26], [-21], [-1], [19], [-6], [-28], [-14], [4], [-27], [30], [-32], [-26], [-18], [-12], [0], [-26], [37], [39], [36], [-27], [0], [-20], [-25], [7], [-1], [11], [-22], [45], [5], [24], [-4], [0], [13], [-38], [-36], [-45], [-6], [8], [30], [26], [-32], [40], [-17], [2], [34], [24], [46], [-18], [42], [-30], [-18], [36], [-38], [6], [23], [3], [-40], [14], [44], [-16], [24], [-17], [-3], [-6], [-39], [-32], [2], [-26], [38], [-8], [31], [18], [-24], [31], [-42], [-27], [-22], [54], [-35], [46], [43], [-49], [-17], [-34], [31], [45], [32], [35], [-8], [32], [56], [-27], [14], [-21], [13], [20], [61], [29], [0], [-24], [-6], [-21], [20], [35], [52], [42], [-58], [22], [-55], [-15], [34], [7], [6], [9], [-50], [-36], [-26], [-3], [20], [-30], [-49], [1], [10], [39], [18], [-13], [-30], [-43], [-6], [-62], [10], [20], [15], [25], [-46], [-48], [36], [14], [61], [-10], [1], [-15], [-67], [64], [-31], [-66], [66], [4], [-1], [-70], [-42], [21], [-47], [-11], [17], [44], [6], [-56], [66], [24], [61], [-58], [22], [-40], [18], [18], [44], [59], [54], [6], [6], [62], [2], [-65], [8], [21], [-57], [36], [1], [-62], [-2], [-40], [1], [-28], [31], [76], [-50], [-78], [36], [34], [22], [46], [70], [-20], [-40], [54], [3], [-66], [24], [32], [-32], [-47], [22], [16], [-42], [-41], [-29], [-11], [-35], [-7], [35], [28], [-68], [-60], [-31], [-76], [70], [-3], [-62], [-27], [78], [49], [7], [49], [-32], [51], [25], [-10], [-50], [-72], [56], [74], [-29], [-26], [-40], [-50], [-22], [42], [12], [48], [34], [-5], [-38], [-10], [-1], [60], [26], [-56], [-42], [-63], [-67], [76], [66], [22], [65], [-22], [-14], [26], [70], [-30], [60], [-19], [-53], [-13], [-10], [-39], [-2], [-22], [-49], [78], [-37], [0], [-34], [24], [64], [57], [81], [-58], [-88], [-19], [60], [-54], [-49], [27], [0], [51], [12], [6], [-5], [45], [0], [-54], [32], [-11], [71], [-43], [31], [92], [92], [32], [56], [78], [-98], [4], [-22], [-92], [34], [-24], [-67], [-64], [20], [82], [-10], [-24], [40], [-46], [11], [-28], [-37], [-14], [12], [2], [84], [-98], [52], [-9], [-4], [-16], [0], [90], [3], [41], [-86], [-38], [29], [-66], [-56], [4], [-48], [-40], [-86], [-65], [-61], [18], [66], [48], [3], [-12], [98], [8], [102], [-105], [-42], [17], [-40], [-26], [6], [72]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6048_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6048_2_a_w();".
// 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_6048_2_a_w(:prec:=1)
    chi := MakeCharacter_6048_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_6048_2_a_w();".
// 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_6048_2_a_w( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6048_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-3, 1]>,<11,R![2, 1]>,<13,R![4, 1]>,<17,R![5, 1]>],Snew);
    return Vf;
end function;