
// Make newform 8048.2.a.i in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8048_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_8048_2_a_i();".
// 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_8048_2_a_i();".
// 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_8048_a();"
function MakeCharacter_8048_a()
    N := 8048;
    order := 1;
    char_gens := [1007, 6037, 2017];
    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_8048_a_Hecke(Kf)
    return MakeCharacter_8048_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], [1], [0], [5], [5], [1], [-4], [4], [9], [2], [-2], [-6], [6], [5], [-1], [-6], [12], [-3], [5], [-6], [-2], [8], [-7], [6], [10], [-8], [4], [0], [-14], [-3], [2], [-7], [-8], [-14], [-8], [-4], [4], [10], [10], [15], [2], [4], [22], [-22], [-21], [8], [6], [-11], [-12], [19], [11], [-16], [-20], [2], [3], [29], [6], [11], [18], [-25], [28], [-31], [8], [-26], [-30], [19], [-10], [22], [-26], [34], [-14], [-30], [3], [1], [29], [0], [-6], [29], [-11], [26], [-18], [26], [0], [-18], [-26], [-3], [-32], [-28], [10], [11], [32], [6], [34], [32], [-6], [1], [21], [3], [2], [-22], [28], [-31], [-6], [10], [14], [24], [-12], [28], [4], [-47], [1], [10], [-44], [2], [7], [-15], [4], [-8], [-11], [41], [-35], [26], [24], [-16], [-4], [-15], [-28], [32], [-31], [-38], [43], [16], [42], [12], [-25], [-26], [-30], [-28], [-18], [-40], [41], [14], [16], [-20], [-8], [-51], [-39], [-15], [28], [20], [36], [46], [44], [24], [-10], [-54], [0], [-60], [-48], [45], [-48], [9], [-30], [-53], [2], [-10], [37], [10], [2], [-26], [-3], [-22], [24], [-47], [-3], [51], [40], [34], [28], [11], [-15], [-45], [-22], [-3], [-9], [24], [46], [53], [26], [-47], [31], [-18], [-46], [-39], [-12], [-36], [-32], [-31], [42], [16], [57], [35], [-27], [-55], [-12], [-30], [-35], [-18], [43], [-52], [-33], [50], [-43], [0], [-42], [38], [24], [-18], [57], [-53], [-14], [-13], [-50], [16], [-32], [7], [-17], [-23], [-1], [-39], [-4], [-20], [-59], [66], [-38], [-18], [17], [0], [44], [-47], [-69], [-55], [32], [58], [-37], [-5], [62], [-76], [-16], [-18], [-50], [-62], [19], [-55], [34], [66], [58], [40], [33], [-36], [-60], [-20], [0], [-47], [-9], [23], [-65], [54], [74], [14], [-40], [46], [63], [36], [-29], [-8], [-56], [80], [-25], [30], [-66], [23], [63], [-38], [46], [-34], [-23], [86], [73], [-65], [55], [15], [-24], [-56], [-76], [16], [-64], [40], [8], [-86], [-23], [18], [16], [60], [42], [-8], [40], [14], [-39], [-62], [-72], [-32], [34], [74], [3], [59], [6], [-45], [-72], [29], [-70], [84], [-37], [-44], [42], [-42], [-56], [-4], [10], [69], [14], [-50], [-4], [-51], [-76], [-64], [50], [-56], [-22], [8], [33], [-70], [-29], [-37], [-24], [74], [2], [-82], [-11], [-88], [-24], [34], [-14], [-44], [62], [-34], [39], [-46], [-38], [72], [-52], [33], [-35], [20], [-40], [-4], [63], [-34], [23], [63], [39], [19], [48], [-77], [-46], [77], [-78], [-71], [-30], [46], [74], [-93], [9], [60], [-12], [18], [35], [45], [19], [21], [-8], [64], [-75], [-29], [-64], [33], [10], [6], [-17], [66], [15], [-2], [-21], [-42], [-66], [-97], [46], [-51], [-1], [-26], [52], [-57], [-2], [-77], [-40], [-18], [-72], [66], [-38], [97], [39], [2], [30], [22], [-7], [60], [-43], [94], [70], [-16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8048_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8048_2_a_i();".
// 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_8048_2_a_i(:prec:=1)
    chi := MakeCharacter_8048_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_8048_2_a_i();".
// 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_8048_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8048_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-1, 1]>,<5,R![0, 1]>,<7,R![-5, 1]>,<13,R![-1, 1]>],Snew);
    return Vf;
end function;