
// Make newform 6048.2.a.u in Magma, downloaded from the LMFDB on 25 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_u();".
// 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_u();".
// 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], [1], [2], [2], [3], [1], [-2], [5], [7], [7], [-8], [-4], [-4], [-4], [4], [-2], [-3], [0], [-4], [-2], [15], [4], [6], [5], [4], [1], [6], [2], [0], [14], [-4], [6], [-22], [6], [22], [22], [-3], [-12], [-6], [-21], [-10], [18], [25], [-10], [-21], [22], [4], [18], [12], [-28], [-20], [-5], [13], [15], [20], [-13], [12], [28], [22], [-17], [18], [-26], [4], [12], [-21], [-3], [14], [-15], [-32], [17], [-27], [-8], [10], [36], [2], [-10], [16], [-30], [15], [-5], [-2], [0], [3], [-18], [25], [17], [-30], [-24], [12], [0], [37], [16], [-2], [10], [-11], [-29], [15], [4], [-18], [26], [-6], [-22], [-14], [-42], [-3], [39], [14], [12], [11], [8], [41], [-28], [-14], [11], [-36], [8], [33], [30], [-6], [33], [-9], [8], [-6], [29], [-36], [16], [32], [-38], [41], [0], [-2], [-6], [-54], [45], [-28], [-35], [26], [-11], [-40], [26], [51], [0], [34], [12], [-21], [-17], [-12], [-38], [-31], [-16], [-46], [14], [8], [-4], [-18], [-28], [-17], [43], [-26], [22], [-6], [-44], [24], [26], [-34], [-23], [-35], [-24], [-12], [-24], [-9], [13], [1], [44], [-54], [-17], [-44], [-62], [-18], [-59], [2], [-9], [-51], [10], [-36], [-41], [60], [-34], [36], [-14], [-34], [-47], [45], [-3], [38], [15], [-46], [42], [-37], [-20], [-52], [-6], [14], [-8], [3], [-12], [-20], [-23], [-65], [-40], [-48], [-28], [-8], [22], [-27], [-39], [-40], [-58], [16], [30], [12], [-2], [-5], [-53], [35], [-51], [-59], [-3], [-15], [26], [-74], [35], [26], [-6], [-42], [68], [-70], [21], [-39], [-9], [17], [31], [4], [61], [54], [42], [3], [-2], [26], [40], [50], [-31], [11], [-59], [-67], [26], [48], [-12], [7], [46], [4], [-7], [-31], [34], [6], [54], [42], [36], [28], [34], [-46], [4], [47], [-21], [34], [46], [-38], [4], [46], [14], [-56], [-37], [-15], [29], [4], [-61], [-45], [50], [72], [-66], [-2], [61], [5], [-49], [-6], [-26], [19], [10], [42], [70], [-37], [86], [-29], [47], [4], [-19], [-3], [-6], [-60], [-88], [-34], [79], [-4], [-58], [-15], [-4], [20], [-12], [57], [37], [44], [0], [-68], [22], [-35], [-8], [34], [31], [78], [-63], [74], [-22], [56], [23], [-69], [-32], [-5], [-67], [90], [34], [-42], [40], [60], [-56], [42], [-78], [22], [-47], [-22], [-12], [-9], [-67], [-39], [-72], [-12], [60], [72], [-38], [-57], [-48], [-27], [-20], [-59], [67], [58], [7], [-73], [-65], [68], [52], [39], [-53], [40], [56], [46], [41], [78], [-95], [-50], [-86], [44], [19], [-36], [14], [-100], [-20], [-76], [-10], [23], [87], [-32], [-12], [79], [41], [69], [52], [-2], [-12], [90], [-54], [43], [95], [-1], [-73], [-6], [-25], [76], [-12], [-8], [16], [68], [26], [-33], [87], [12], [-18], [-6], [16], [-76], [3], [-36], [-54], [82], [-52], [-100], [-39], [-90]];
    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_u();".
// 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_u(: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_u();".
// 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_u( : 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![-1, 1]>,<13,R![-2, 1]>,<17,R![-2, 1]>],Snew);
    return Vf;
end function;