
// Make newform 6046.2.a.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6046_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_6046_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_6046_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_6046_a();"
function MakeCharacter_6046_a()
    N := 6046;
    order := 1;
    char_gens := [5];
    v := [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_6046_a_Hecke(Kf)
    return MakeCharacter_6046_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], [2], [-2], [2], [-5], [-2], [4], [-6], [6], [-3], [9], [5], [6], [-7], [4], [10], [-4], [5], [-2], [10], [-4], [-11], [2], [18], [3], [18], [19], [-8], [-7], [14], [-6], [0], [-3], [14], [-14], [-16], [17], [18], [12], [18], [0], [12], [-7], [9], [-2], [4], [-12], [1], [-6], [-10], [-10], [-9], [-20], [8], [21], [0], [-9], [-2], [10], [9], [14], [20], [19], [-4], [9], [30], [-12], [18], [-18], [-2], [8], [9], [5], [-3], [-30], [-5], [-28], [10], [-12], [-7], [34], [-10], [27], [21], [2], [41], [-26], [-26], [0], [-26], [-5], [-42], [14], [-24], [8], [36], [-24], [0], [16], [-26], [-26], [12], [-11], [-26], [14], [-22], [-12], [11], [32], [-22], [-31], [44], [11], [43], [32], [27], [-4], [48], [18], [-4], [2], [24], [-14], [-41], [-21], [-17], [-32], [16], [32], [-30], [47], [28], [8], [12], [5], [28], [-19], [-1], [-43], [9], [20], [-10], [4], [8], [-24], [18], [8], [26], [36], [-6], [52], [15], [47], [-18], [20], [-36], [11], [-30], [-51], [10], [18], [21], [40], [17], [28], [-19], [-47], [12], [-14], [-33], [-43], [-4], [0], [-9], [10], [36], [16], [-8], [44], [1], [0], [-12], [-33], [33], [50], [-25], [6], [-37], [-34], [-4], [14], [-39], [-47], [49], [8], [39], [-4], [4], [3], [-41], [-51], [27], [-32], [14], [12], [-4], [-40], [60], [-10], [-28], [38], [66], [-17], [-41], [-3], [-8], [22], [34], [-56], [-71], [70], [-40], [-34], [0], [-44], [62], [-28], [-46], [-12], [-28], [-34], [14], [-20], [-24], [-64], [-56], [27], [6], [-44], [0], [-30], [-55], [-68], [65], [51], [69], [2], [-43], [4], [-66], [62], [-30], [-24], [42], [39], [-72], [-22], [-55], [-23], [-7], [-1], [39], [60], [26], [-4], [49], [-78], [13], [4], [-28], [68], [-20], [-17], [-12], [-53], [-47], [28], [4], [-37], [22], [-29], [-6], [49], [4], [-52], [-11], [18], [42], [59], [54], [62], [4], [35], [-12], [46], [-48], [-17], [-26], [-1], [13], [-78], [-28], [24], [-1], [-8], [-76], [-3], [-40], [-66], [73], [14], [54], [30], [14], [-72], [15], [-16], [-75], [56], [45], [-46], [13], [48], [48], [75], [-82], [-74], [20], [-84], [33], [-27], [64], [72], [4], [-45], [69], [-79], [-74], [31], [53], [-52], [-24], [49], [-80], [-21], [25], [-54], [-84], [71], [20], [19], [-78], [57], [88], [-95], [21], [84], [42], [86], [35], [-9], [66], [32], [75], [95], [-9], [-49], [44], [-88], [12], [28], [78], [-6], [13], [-33], [5], [3], [16], [18], [14], [-12], [70], [-23], [-18], [-88], [-18], [28], [73], [24], [83], [-8], [36], [56], [22], [18], [-10], [11], [30], [-36], [-56], [-20], [18], [-24], [54], [21], [77], [74], [30], [-59], [-90], [63], [-57], [96], [-48], [84], [-106], [-36], [30], [-66], [-58], [98], [45], [39], [-19], [-45], [-43], [100], [90], [106], [-8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6046_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6046_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_6046_2_a_a(:prec:=1)
    chi := MakeCharacter_6046_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_6046_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_6046_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6046_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 1]>,<11,R![5, 1]>],Snew);
    return Vf;
end function;