
// Make newform 6042.2.a.b in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6042_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_6042_2_a_b();".
// 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_6042_2_a_b();".
// 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_6042_a();"
function MakeCharacter_6042_a()
    N := 6042;
    order := 1;
    char_gens := [2015, 4771, 2281];
    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_6042_a_Hecke(Kf)
    return MakeCharacter_6042_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], [-1], [-1], [-3], [-4], [-2], [-4], [1], [1], [-1], [-7], [-8], [2], [-1], [0], [-1], [-9], [-14], [-5], [-6], [-8], [-8], [12], [5], [14], [-17], [-3], [9], [-17], [-9], [7], [-10], [-13], [6], [12], [-17], [2], [17], [-2], [8], [0], [7], [-20], [6], [22], [-9], [4], [-2], [13], [-15], [2], [12], [-24], [20], [-6], [13], [-19], [-15], [2], [-13], [-32], [-26], [-22], [6], [22], [-2], [30], [11], [18], [2], [-14], [7], [24], [5], [20], [10], [-19], [0], [-6], [24], [12], [-39], [3], [-22], [-16], [-37], [6], [-38], [12], [6], [8], [39], [0], [33], [-4], [9], [36], [42], [-32], [3], [-8], [35], [30], [-42], [-36], [-2], [-36], [-48], [44], [5], [26], [-28], [18], [5], [-34], [26], [37], [0], [6], [36], [-12], [22], [-10], [-20], [-44], [-27], [10], [41], [0], [41], [12], [24], [-14], [-9], [-50], [-12], [12], [-15], [-32], [-39], [16], [-33], [24], [-18], [1], [-53], [44], [25], [12], [-33], [6], [47], [-34], [-12], [32], [27], [-6], [-20], [3], [-45], [-40], [13], [-40], [51], [-56], [-27], [-18], [-19], [10], [-4], [-8], [2], [-14], [-44], [49], [-10], [-8], [2], [-21], [-35], [-62], [20], [43], [2], [-54], [6], [64], [-6], [8], [-58], [-28], [-27], [23], [-63], [-45], [56], [32], [-58], [-33], [-26], [23], [55], [3], [61], [22], [-5], [-35], [-43], [-16], [-19], [-41], [-30], [-20], [-18], [-51], [22], [-4], [9], [-29], [-6], [19], [-49], [-10], [-72], [21], [-10], [8], [-42], [-58], [-15], [-30], [-54], [-14], [0], [-9], [35], [-38], [0], [-6], [26], [30], [9], [5], [-38], [-71], [-39], [2], [0], [-66], [6], [10], [-54], [-2], [-45], [30], [0], [-58], [38], [-33], [22], [-16], [-15], [28], [34], [-14], [76], [-53], [-60], [9], [18], [-22], [27], [-53], [52], [44], [-44], [-48], [4], [-2], [30], [84], [-9], [36], [24], [40], [-6], [-52], [42], [-60], [-52], [16], [48], [48], [0], [-16], [-15], [10], [-63], [-58], [-62], [44], [-62], [-16], [19], [64], [70], [-39], [40], [0], [-14], [48], [-42], [32], [24], [67], [-75], [79], [-64], [-1], [-48], [74], [35], [12], [-73], [-42], [25], [66], [55], [24], [36], [50], [2], [-59], [16], [11], [-27], [-40], [37], [-12], [-8], [22], [-66], [-40], [17], [-3], [75], [-91], [42], [-34], [-70], [-7], [11], [10], [-11], [-90], [40], [-3], [37], [62], [3], [38], [-50], [-39], [66], [31], [-10], [33], [35], [87], [70], [100], [-52], [34], [4], [-14], [44], [48], [53], [-3], [70], [-42], [-80], [-80], [52], [-56], [27], [-62], [38], [14], [-51], [9], [-29], [58], [17], [-30], [-22], [-20], [6], [58], [32], [-44], [-93], [-62], [-16], [-40], [68], [40], [-8], [-12], [80], [68], [10], [-10], [-73], [10], [30], [48], [32], [37], [40], [5], [-54], [-28], [-9], [-52], [-86], [46], [48], [72], [19]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6042_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6042_2_a_b();".
// 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_6042_2_a_b(:prec:=1)
    chi := MakeCharacter_6042_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_6042_2_a_b();".
// 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_6042_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6042_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, 1]>,<7,R![3, 1]>,<11,R![4, 1]>],Snew);
    return Vf;
end function;