
// Make newform 6042.2.a.m in Magma, downloaded from the LMFDB on 24 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_m();".
// 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_m();".
// 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], [2], [4], [0], [4], [-6], [-1], [0], [6], [4], [-4], [10], [-4], [-8], [1], [-14], [-4], [4], [0], [-10], [12], [-6], [0], [18], [14], [12], [-10], [-6], [12], [-4], [-12], [-12], [16], [-22], [0], [4], [12], [-8], [-14], [-24], [-22], [16], [-18], [6], [-8], [-8], [2], [2], [-22], [-4], [24], [10], [6], [-18], [-12], [6], [-20], [28], [28], [16], [22], [8], [8], [22], [-10], [0], [18], [12], [-12], [24], [-24], [20], [-6], [-36], [-24], [-6], [12], [18], [18], [30], [30], [-32], [-10], [26], [34], [-40], [-22], [-14], [-22], [0], [-28], [14], [30], [12], [-36], [30], [-32], [40], [-2], [24], [18], [-28], [30], [-16], [-2], [0], [-42], [-24], [-2], [-6], [8], [-12], [-36], [10], [42], [44], [-32], [50], [0], [40], [6], [26], [-26], [36], [30], [-36], [4], [40], [-30], [-40], [-36], [-42], [22], [12], [22], [-46], [4], [18], [48], [-20], [-26], [52], [-16], [6], [-44], [-36], [-36], [4], [-36], [-44], [0], [-8], [-48], [-44], [0], [-10], [-30], [26], [-22], [32], [12], [44], [18], [-14], [0], [-46], [-42], [14], [6], [56], [-42], [-24], [6], [40], [-6], [-12], [-34], [48], [-34], [-26], [-28], [50], [-48], [64], [30], [-44], [-4], [42], [-56], [62], [-14], [-60], [-22], [-14], [62], [50], [2], [-8], [-48], [42], [0], [-30], [58], [16], [-10], [-8], [10], [26], [-60], [-2], [30], [-26], [-12], [36], [46], [-22], [12], [-60], [-6], [-30], [-4], [18], [-46], [-30], [56], [-18], [24], [22], [-50], [28], [-32], [22], [-6], [36], [28], [-6], [6], [-24], [64], [24], [28], [36], [46], [-16], [32], [6], [-28], [-72], [8], [-18], [-6], [56], [70], [-18], [-16], [64], [-24], [-2], [14], [24], [66], [-28], [14], [62], [-28], [-30], [-30], [-12], [-2], [-50], [-20], [66], [-24], [38], [38], [-20], [-4], [66], [-20], [-8], [76], [-60], [80], [40], [16], [-38], [-18], [-4], [-46], [18], [-56], [-30], [-28], [28], [66], [10], [30], [-60], [76], [-18], [-22], [44], [-78], [-80], [-30], [54], [72], [24], [28], [8], [-42], [66], [32], [-8], [-46], [42], [-32], [-18], [10], [-28], [6], [66], [-28], [44], [18], [-20], [44], [24], [-6], [28], [26], [4], [20], [36], [-30], [-32], [88], [30], [42], [46], [-36], [-30], [-92], [70], [30], [26], [-52], [-40], [6], [-60], [58], [78], [40], [-56], [-6], [24], [-6], [48], [48], [-10], [40], [12], [48], [28], [54], [46], [24], [-78], [-88], [-12], [0], [-14], [88], [-10], [-24], [-24], [14], [-56], [-38], [90], [38], [-14], [6], [0], [84], [18], [8], [80], [8], [42], [-30], [-32], [-36], [-24], [-70], [-70], [-86], [68], [54], [24], [-36], [-22], [-70], [18], [-62], [-40], [42], [-56], [24], [-22], [-54], [-68], [60], [-10], [66], [-20], [-26], [48], [72], [2], [38], [-92], [-90], [70], [-18], [-32], [22], [68], [-80]];
    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_m();".
// 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_m(: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_m();".
// 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_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6042_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<7,R![-4, 1]>,<11,R![0, 1]>],Snew);
    return Vf;
end function;