
// Make newform 6042.2.a.h in Magma, downloaded from the LMFDB on 25 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_h();".
// 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_h();".
// 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], [0], [0], [2], [6], [6], [-1], [-8], [-2], [8], [6], [2], [-4], [6], [-1], [8], [14], [12], [0], [2], [4], [0], [-10], [-10], [0], [-4], [-8], [0], [2], [-4], [-10], [0], [-12], [-22], [20], [-14], [20], [8], [-10], [12], [-16], [-4], [16], [-18], [20], [-18], [14], [8], [-14], [-20], [28], [30], [24], [6], [28], [-6], [20], [-2], [-26], [20], [6], [2], [6], [18], [18], [2], [16], [-14], [10], [-24], [-24], [-12], [-20], [28], [-24], [20], [18], [18], [-14], [-8], [20], [-32], [-2], [-2], [0], [-22], [22], [2], [16], [38], [-36], [2], [20], [36], [12], [-2], [42], [-26], [-2], [-34], [40], [12], [30], [20], [18], [-30], [-10], [0], [-4], [-22], [-38], [12], [-32], [8], [-30], [32], [-18], [-30], [28], [-38], [-14], [-30], [-24], [28], [20], [46], [-24], [28], [-30], [-52], [-48], [-26], [-42], [24], [-38], [-26], [0], [-18], [12], [-2], [-24], [12], [-44], [-44], [-56], [38], [42], [-4], [-48], [-2], [-8], [12], [56], [34], [24], [-32], [18], [2], [46], [-10], [54], [52], [-52], [2], [-4], [2], [-6], [12], [-6], [36], [32], [-18], [-26], [-36], [-54], [-28], [-42], [32], [10], [8], [-36], [-40], [-6], [26], [-18], [42], [26], [-14], [-54], [-30], [44], [8], [-30], [-48], [-6], [-58], [-30], [24], [-26], [18], [-40], [-56], [-28], [-10], [-40], [-4], [-36], [30], [-44], [18], [-54], [32], [12], [4], [34], [-24], [30], [60], [48], [16], [-32], [-6], [-40], [36], [-26], [-2], [8], [-26], [32], [-54], [14], [42], [10], [64], [-48], [-70], [6], [-58], [40], [-4], [56], [20], [-24], [-18], [60], [-64], [12], [50], [-34], [30], [-22], [-14], [-8], [4], [66], [70], [30], [10], [14], [16], [16], [-46], [-80], [70], [-28], [-34], [-22], [36], [-30], [-32], [52], [-66], [-16], [-46], [40], [66], [2], [-18], [-78], [48], [-28], [44], [-58], [-36], [42], [6], [-42], [-60], [-82], [50], [6], [-34], [-28], [-46], [68], [-22], [-12], [-36], [-12], [62], [82], [56], [56], [-52], [-48], [8], [-10], [6], [54], [34], [0], [-54], [26], [-4], [52], [-80], [-52], [30], [38], [-38], [2], [-6], [-32], [0], [52], [-82], [4], [64], [-54], [-38], [-62], [0], [50], [60], [24], [34], [30], [-42], [48], [16], [32], [-30], [-20], [-26], [48], [62], [-34], [0], [70], [-92], [-16], [18], [-4], [-62], [26], [0], [-28], [-60], [16], [26], [64], [8], [-4], [-32], [16], [4], [-44], [-64], [-42], [-64], [-66], [66], [-64], [-34], [-12], [-6], [-34], [68], [72], [86], [14], [48], [-42], [-58], [36], [48], [-50], [4], [-4], [-12], [-26], [86], [-80], [-40], [-50], [6], [6], [-102], [30], [-42], [80], [62], [54], [78], [32], [54], [-42], [-26], [58], [94], [78], [36], [44], [-30], [-2], [0], [26], [60], [-56], [-74], [48], [-96], [-76], [-50], [-98], [-36], [-10], [28], [24]];
    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_h();".
// 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_h(: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_h();".
// 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_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6042_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![0, 1]>,<11,R![-2, 1]>],Snew);
    return Vf;
end function;