
// Make newform 6042.2.a.c 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_c();".
// 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_c();".
// 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], [0], [-4], [-2], [2], [1], [4], [2], [8], [-2], [-10], [-4], [0], [-1], [12], [-2], [4], [0], [10], [-8], [0], [14], [2], [10], [0], [12], [10], [6], [16], [-4], [-10], [-12], [6], [16], [-10], [-4], [-8], [2], [12], [-14], [-20], [6], [22], [0], [16], [4], [28], [6], [-10], [12], [-30], [8], [-18], [4], [26], [24], [-10], [-10], [4], [34], [32], [24], [10], [10], [-24], [-34], [12], [-10], [-2], [28], [24], [2], [20], [-8], [26], [6], [30], [-30], [-24], [-6], [24], [-22], [-4], [32], [30], [34], [-18], [24], [-28], [-36], [-36], [-24], [-4], [36], [18], [-18], [40], [30], [40], [2], [-12], [6], [12], [-14], [-12], [42], [8], [38], [-28], [14], [30], [-4], [-16], [-34], [-20], [-24], [-18], [36], [-42], [-14], [26], [-20], [-20], [42], [22], [44], [0], [14], [-12], [48], [-44], [-42], [-26], [-30], [-6], [12], [34], [-42], [16], [-6], [24], [36], [34], [16], [-10], [46], [36], [24], [6], [14], [44], [24], [32], [0], [-24], [-14], [-6], [42], [-52], [-26], [8], [-36], [-2], [-48], [-12], [38], [-26], [38], [28], [2], [40], [34], [-32], [-22], [-44], [14], [0], [-50], [-32], [-28], [-14], [-10], [-24], [30], [-26], [24], [50], [56], [26], [-24], [-28], [-30], [24], [-10], [-22], [14], [6], [-32], [2], [-8], [18], [22], [52], [-14], [-8], [44], [-22], [-28], [34], [30], [-56], [-36], [-24], [10], [56], [-42], [-8], [-6], [-14], [-40], [14], [0], [-12], [-10], [-58], [0], [68], [-48], [6], [-24], [4], [-42], [-36], [-4], [-14], [-42], [-12], [56], [-12], [-12], [-16], [-62], [-2], [-60], [32], [60], [-24], [0], [-2], [-30], [-32], [54], [-78], [-60], [38], [-16], [-6], [-62], [32], [-36], [-50], [82], [-38], [52], [-14], [-54], [36], [-18], [-46], [36], [58], [40], [-22], [64], [-12], [-50], [34], [-36], [0], [-24], [36], [-18], [-20], [72], [26], [-6], [-24], [-34], [46], [-12], [-42], [-84], [14], [-30], [28], [-54], [-28], [-68], [50], [82], [-64], [52], [-68], [-50], [-60], [46], [-72], [-26], [48], [18], [14], [-48], [28], [78], [-56], [-40], [-46], [-30], [56], [-22], [-30], [80], [54], [-26], [24], [28], [-48], [-42], [-10], [-46], [-8], [-44], [-52], [72], [-82], [-74], [-30], [64], [-38], [6], [26], [8], [66], [-84], [-10], [-24], [-40], [-34], [20], [38], [70], [-32], [54], [-74], [-60], [-80], [38], [0], [-58], [70], [-24], [-36], [4], [62], [-54], [-40], [30], [4], [-20], [-16], [10], [88], [-2], [-28], [24], [-46], [-66], [70], [18], [-14], [-32], [58], [-80], [-24], [16], [38], [68], [-24], [-90], [-86], [-44], [68], [72], [-22], [76], [-42], [-32], [86], [22], [102], [4], [-10], [-70], [-40], [46], [-2], [24], [-28], [-22], [22], [44], [-28], [58], [6], [-96], [-100], [-26], [-32], [-46], [-6], [92], [60], [2], [58], [76], [-30], [-36], [100]];
    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_c();".
// 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_c(: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_c();".
// 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_c( : 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![0, 1]>,<11,R![4, 1]>],Snew);
    return Vf;
end function;