
// Make newform 6040.2.a.h in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6040_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_6040_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_6040_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_6040_a();"
function MakeCharacter_6040_a()
    N := 6040;
    order := 1;
    char_gens := [1511, 3021, 2417, 761];
    v := [1, 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_6040_a_Hecke(Kf)
    return MakeCharacter_6040_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 := [[0], [1], [1], [4], [-3], [1], [-6], [-5], [3], [-3], [-7], [-2], [0], [2], [-10], [-2], [3], [-10], [1], [-2], [1], [-4], [-11], [8], [-8], [0], [-14], [9], [18], [-6], [8], [-4], [6], [20], [8], [-1], [-13], [1], [8], [-6], [-26], [10], [-16], [14], [-6], [18], [10], [16], [24], [15], [-11], [9], [21], [9], [11], [-21], [-1], [-10], [-6], [-12], [7], [-14], [-10], [33], [-20], [-27], [12], [3], [18], [-1], [-19], [-18], [20], [21], [18], [-24], [-16], [-8], [-38], [-20], [10], [18], [-30], [-14], [19], [-28], [-10], [-16], [-18], [-32], [28], [2], [38], [-36], [12], [-2], [12], [-23], [27], [-23], [-30], [25], [-20], [10], [19], [-12], [-28], [39], [32], [-23], [0], [-26], [-3], [26], [2], [-3], [-4], [18], [-24], [-43], [-4], [26], [-34], [31], [6], [-50], [1], [28], [-28], [9], [-30], [-36], [12], [4], [-30], [-26], [-24], [-4], [-12], [44], [-16], [18], [-22], [18], [25], [-4], [26], [-1], [-44], [48], [25], [0], [32], [11], [20], [-35], [14], [16], [-6], [-8], [55], [6], [-23], [44], [-2], [56], [-55], [44], [-2], [49], [-20], [-26], [-7], [56], [26], [42], [61], [-15], [37], [6], [-35], [65], [-50], [-48], [-21], [-24], [-2], [28], [-49], [-9], [35], [-36], [-36], [18], [21], [-58], [-33], [-52], [-16], [-4], [-33], [-62], [-62], [32], [20], [10], [16], [3], [57], [8], [-49], [-12], [-20], [20], [12], [-10], [-13], [59], [44], [-61], [-9], [4], [-63], [58], [-24], [-10], [18], [-24], [56], [24], [54], [-55], [-22], [-60], [0], [-38], [46], [-29], [-20], [-17], [39], [63], [27], [-63], [-18], [5], [-17], [-48], [43], [-11], [-23], [27], [-22], [23], [18], [24], [58], [5], [72], [14], [-26], [60], [17], [-32], [4], [52], [60], [2], [14], [26], [42], [46], [-14], [-1], [-30], [3], [-17], [70], [8], [-84], [48], [40], [-57], [30], [16], [-56], [-77], [33], [12], [54], [-78], [-4], [31], [-36], [82], [33], [36], [-26], [-1], [-12], [49], [14], [25], [-16], [42], [37], [-8], [56], [-45], [-30], [-66], [30], [-30], [12], [76], [58], [-69], [29], [-39], [-48], [-12], [13], [-28], [-16], [68], [30], [-20], [-68], [-69], [-46], [40], [42], [-9], [-9], [52], [-68], [63], [20], [-47], [76], [65], [-72], [87], [-68], [60], [48], [-9], [-79], [-64], [67], [-18], [1], [19], [-74], [-51], [48], [4], [18], [-6], [33], [12], [54], [-30], [-85], [-12], [-39], [21], [-69], [-85], [0], [-43], [26], [-30], [-64], [-45], [-24], [-36], [46], [-53], [-18], [-4], [11], [-68], [-2], [8], [36], [-62], [32], [20], [9], [-64], [41], [-6], [40], [31], [26], [53], [-47], [20], [84], [-55], [57], [-4], [-28], [4], [40], [10], [20], [4], [-84], [-63], [-37], [-76], [-64], [-30], [22], [-49], [-26], [-29], [-74], [-69], [-41], [-90], [22], [-32], [-90], [-20], [-33], [18], [-40]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6040_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6040_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_6040_2_a_h(:prec:=1)
    chi := MakeCharacter_6040_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_6040_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_6040_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6040_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-1, 1]>,<7,R![-4, 1]>,<11,R![3, 1]>],Snew);
    return Vf;
end function;