
// Make newform 5040.2.a.e in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5040_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_5040_2_a_e();".
// 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_5040_2_a_e();".
// 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_5040_a();"
function MakeCharacter_5040_a()
    N := 5040;
    order := 1;
    char_gens := [3151, 3781, 2801, 2017, 3601];
    v := [1, 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_5040_a_Hecke(Kf)
    return MakeCharacter_5040_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], [0], [-1], [-1], [0], [-4], [-6], [-2], [6], [0], [-2], [2], [-6], [4], [0], [6], [12], [-10], [4], [12], [-4], [-8], [12], [-6], [8], [6], [4], [6], [14], [-6], [16], [-12], [18], [-14], [24], [-8], [-4], [4], [12], [18], [-12], [-10], [0], [-22], [-6], [-14], [-8], [28], [0], [26], [6], [0], [2], [12], [18], [-30], [-6], [10], [2], [24], [28], [6], [-8], [24], [-4], [30], [16], [2], [-18], [14], [-18], [24], [4], [14], [-20], [0], [36], [8], [12], [14], [36], [-34], [-36], [-28], [10], [-18], [-24], [-10], [6], [-8], [0], [-24], [-8], [36], [28], [24], [18], [-42], [-20], [-34], [4], [-18], [12], [-12], [-32], [20], [36], [-18], [0], [38], [28], [-46], [-42], [10], [40], [-24], [40], [-12], [30], [12], [-10], [38], [6], [6], [-26], [0], [26], [0], [16], [-4], [16], [30], [-44], [38], [-6], [-34], [-30], [4], [18], [-12], [-38], [12], [-8], [-18], [14], [0], [-40], [-42], [34], [30], [-22], [-42], [-20], [0], [28], [-24], [28], [-30], [56], [18], [18], [-6], [40], [-36], [-18], [-36], [-20], [8], [-34], [6], [-48], [50], [12], [26], [22], [30], [4], [12], [40], [-22], [40], [-12], [38], [-54], [-30], [-18], [50], [4], [-58], [-48], [56], [-42], [52], [-42], [18], [6], [26], [14], [18], [-60], [-60], [-62], [-16], [-22], [-36], [18], [-2], [54], [36], [-62], [38], [66], [-56], [-60], [-24], [-10], [40], [30], [18], [-42], [-22], [46], [60], [-56], [-48], [-34], [-54], [60], [-68], [60], [-58], [70], [-56], [-36], [64], [-60], [-10], [18], [60], [24], [66], [10], [40], [14], [30], [72], [16], [12], [4], [-66], [-22], [-18], [66], [-10], [42], [-24], [2], [28], [-30], [20], [64], [78], [50], [-52], [30], [46], [36], [6], [-68], [-18], [-22], [-44], [80], [76], [32], [-56], [6], [-22], [-58], [36], [36], [-32], [60], [26], [28], [48], [38], [6], [46], [-6], [-60], [-12], [6], [-36], [-34], [6], [46], [42], [-12], [-44], [56], [30], [-20], [66], [40], [-22], [42], [50], [84], [14], [36], [-24], [12], [4], [-66], [26], [60], [12], [-88], [-12], [-38], [-34], [78], [-32], [-42], [-70], [-8], [52], [42], [66], [-10], [-18], [-14], [12], [28], [-84], [38], [66], [74], [16], [-10], [78], [42], [-8], [30], [60], [-22], [-44], [72], [54], [-50], [62], [-60], [64], [50], [54], [-24], [-84], [66], [6], [62], [-30], [-18], [36], [-20], [62], [-30], [88], [-58], [12], [-38], [-90], [96], [10], [2], [12], [-60], [-4], [30], [-28], [78], [66], [-8], [6], [34], [-24], [-20], [-52], [4], [0], [-34], [-42], [-60], [-8], [12], [-94], [-50], [6], [64], [-24], [50], [66], [40], [78], [42], [70], [98], [0], [-32], [-36], [8], [-54], [78], [88], [50], [66], [-24], [-80], [18], [24], [-36], [-64], [-66], [-36], [20], [-66], [66], [12], [70], [24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5040_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5040_2_a_e();".
// 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_5040_2_a_e(:prec:=1)
    chi := MakeCharacter_5040_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_5040_2_a_e();".
// 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_5040_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5040_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![0, 1]>,<13,R![4, 1]>,<17,R![6, 1]>,<19,R![2, 1]>,<29,R![0, 1]>],Snew);
    return Vf;
end function;