
// Make newform 6240.2.a.l in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6240_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_6240_2_a_l();".
// 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_6240_2_a_l();".
// 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_6240_a();"
function MakeCharacter_6240_a()
    N := 6240;
    order := 1;
    char_gens := [1951, 2341, 2081, 2497, 5761];
    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_6240_a_Hecke(Kf)
    return MakeCharacter_6240_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], [2], [0], [-1], [4], [2], [6], [0], [4], [-2], [-6], [4], [4], [-10], [8], [6], [-8], [0], [16], [0], [-4], [6], [-8], [-4], [8], [4], [-4], [0], [8], [18], [6], [4], [10], [-20], [-10], [4], [-16], [6], [-18], [-14], [24], [-4], [-22], [8], [16], [-2], [4], [4], [12], [16], [-22], [6], [-8], [18], [20], [20], [-10], [6], [-20], [-6], [0], [32], [6], [-2], [-6], [6], [-24], [28], [30], [0], [16], [14], [30], [24], [12], [6], [18], [-26], [-22], [8], [16], [22], [32], [-24], [30], [-16], [30], [6], [12], [0], [-6], [18], [-10], [-6], [-10], [6], [4], [40], [-20], [6], [-32], [-2], [12], [-20], [12], [-22], [-8], [-42], [8], [-2], [30], [-26], [0], [34], [8], [-6], [-6], [-6], [-8], [-30], [-6], [4], [-6], [-12], [12], [-20], [-28], [-18], [-26], [24], [16], [-22], [18], [-30], [-6], [-36], [6], [10], [18], [26], [0], [-36], [-18], [-16], [6], [32], [-24], [24], [18], [-10], [-20], [-42], [-20], [-12], [40], [14], [-54], [-38], [28], [-36], [-30], [-26], [18], [-24], [40], [22], [-14], [30], [-12], [-52], [-12], [28], [24], [-18], [10], [10], [44], [34], [-46], [34], [-6], [-6], [8], [16], [-30], [-64], [-34], [8], [-6], [-28], [-64], [26], [48], [24], [-26], [-10], [-30], [-38], [54], [8], [-38], [14], [0], [-30], [-20], [-56], [-54], [32], [14], [48], [28], [-12], [-32], [-54], [-28], [70], [0], [6], [18], [-48], [18], [-30], [-48], [26], [-4], [12], [-68], [72], [22], [16], [60], [30], [-44], [-72], [22], [-10], [22], [-20], [-36], [32], [-68], [72], [34], [24], [-66], [68], [-6], [42], [-10], [-50], [-68], [-70], [-74], [-36], [46], [4], [46], [-64], [4], [-28], [-52], [58], [-54], [-20], [-66], [22], [-16], [-14], [-38], [20], [44], [16], [26], [74], [-20], [8], [-2], [30], [34], [-32], [18], [-12], [80], [84], [14], [10], [16], [-22], [-56], [36], [-6], [12], [66], [28], [-16], [66], [-6], [-12], [-58], [66], [-24], [-48], [0], [68], [52], [10], [-64], [-38], [-42], [-66], [-90], [-4], [-68], [-22], [-72], [-80], [24], [-6], [68], [8], [-60], [-34], [-46], [-2], [-50], [8], [-54], [-34], [-32], [74], [16], [12], [26], [4], [16], [-34], [-66], [88], [58], [-8], [10], [-80], [-38], [34], [-10], [48], [0], [-22], [14], [-20], [18], [28], [-46], [68], [0], [-40], [-12], [20], [2], [66], [26], [20], [-84], [-34], [-38], [30], [-54], [58], [28], [-32], [-48], [56], [-62], [-20], [16], [-52], [46], [14], [-70], [-42], [-34], [-38], [38], [-36], [8], [-14], [8], [-102], [26], [-34], [40], [92], [16], [22], [44], [66], [24], [-90], [8], [-40], [-18], [-62], [46], [56], [78], [-2], [84], [44], [78], [14], [88], [92], [-38], [-24], [-24], [-96], [-86], [66], [88], [-46], [36], [-50], [56], [38], [-44], [-66], [-34], [56]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6240_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6240_2_a_l();".
// 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_6240_2_a_l(:prec:=1)
    chi := MakeCharacter_6240_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_6240_2_a_l();".
// 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_6240_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6240_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-2, 1]>,<11,R![0, 1]>,<17,R![-4, 1]>,<19,R![-2, 1]>],Snew);
    return Vf;
end function;