
// Make newform 4010.2.a.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4010_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_4010_2_a_f();".
// 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_4010_2_a_f();".
// 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_4010_a();"
function MakeCharacter_4010_a()
    N := 4010;
    order := 1;
    char_gens := [2407, 3211];
    v := [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_4010_a_Hecke(Kf)
    return MakeCharacter_4010_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], [2], [1], [4], [-4], [4], [-4], [4], [6], [-6], [8], [-4], [-10], [4], [8], [-12], [12], [-10], [-2], [4], [10], [-8], [-4], [-6], [8], [2], [20], [-2], [-14], [6], [2], [-12], [12], [0], [-18], [-8], [4], [-6], [-18], [-22], [-12], [-10], [-8], [-16], [10], [20], [-12], [20], [14], [-10], [20], [-24], [-30], [12], [30], [24], [-6], [8], [8], [-2], [6], [0], [20], [0], [22], [-12], [-20], [2], [-18], [14], [-24], [0], [-18], [-26], [36], [8], [18], [2], [1], [-26], [28], [26], [32], [-2], [8], [-18], [-6], [22], [-22], [-22], [26], [8], [-8], [-12], [4], [-24], [-26], [10], [22], [-42], [16], [6], [16], [10], [-36], [38], [0], [4], [8], [-22], [-8], [-44], [-28], [12], [-48], [-26], [-20], [-14], [-14], [-4], [-18], [-44], [50], [18], [-44], [-2], [-26], [24], [-2], [2], [36], [34], [-16], [42], [14], [54], [30], [18], [42], [18], [-12], [-34], [-34], [24], [2], [12], [22], [6], [-44], [58], [-48], [-26], [-12], [-6], [30], [32], [-40], [-14], [56], [18], [-4], [54], [14], [-12], [-58], [-36], [4], [62], [46], [44], [-48], [-38], [8], [-64], [40], [6], [-32], [18], [48], [-14], [-20], [12], [14], [0], [4], [2], [-34], [36], [-30], [24], [2], [-56], [-12], [50], [-28], [6], [-2], [-2], [-22], [20], [-18], [40], [12], [46], [44], [4], [52], [-28], [-10], [-28], [-46], [26], [-60], [-18], [-24], [-42], [26], [-62], [4], [36], [10], [-16], [-22], [-24], [24], [30], [8], [68], [46], [56], [-46], [52], [44], [14], [40], [58], [-14], [34], [0], [48], [48], [-12], [-22], [10], [6], [-48], [-38], [72], [-20], [46], [-58], [-30], [-58], [10], [-14], [0], [2], [-2], [-52], [64], [18], [40], [30], [-46], [-6], [16], [-50], [26], [46], [-60], [26], [34], [-42], [40], [-10], [-36], [-4], [-82], [74], [-36], [-6], [-4], [-54], [-10], [52], [80], [20], [58], [72], [14], [54], [40], [72], [-56], [46], [-10], [-12], [22], [68], [80], [4], [74], [-4], [-56], [-60], [-8], [36], [70], [-32], [52], [-84], [-6], [66], [32], [-28], [-58], [36], [44], [16], [58], [20], [32], [-42], [12], [24], [14], [40], [28], [-62], [-84], [22], [54], [72], [-38], [48], [54], [-22], [-28], [-10], [-10], [82], [-38], [82], [-32], [-6], [24], [14], [10], [-64], [-18], [-20], [26], [94], [82], [-50], [48], [-40], [-60], [90], [54], [28], [-70], [-88], [60], [86], [-32], [-60], [2], [-42], [-4], [16], [74], [90], [80], [46], [-60], [0], [72], [-78], [28], [34], [60], [44], [-60], [-60], [-52], [44], [-68], [62], [6], [-62], [-54], [-48], [56], [44], [-14], [0], [54], [-60], [70], [34], [-60], [48], [-36], [-42], [-8], [58], [-46], [92], [56], [-44], [-92], [-72], [-52], [-94], [42], [0], [-32], [90], [18], [-30], [-78], [22], [72], [22], [-32], [24], [-86], [20], [-36]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4010_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4010_2_a_f();".
// 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_4010_2_a_f(:prec:=1)
    chi := MakeCharacter_4010_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_4010_2_a_f();".
// 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_4010_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4010_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 1]>,<7,R![-4, 1]>,<11,R![4, 1]>],Snew);
    return Vf;
end function;