
// Make newform 4114.2.a.c in Magma, downloaded from the LMFDB on 12 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4114_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_4114_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_4114_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_4114_a();"
function MakeCharacter_4114_a()
    N := 4114;
    order := 1;
    char_gens := [3027, 2179];
    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_4114_a_Hecke(Kf)
    return MakeCharacter_4114_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], [0], [0], [2], [0], [2], [1], [4], [6], [4], [-2], [-4], [2], [4], [0], [2], [4], [0], [12], [2], [-2], [14], [-12], [6], [-2], [10], [20], [-12], [4], [-14], [12], [-12], [18], [20], [-2], [-4], [-18], [-8], [14], [4], [-4], [-12], [-8], [-22], [24], [10], [-4], [4], [8], [-18], [14], [-12], [18], [-4], [2], [-12], [-24], [-4], [-12], [-14], [-12], [22], [-4], [6], [22], [-4], [-28], [-2], [32], [-14], [6], [8], [-10], [26], [-32], [16], [-18], [28], [14], [-38], [0], [-22], [30], [-2], [10], [12], [-10], [-6], [18], [-20], [-36], [10], [-34], [-4], [-4], [-18], [30], [-26], [-4], [0], [20], [2], [-28], [-42], [0], [-14], [12], [-34], [-4], [-26], [-26], [-18], [-46], [-24], [-4], [22], [-24], [-32], [-16], [-36], [42], [-6], [24], [0], [44], [50], [8], [-30], [-16], [-46], [-36], [6], [18], [22], [-26], [-34], [54], [-20], [34], [-30], [-16], [0], [30], [12], [-42], [-6], [-4], [2], [-12], [36], [-20], [42], [-28], [-38], [-12], [26], [-4], [54], [-30], [48], [20], [58], [-8], [12], [42], [-14], [-30], [-8], [26], [-52], [-20], [2], [18], [34], [-56], [-54], [64], [-4], [24], [46], [-4], [-36], [-44], [-10], [-24], [34], [48], [-44], [2], [-50], [42], [-16], [-20], [6], [-24], [-30], [46], [-52], [22], [48], [56], [-10], [-50], [-18], [-44], [6], [-60], [36], [-42], [68], [66], [38], [58], [28], [-62], [70], [-32], [22], [58], [-30], [-42], [-42], [30], [-2], [28], [10], [18], [62], [-20], [-24], [-46], [-56], [0], [10], [-12], [-72], [22], [72], [76], [52], [32], [4], [16], [22], [30], [-18], [-50], [8], [44], [-24], [22], [6], [-40], [-54], [30], [36], [20], [32], [-60], [-18], [54], [-20], [24], [44], [54], [-44], [18], [-18], [48], [6], [48], [36], [26], [56], [-70], [-4], [-60], [-22], [18], [12], [-60], [46], [-70], [-62], [-4], [-76], [-2], [-24], [-80], [50], [40], [48], [6], [-44], [64], [36], [-20], [-14], [16], [12], [74], [-66], [74], [48], [40], [-10], [-12], [4], [60], [38], [46], [-40], [66], [36], [16], [-30], [20], [-18], [-86], [-30], [24], [-34], [-2], [-28], [-30], [-70], [-36], [-44], [-22], [-48], [-28], [28], [-14], [-44], [40], [28], [42], [30], [10], [8], [66], [-78], [-48], [44], [26], [40], [68], [36], [38], [-60], [4], [78], [-10], [22], [46], [-42], [-84], [88], [-6], [48], [-56], [30], [8], [-8], [68], [-26], [60], [-60], [-38], [60], [76], [54], [-6], [-12], [-16], [-16], [50], [14], [-78], [46], [24], [-58], [-70], [102], [40], [50], [-64], [-26], [48], [92], [42], [36], [12], [-4], [-40], [26], [-100], [102], [-52], [74], [80], [42], [32], [78], [-18], [-58], [70], [-34], [84], [-72], [62], [-42], [20], [-52], [-22], [4], [70], [14], [-82], [-76], [-6], [28], [42], [60], [-34], [-66], [-76], [-26], [20], [-38]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4114_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4114_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_4114_2_a_c(:prec:=1)
    chi := MakeCharacter_4114_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_4114_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_4114_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4114_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<5,R![0, 1]>,<7,R![-2, 1]>],Snew);
    return Vf;
end function;