
// Make newform 4620.2.a.j in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4620_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_4620_2_a_j();".
// 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_4620_2_a_j();".
// 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_4620_a();"
function MakeCharacter_4620_a()
    N := 4620;
    order := 1;
    char_gens := [2311, 1541, 3697, 661, 2521];
    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_4620_a_Hecke(Kf)
    return MakeCharacter_4620_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], [1], [-1], [2], [-6], [8], [-6], [6], [2], [2], [0], [8], [-12], [6], [6], [8], [2], [0], [-10], [8], [12], [6], [2], [12], [-4], [-12], [2], [-6], [-4], [12], [18], [-4], [6], [-16], [14], [2], [24], [-18], [0], [2], [0], [20], [0], [-10], [20], [8], [12], [2], [0], [0], [-28], [30], [18], [24], [6], [-16], [-16], [-6], [-28], [-18], [20], [-18], [-10], [18], [20], [32], [-24], [8], [-18], [24], [-28], [-4], [-16], [24], [-6], [14], [-18], [-4], [-6], [26], [-24], [14], [8], [18], [-18], [-16], [-24], [-34], [-24], [-24], [2], [36], [32], [24], [-42], [-6], [-28], [-34], [-28], [0], [-36], [6], [-4], [26], [24], [-42], [24], [-4], [32], [-40], [18], [26], [20], [30], [-40], [-36], [-18], [12], [-34], [-16], [18], [6], [-10], [-18], [-22], [18], [-28], [2], [20], [24], [-40], [-10], [24], [-16], [6], [44], [30], [30], [-28], [-6], [-34], [0], [-34], [6], [-34], [-42], [50], [-30], [-40], [-30], [38], [-24], [50], [36], [56], [-54], [2], [-24], [18], [-12], [8], [18], [-30], [12], [20], [-10], [-34], [-42], [-60], [14], [0], [44], [2], [-30], [20], [42], [-40], [-16], [26], [36], [14], [-18], [-18], [-42], [-4], [-28], [14], [-12], [-34], [0], [20], [-54], [36], [18], [2], [-22], [-30], [-24], [-30], [-40], [26], [-16], [-42], [-30], [-34], [12], [-66], [50], [-52], [6], [38], [-12], [-24], [-22], [-4], [-24], [18], [-18], [38], [68], [-18], [-58], [-12], [2], [30], [24], [-64], [36], [-58], [-40], [8], [-18], [44], [-48], [14], [-36], [48], [30], [30], [20], [-64], [14], [54], [-60], [-16], [66], [44], [-48], [32], [0], [-30], [50], [-30], [-36], [38], [-76], [-6], [62], [-16], [48], [-40], [2], [-6], [62], [30], [42], [56], [24], [14], [2], [-10], [8], [2], [-64], [42], [26], [38], [24], [24], [-4], [-24], [56], [68], [-36], [-34], [12], [-46], [12], [54], [36], [72], [0], [-64], [-24], [14], [54], [-48], [-28], [26], [-36], [-16], [18], [-40], [74], [42], [62], [60], [20], [48], [54], [30], [74], [-12], [8], [78], [0], [-34], [-18], [-40], [38], [36], [2], [-60], [-70], [-52], [80], [-48], [-12], [-22], [42], [-64], [84], [20], [60], [62], [54], [-70], [56], [-10], [30], [-36], [8], [54], [-12], [74], [-34], [-60], [66], [32], [50], [30], [8], [74], [-6], [42], [12], [60], [42], [-64], [-36], [-30], [-12], [44], [-82], [78], [-4], [62], [60], [38], [-24], [-66], [-88], [-10], [78], [72], [-34], [-84], [-34], [-78], [6], [-88], [72], [8], [84], [-100], [-22], [8], [0], [-82], [6], [24], [8], [-36], [44], [56], [54], [-4], [-18], [68], [18], [8], [-42], [-60], [-76], [62], [42], [8], [-42], [74], [72], [-78], [-100], [104], [-102], [96], [-76], [-18], [-48], [30], [-82], [90], [-96], [-58], [78], [30], [6], [38], [-96]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4620_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4620_2_a_j();".
// 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_4620_2_a_j(:prec:=1)
    chi := MakeCharacter_4620_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_4620_2_a_j();".
// 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_4620_2_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4620_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<13,R![-2, 1]>,<17,R![6, 1]>,<19,R![-8, 1]>],Snew);
    return Vf;
end function;