
// Make newform 4830.2.a.f in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4830_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_4830_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_4830_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_4830_a();"
function MakeCharacter_4830_a()
    N := 4830;
    order := 1;
    char_gens := [3221, 967, 2761, 1891];
    v := [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_4830_a_Hecke(Kf)
    return MakeCharacter_4830_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], [-1], [1], [1], [-2], [-6], [0], [6], [-1], [4], [-2], [-2], [2], [-10], [6], [6], [4], [-6], [6], [-2], [6], [-8], [-14], [-2], [-14], [-10], [-16], [4], [2], [-14], [-2], [12], [-2], [16], [6], [-8], [-6], [0], [6], [20], [-24], [-6], [-8], [2], [6], [-8], [12], [8], [10], [10], [-22], [6], [-12], [28], [2], [24], [-14], [-18], [-24], [-20], [-28], [-10], [-20], [20], [-22], [-6], [-20], [-20], [-36], [-16], [-2], [0], [-24], [-26], [4], [8], [-14], [18], [-24], [-6], [16], [38], [-12], [-6], [-10], [4], [-18], [-28], [6], [14], [6], [24], [-2], [16], [-28], [-20], [-18], [-2], [-4], [-6], [-32], [6], [42], [-16], [40], [-14], [28], [-30], [30], [-42], [4], [42], [18], [10], [16], [4], [40], [-6], [-2], [-38], [-14], [-2], [-42], [-4], [36], [42], [-26], [-4], [-16], [-18], [36], [8], [24], [2], [-34], [8], [-46], [-40], [2], [-6], [32], [-36], [-2], [-12], [52], [48], [-22], [30], [4], [24], [52], [2], [-24], [26], [-22], [-4], [-48], [30], [22], [30], [-36], [2], [-26], [20], [-22], [-16], [8], [10], [34], [0], [-6], [28], [-32], [48], [-10], [-14], [-52], [32], [40], [6], [58], [36], [-52], [30], [-24], [42], [8], [44], [26], [0], [38], [28], [20], [-14], [-44], [56], [-46], [-10], [36], [66], [-54], [2], [-2], [-28], [-48], [0], [-40], [-60], [2], [56], [-34], [-10], [-8], [6], [-36], [-64], [2], [-18], [-48], [54], [-46], [-36], [18], [16], [-20], [10], [0], [10], [-56], [40], [32], [46], [-56], [-50], [52], [-52], [-8], [6], [-32], [-40], [-44], [-8], [32], [-22], [26], [16], [72], [0], [-4], [48], [-62], [-18], [8], [16], [24], [52], [-22], [48], [4], [58], [40], [0], [72], [30], [46], [-10], [-54], [-26], [22], [-2], [20], [-58], [50], [32], [78], [8], [20], [10], [38], [-48], [-54], [64], [72], [-30], [28], [-50], [56], [42], [-18], [14], [30], [18], [10], [-36], [-4], [-66], [84], [-24], [-60], [28], [46], [2], [32], [60], [28], [-82], [68], [-14], [-48], [52], [66], [54], [-4], [-4], [-40], [8], [76], [26], [-6], [42], [46], [-28], [-46], [-22], [6], [-76], [92], [36], [-56], [46], [-2], [22], [-22], [12], [36], [84], [34], [-28], [-30], [16], [-12], [42], [-14], [32], [-46], [48], [-24], [12], [-64], [6], [16], [-14], [-32], [-16], [-22], [-66], [-48], [12], [2], [32], [-38], [-42], [0], [-18], [-36], [-82], [20], [40], [-10], [-24], [-44], [-80], [-54], [8], [68], [76], [-48], [34], [30], [-42], [18], [-6], [-22], [-46], [30], [-74], [48], [-14], [-82], [80], [50], [-84], [-12], [-84], [-48], [68], [-60], [58], [44], [-52], [100], [18], [96], [72], [42], [-58], [10], [-86], [-76], [-68], [46], [18], [76], [-72], [-92], [50], [66], [52], [-8], [36], [2], [70], [-96], [-84], [58], [84], [-84], [74], [-48], [104]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4830_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4830_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_4830_2_a_f(:prec:=1)
    chi := MakeCharacter_4830_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_4830_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_4830_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4830_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![2, 1]>,<13,R![6, 1]>,<17,R![0, 1]>,<19,R![-6, 1]>,<29,R![-4, 1]>],Snew);
    return Vf;
end function;