
// Make newform 4830.2.a.s in Magma, downloaded from the LMFDB on 25 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_s();".
// 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_s();".
// 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], [-4], [-2], [6], [-4], [-1], [2], [4], [-6], [10], [4], [-12], [-6], [-4], [-14], [-12], [-12], [10], [-8], [-4], [-6], [2], [14], [8], [0], [-6], [-14], [-12], [12], [10], [-4], [10], [0], [-14], [12], [-20], [10], [4], [-22], [8], [-14], [-18], [24], [-28], [-16], [-28], [-22], [26], [-28], [-18], [24], [18], [-16], [-10], [-20], [-14], [6], [24], [26], [20], [24], [10], [-18], [-28], [30], [28], [-30], [26], [-32], [8], [-22], [-8], [0], [26], [-2], [-34], [10], [-40], [26], [0], [-30], [36], [-20], [-22], [14], [30], [36], [4], [24], [-20], [28], [-20], [24], [14], [-30], [-32], [14], [20], [42], [36], [-10], [32], [2], [12], [34], [12], [-14], [-8], [42], [-6], [-28], [-8], [-26], [24], [12], [30], [4], [26], [10], [26], [-12], [44], [2], [-22], [-16], [-40], [-14], [36], [-8], [0], [10], [42], [-26], [-54], [8], [18], [-22], [-20], [2], [-44], [16], [-38], [32], [22], [18], [-36], [-16], [18], [-6], [-28], [4], [28], [-16], [-40], [-30], [10], [50], [12], [18], [-44], [0], [-38], [-16], [56], [-18], [42], [34], [-52], [58], [8], [14], [60], [50], [36], [-54], [-24], [18], [4], [-48], [-6], [2], [24], [-14], [18], [16], [10], [0], [34], [-4], [-8], [-66], [-8], [-10], [-30], [2], [38], [28], [18], [-28], [-10], [-26], [-56], [34], [-64], [20], [10], [-28], [42], [62], [-48], [-60], [-32], [-18], [20], [66], [-8], [-34], [-50], [-24], [2], [-8], [-36], [-34], [-58], [36], [56], [76], [26], [4], [-8], [18], [-40], [-16], [54], [66], [36], [-24], [64], [-60], [-40], [-26], [2], [-48], [-8], [-40], [-32], [-64], [-46], [-46], [24], [-58], [10], [-36], [-78], [8], [18], [-22], [72], [48], [74], [42], [42], [-44], [50], [58], [4], [-2], [-6], [12], [-14], [8], [50], [-64], [4], [70], [-46], [48], [-28], [-16], [0], [-6], [36], [36], [-34], [18], [20], [66], [18], [-36], [26], [0], [34], [-78], [40], [-14], [28], [-44], [18], [-22], [56], [20], [-40], [-86], [4], [-14], [-40], [18], [36], [-54], [38], [12], [-72], [-58], [-68], [-28], [58], [2], [52], [14], [-18], [-28], [-6], [-50], [8], [-12], [-40], [2], [38], [-66], [-44], [-60], [32], [36], [2], [-26], [42], [-48], [50], [-62], [70], [56], [-22], [84], [-30], [-4], [0], [-22], [-68], [-38], [58], [24], [66], [-30], [24], [-24], [-14], [48], [10], [-86], [-56], [-20], [-12], [18], [-62], [-40], [-54], [-60], [-68], [8], [-22], [-32], [-6], [-60], [-88], [58], [-6], [42], [10], [34], [-12], [-30], [44], [4], [0], [-2], [76], [72], [-78], [-46], [-4], [-20], [-48], [14], [0], [-6], [-32], [-78], [-30], [18], [0], [-34], [70], [-20], [2], [-30], [80], [36], [82], [-18], [-32], [88], [38], [22], [-60], [-64], [38], [-24], [2], [58], [0], [-84], [-70], [18], [32], [-54], [-4], [-8]];
    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_s();".
// 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_s(: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_s();".
// 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_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4830_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![4, 1]>,<13,R![2, 1]>,<17,R![-6, 1]>,<19,R![4, 1]>,<29,R![-2, 1]>],Snew);
    return Vf;
end function;