
// Make newform 4830.2.a.z in Magma, downloaded from the LMFDB on 20 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_z();".
// 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_z();".
// 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], [0], [-6], [-6], [4], [1], [-6], [0], [-10], [2], [-8], [8], [-10], [-4], [14], [-16], [-12], [-14], [-8], [4], [14], [14], [2], [8], [20], [-2], [-14], [12], [-4], [-6], [4], [6], [0], [-18], [4], [-8], [-18], [-12], [6], [0], [18], [-10], [8], [-20], [-12], [-4], [-26], [-6], [12], [2], [-24], [18], [0], [18], [-16], [18], [-2], [-16], [6], [12], [4], [14], [22], [-28], [-26], [12], [-10], [18], [8], [-32], [-10], [-12], [-24], [-18], [-22], [-2], [-22], [-8], [6], [-32], [-2], [16], [-4], [26], [22], [42], [20], [36], [0], [-20], [20], [28], [-8], [-6], [-18], [16], [-26], [12], [30], [36], [22], [36], [26], [-20], [34], [12], [26], [-12], [-2], [34], [-4], [-16], [-10], [-24], [-16], [6], [-24], [38], [-22], [6], [44], [4], [-42], [38], [36], [-48], [-50], [-12], [16], [16], [46], [26], [26], [22], [8], [-2], [10], [-52], [-6], [4], [-12], [6], [48], [18], [42], [-52], [-8], [-22], [-42], [4], [-48], [24], [40], [24], [10], [-26], [-18], [-52], [-6], [28], [24], [-22], [-32], [8], [18], [-30], [22], [40], [-50], [-8], [-10], [8], [54], [-52], [-38], [-32], [46], [-44], [48], [34], [-38], [16], [30], [-38], [-64], [-38], [-44], [-30], [20], [-52], [-6], [52], [-22], [58], [62], [18], [-16], [6], [8], [-54], [-14], [56], [30], [-40], [-4], [-54], [4], [-22], [18], [56], [-36], [4], [34], [28], [-62], [48], [54], [22], [40], [66], [8], [28], [-42], [42], [-28], [-56], [-52], [-62], [-4], [16], [-14], [0], [24], [-38], [-42], [-4], [44], [4], [-20], [4], [-10], [2], [-4], [-4], [72], [-36], [24], [-26], [70], [-24], [2], [22], [28], [38], [0], [-2], [10], [-64], [-36], [6], [6], [-30], [28], [6], [-66], [-24], [-66], [-18], [-8], [-74], [24], [58], [4], [-12], [22], [82], [0], [0], [64], [0], [-26], [-52], [60], [38], [-58], [0], [42], [-2], [-36], [-54], [56], [-62], [-34], [16], [6], [-84], [28], [-34], [62], [-16], [-36], [12], [-54], [-52], [46], [48], [-70], [-88], [-10], [-26], [-36], [0], [10], [44], [-60], [62], [-30], [-28], [46], [2], [28], [-14], [18], [68], [4], [-48], [-26], [-50], [-26], [-56], [60], [-84], [-20], [-66], [26], [26], [-24], [58], [66], [-30], [-80], [-18], [36], [-10], [12], [-88], [-42], [-60], [90], [-54], [-8], [-58], [-54], [24], [-16], [2], [40], [62], [-62], [32], [-40], [-68], [-30], [70], [56], [-94], [44], [76], [-64], [-82], [40], [-78], [44], [-24], [22], [-90], [66], [-58], [90], [76], [34], [28], [-80], [40], [18], [-40], [-8], [10], [-50], [4], [-20], [-72], [30], [-80], [54], [-76], [-30], [-66], [18], [-88], [90], [2], [48], [70], [-30], [-56], [100], [-62], [6], [-84], [-60], [62], [-78], [12], [68], [2], [24], [6], [70], [-16], [-20], [90], [-18], [-20], [-22], [4], [-100]];
    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_z();".
// 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_z(: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_z();".
// 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_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4830_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![0, 1]>,<13,R![6, 1]>,<17,R![6, 1]>,<19,R![-4, 1]>,<29,R![6, 1]>],Snew);
    return Vf;
end function;