
// Make newform 4830.2.a.bg in Magma, downloaded from the LMFDB on 22 September 2025.
// 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_bg();".
// 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_bg();".
// 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 := 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]];
    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], [-4], [-6], [1], [0], [-6], [6], [-6], [6], [2], [2], [-12], [-14], [-2], [6], [-14], [-16], [-2], [10], [2], [10], [8], [-4], [18], [6], [-2], [12], [-6], [12], [-2], [0], [-2], [-8], [-14], [0], [4], [10], [12], [10], [-14], [4], [12], [4], [14], [-6], [-10], [-18], [-20], [-8], [-14], [-24], [30], [2], [-16], [8], [0], [6], [-12], [-16], [10], [-10], [-28], [36], [-28], [-8], [-26], [20], [-8], [-26], [20], [0], [-6], [-14], [20], [34], [-4], [6], [-24], [2], [26], [-20], [-18], [12], [2], [22], [10], [-24], [-26], [-36], [-20], [12], [26], [42], [24], [2], [-32], [-30], [-18], [-4], [16], [30], [-12], [26], [38], [38], [-24], [10], [-18], [-34], [-8], [48], [28], [6], [-14], [18], [-14], [46], [-2], [-28], [-8], [-30], [38], [-16], [-32], [50], [44], [-8], [-40], [-22], [-42], [-40], [-38], [-4], [-46], [-22], [12], [0], [46], [-36], [20], [48], [-14], [-18], [40], [-32], [-12], [-18], [-24], [30], [10], [8], [24], [30], [-2], [-18], [-4], [22], [-26], [-48], [46], [0], [-48], [-22], [50], [-44], [-30], [-44], [-36], [-56], [2], [-2], [52], [-36], [32], [46], [34], [0], [52], [-2], [0], [26], [-16], [-16], [26], [-36], [2], [4], [4], [6], [-12], [12], [-22], [14], [24], [30], [-38], [-26], [-34], [28], [20], [20], [44], [-52], [10], [-4], [38], [-30], [56], [-22], [-56], [48], [-54], [-10], [32], [50], [-38], [-24], [42], [-16], [36], [50], [-20], [-30], [-32], [36], [-24], [26], [8], [-42], [-64], [-28], [-16], [74], [-4], [-60], [-28], [36], [4], [2], [26], [28], [-36], [-68], [20], [-56], [34], [-30], [-24], [-40], [20], [24], [-70], [20], [-56], [62], [-16], [0], [-40], [50], [-42], [10], [2], [66], [14], [58], [76], [-2], [-46], [-32], [50], [44], [-68], [10], [-10], [-12], [-42], [-32], [40], [2], [4], [30], [80], [-74], [34], [30], [30], [-42], [6], [72], [-4], [46], [72], [-12], [36], [-52], [-10], [54], [-40], [28], [20], [-50], [-28], [-14], [-36], [68], [6], [30], [64], [-4], [-64], [16], [-68], [-46], [-14], [42], [-6], [20], [46], [-86], [-78], [76], [4], [-44], [-72], [-22], [-50], [-86], [-90], [36], [12], [-84], [-30], [48], [-46], [8], [84], [-34], [-90], [-8], [6], [4], [24], [-20], [0], [46], [-68], [10], [60], [-64], [-22], [-42], [-80], [-48], [46], [-80], [34], [70], [32], [-34], [-36], [-66], [-64], [-40], [22], [-12], [8], [-96], [-6], [28], [-68], [-20], [20], [-30], [90], [34], [2], [78], [-38], [-10], [90], [-70], [-24], [-38], [70], [32], [-30], [24], [-96], [100], [-4], [-4], [88], [14], [28], [-24], [-36], [94], [32], [36], [6], [50], [2], [50], [-32], [76], [-94], [-26], [76], [-56], [-4], [-66], [-6], [-64], [12], [-68], [-30], [-38], [16], [-4], [-66], [0], [84], [-22], [-12], [68]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4830_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4830_2_a_bg();".
// 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_bg(: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_bg();".
// 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_bg( : 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![4, 1]>,<19,R![6, 1]>,<29,R![0, 1]>],Snew);
    return Vf;
end function;