
// Make newform 4830.2.a.ba in Magma, downloaded from the LMFDB on 24 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_ba();".
// 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_ba();".
// 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], [-2], [-6], [0], [-1], [-10], [8], [-6], [-6], [0], [8], [-14], [4], [-14], [16], [-8], [-6], [0], [0], [-14], [2], [-2], [-16], [0], [-6], [18], [8], [-4], [18], [-12], [18], [16], [-22], [12], [0], [6], [-20], [-6], [-8], [18], [6], [-8], [20], [16], [-8], [10], [-6], [-8], [-6], [-16], [2], [-8], [-18], [0], [-10], [-14], [-8], [-22], [4], [16], [26], [14], [-20], [26], [-12], [14], [-14], [16], [16], [-38], [16], [0], [-6], [30], [18], [-6], [-24], [26], [8], [-14], [40], [4], [18], [10], [-2], [-24], [32], [16], [16], [-12], [-4], [-16], [-34], [-6], [16], [22], [4], [34], [-24], [-22], [32], [-14], [-12], [18], [0], [10], [0], [34], [-6], [-32], [32], [34], [-16], [-24], [38], [-32], [-6], [-30], [-22], [44], [44], [-46], [-38], [8], [-40], [-38], [-20], [-24], [40], [34], [-6], [-54], [-30], [48], [42], [-54], [-44], [6], [-40], [32], [14], [24], [-26], [-22], [44], [48], [-2], [10], [52], [48], [16], [-40], [-40], [18], [-46], [-46], [-36], [-30], [-8], [56], [-22], [0], [8], [14], [42], [-26], [40], [-2], [8], [-46], [-48], [50], [52], [-42], [-32], [-22], [-32], [8], [-42], [-38], [-40], [-30], [38], [0], [10], [-32], [34], [60], [64], [6], [24], [10], [18], [-30], [-14], [24], [-46], [0], [14], [-30], [-24], [54], [32], [36], [26], [28], [50], [-34], [-64], [-48], [-40], [-14], [-48], [18], [-24], [-26], [22], [16], [-30], [-40], [-28], [-42], [42], [0], [64], [-28], [14], [24], [8], [-54], [56], [-40], [-6], [-54], [12], [-40], [16], [12], [72], [-18], [34], [32], [56], [8], [8], [-64], [-22], [10], [-56], [-38], [14], [-52], [18], [-16], [-2], [74], [8], [-40], [-42], [-62], [-14], [-40], [-46], [18], [64], [6], [-50], [-40], [-46], [24], [-78], [48], [-52], [-66], [26], [0], [64], [48], [64], [-22], [-4], [-40], [-30], [26], [-24], [-46], [-22], [64], [-6], [-40], [-18], [-62], [-48], [22], [-84], [12], [26], [50], [-64], [-12], [-80], [50], [-76], [-30], [56], [-66], [0], [2], [-14], [-60], [0], [-22], [68], [-32], [42], [34], [-40], [42], [-42], [16], [-30], [58], [32], [20], [32], [42], [-26], [-74], [-64], [-68], [40], [60], [-14], [50], [10], [-56], [30], [50], [-18], [8], [2], [-20], [-34], [4], [-24], [-6], [-28], [-54], [-74], [-32], [-46], [-54], [0], [48], [66], [64], [50], [-6], [-72], [-96], [28], [74], [-50], [24], [82], [12], [-20], [24], [90], [48], [-58], [-36], [-72], [-70], [50], [-22], [-86], [-46], [-64], [-30], [-80], [-56], [-32], [-50], [8], [-56], [-62], [62], [12], [-84], [-16], [74], [-24], [-22], [-56], [-10], [70], [-30], [0], [18], [-34], [-48], [-86], [-30], [80], [68], [-14], [94], [-64], [8], [-30], [62], [96], [-56], [10], [16], [-46], [-14], [48], [68], [26], [46], [0], [-6], [4], [-40]];
    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_ba();".
// 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_ba(: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_ba();".
// 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_ba( : 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![2, 1]>,<17,R![6, 1]>,<19,R![0, 1]>,<29,R![10, 1]>],Snew);
    return Vf;
end function;