
// Make newform 4830.2.a.b in Magma, downloaded from the LMFDB on 19 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_b();".
// 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_b();".
// 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], [6], [2], [0], [1], [2], [0], [10], [6], [8], [-12], [-6], [12], [10], [-16], [4], [-6], [-16], [-4], [-2], [18], [-6], [16], [-12], [-10], [-18], [0], [12], [-2], [8], [10], [0], [-10], [8], [-12], [-6], [4], [18], [-12], [10], [-6], [16], [28], [-16], [-12], [10], [-2], [4], [10], [12], [-22], [16], [18], [-16], [2], [26], [-4], [2], [4], [-32], [-22], [2], [20], [2], [4], [18], [26], [-12], [16], [26], [4], [12], [10], [38], [-30], [26], [-20], [-18], [-12], [18], [-40], [4], [-22], [18], [2], [-24], [12], [8], [24], [12], [-4], [28], [34], [14], [4], [-10], [24], [-46], [-28], [-6], [12], [2], [-4], [-30], [36], [26], [-32], [26], [22], [-24], [-16], [10], [44], [12], [34], [-36], [18], [-6], [-22], [12], [-8], [-6], [-26], [24], [-16], [30], [-36], [-8], [-8], [-38], [14], [18], [-30], [-28], [2], [-6], [-56], [-22], [32], [-52], [-14], [40], [14], [18], [0], [24], [-38], [46], [24], [-12], [-24], [36], [-32], [54], [-54], [-46], [-28], [-10], [-8], [-52], [-18], [36], [16], [-50], [34], [-30], [-44], [26], [-60], [58], [-16], [6], [44], [2], [0], [18], [-8], [60], [-22], [42], [-40], [42], [2], [28], [26], [24], [50], [-4], [-36], [2], [28], [26], [66], [-38], [-6], [12], [-14], [56], [-26], [-46], [28], [18], [-64], [-36], [2], [-64], [-54], [-38], [-8], [68], [24], [42], [48], [62], [-48], [-14], [-50], [0], [18], [-40], [-28], [14], [66], [-36], [-40], [28], [58], [0], [8], [2], [28], [-12], [58], [-54], [-52], [-24], [12], [48], [0], [-26], [2], [-32], [-32], [-60], [52], [-8], [-54], [-42], [-48], [-54], [66], [-12], [46], [-36], [58], [-54], [-40], [68], [-14], [38], [26], [-64], [66], [-2], [80], [42], [-22], [56], [-22], [-32], [18], [16], [-36], [-74], [42], [28], [4], [-40], [-68], [26], [68], [36], [-22], [42], [56], [38], [-30], [-28], [-34], [28], [-22], [2], [64], [-54], [36], [-28], [10], [50], [40], [-44], [-68], [18], [60], [-6], [-4], [74], [68], [-30], [34], [-56], [0], [-70], [84], [-12], [34], [-22], [-80], [42], [66], [16], [-66], [-38], [-60], [84], [24], [-6], [-18], [18], [40], [60], [-20], [-28], [-18], [50], [-30], [-80], [-38], [86], [-6], [-80], [34], [36], [-6], [40], [-48], [50], [-64], [-38], [-30], [-16], [-58], [-94], [-24], [36], [38], [64], [10], [78], [-24], [20], [-28], [10], [-14], [-16], [74], [44], [-16], [24], [-78], [-40], [74], [-4], [44], [-14], [-58], [-22], [-62], [-42], [-72], [46], [-72], [12], [72], [-82], [-56], [68], [50], [-14], [84], [52], [12], [-30], [8], [-42], [-60], [18], [2], [38], [72], [-54], [26], [-16], [42], [58], [-12], [4], [34], [42], [-52], [-44], [58], [-14], [-92], [-32], [-86], [-20], [2], [22], [48], [-44], [-22], [98], [4], [66], [-8], [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_b();".
// 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_b(: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_b();".
// 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_b( : 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![-6, 1]>,<17,R![-2, 1]>,<19,R![0, 1]>,<29,R![-2, 1]>],Snew);
    return Vf;
end function;