
// Make newform 4830.2.a.bi 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_bi();".
// 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_bi();".
// 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], [2], [-4], [-1], [-2], [8], [6], [2], [4], [8], [-2], [-4], [14], [-4], [0], [2], [0], [12], [-6], [18], [-10], [-8], [-12], [-10], [-14], [8], [-12], [-6], [12], [6], [-24], [-2], [-4], [16], [6], [12], [-10], [0], [-14], [22], [0], [4], [8], [12], [6], [10], [-24], [2], [-4], [2], [8], [-2], [-8], [22], [18], [-4], [6], [-12], [0], [-6], [-18], [-36], [-22], [12], [-26], [-14], [24], [16], [6], [20], [-24], [-10], [-34], [-6], [-6], [-12], [-2], [0], [2], [32], [12], [-30], [2], [-18], [-40], [-20], [-16], [-32], [-12], [4], [16], [-2], [-6], [-4], [14], [12], [6], [28], [-30], [4], [-22], [-4], [-14], [0], [42], [8], [-26], [-38], [-28], [8], [-6], [4], [32], [30], [12], [38], [-14], [22], [28], [-28], [14], [14], [-8], [-8], [-34], [4], [24], [48], [-26], [-14], [-14], [-10], [4], [30], [-6], [28], [-10], [-32], [-12], [-10], [40], [6], [-6], [-36], [-16], [14], [-30], [-52], [0], [20], [0], [-24], [-54], [26], [14], [-60], [-54], [32], [-36], [18], [32], [16], [6], [-30], [-2], [-12], [22], [-8], [-46], [-24], [-6], [-36], [38], [24], [-50], [8], [20], [-26], [42], [16], [-10], [-2], [52], [-22], [56], [26], [-36], [-36], [-50], [12], [58], [34], [-34], [18], [0], [-18], [8], [-58], [-14], [60], [38], [40], [68], [-54], [-68], [2], [-58], [56], [36], [32], [58], [-24], [-6], [-40], [-18], [-10], [32], [50], [16], [-60], [-26], [42], [24], [8], [36], [-50], [-36], [48], [26], [28], [40], [2], [30], [4], [-16], [-4], [44], [-64], [46], [66], [-32], [-56], [36], [-76], [-32], [14], [-62], [8], [58], [22], [60], [30], [76], [62], [-78], [32], [12], [-50], [-2], [-14], [-36], [-18], [-70], [-44], [70], [38], [12], [26], [32], [58], [-16], [12], [-34], [-38], [52], [24], [40], [-48], [22], [-52], [-40], [-6], [78], [0], [42], [-18], [12], [-70], [44], [-82], [-34], [24], [-2], [60], [-12], [26], [54], [-80], [-12], [20], [50], [60], [-2], [-24], [38], [-8], [6], [-6], [-52], [40], [-70], [4], [72], [34], [2], [60], [50], [-66], [40], [42], [-62], [-52], [-20], [0], [-2], [-34], [78], [40], [52], [36], [12], [6], [50], [74], [-48], [54], [-38], [-42], [-72], [38], [44], [46], [68], [0], [-10], [-92], [-38], [-18], [80], [14], [42], [-80], [60], [-46], [24], [22], [-78], [48], [52], [52], [-86], [-58], [72], [-6], [-68], [12], [-48], [86], [64], [-34], [36], [96], [-14], [-62], [82], [-98], [-6], [48], [34], [12], [32], [32], [-10], [84], [24], [-78], [-66], [36], [-28], [8], [-14], [-40], [-6], [20], [22], [22], [-94], [-32], [-54], [-42], [-32], [-98], [-30], [52], [-12], [-70], [6], [84], [12], [66], [30], [40], [48], [-30], [96], [14], [6], [48], [-20], [34], [-58], [28], [10], [-52], [96]];
    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_bi();".
// 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_bi(: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_bi();".
// 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_bi( : 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![-2, 1]>,<19,R![4, 1]>,<29,R![2, 1]>],Snew);
    return Vf;
end function;