
// Make newform 4851.2.a.o in Magma, downloaded from the LMFDB on 26 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_4851_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_4851_2_a_o();".
// 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_4851_2_a_o();".
// 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_4851_a();"
function MakeCharacter_4851_a()
    N := 4851;
    order := 1;
    char_gens := [4313, 199, 442];
    v := [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_4851_a_Hecke( : Kf := Kf)
    return MakeCharacter_4851_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], [0], [-4], [0], [1], [2], [2], [6], [-4], [6], [-4], [-6], [-10], [6], [-8], [0], [4], [6], [8], [0], [2], [-10], [12], [0], [-2], [-14], [-8], [-12], [-2], [-12], [-10], [-12], [-4], [-10], [2], [14], [10], [-20], [0], [-6], [24], [-10], [-16], [-14], [-2], [-12], [-6], [8], [-12], [-6], [-6], [0], [26], [8], [8], [32], [-16], [-2], [-2], [-6], [2], [-18], [22], [12], [34], [4], [4], [14], [-20], [-18], [24], [-8], [-16], [34], [4], [-20], [-36], [14], [-28], [-6], [-32], [34], [-24], [2], [-10], [4], [32], [-18], [6], [16], [36], [-32], [-4], [28], [-16], [-40], [-24], [-12], [-38], [22], [38], [-38], [-28], [-18], [-34], [-18], [-4], [-26], [28], [22], [10], [-46], [12], [-44], [4], [24], [4], [28], [16], [-44], [50], [26], [-6], [32], [-20], [-22], [-10], [24], [-12], [-6], [-34], [16], [-20], [2], [42], [-2], [-24], [-22], [28], [30], [-34], [22], [-48], [-28], [2], [20], [-50], [-14], [36], [-36], [42], [-20], [-32], [16], [-12], [-48], [-14], [-12], [-2], [-30], [48], [-26], [-50], [4], [36], [-36], [-44], [-50], [14], [48], [-4], [-34], [8], [14], [-56], [-44], [-10], [32], [34], [-34], [44], [4], [-30], [-30], [-24], [-24], [6], [-36], [-10], [40], [34], [4], [-48], [-36], [-12], [48], [-46], [-14], [-6], [16], [42], [42], [-42], [2], [40], [20], [0], [12], [-66], [8], [-6], [0], [36], [-28], [0], [34], [2], [-66], [72], [-24], [-50], [42], [-12], [-40], [36], [26], [36], [12], [62], [4], [46], [14], [22], [62], [56], [8], [66], [-30], [16], [68], [52], [58], [-12], [54], [14], [48], [28], [40], [6], [64], [-50], [22], [24], [-2], [30], [4], [22], [-18], [12], [-58], [10], [60], [34], [-42], [0], [32], [24], [-36], [26], [-18], [-50], [-36], [-70], [2], [-38], [-20], [-48], [-62], [10], [36], [56], [68], [-32], [22], [-70], [60], [66], [54], [-28], [14], [-52], [28], [-78], [-12], [-54], [-2], [20], [8], [-60], [-26], [-18], [18], [-74], [52], [32], [-38], [-72], [54], [-12], [90], [-24], [-44], [66], [-16], [-48], [10], [-44], [-56], [2], [-82], [38], [-38], [-2], [-52], [34], [34], [-24], [-32], [-24], [70], [-50], [36], [10], [20], [54], [60], [34], [-10], [-70], [58], [-22], [72], [-42], [32], [56], [-44], [38], [-28], [96], [-24], [22], [6], [48], [90], [34], [18], [24], [52], [38], [32], [-46], [30], [-36], [-44], [0], [-86], [78], [38], [-6], [-84], [-36], [16], [70], [14], [10], [-8], [-16], [22], [42], [66], [-76], [-96], [86], [-82], [-22], [0], [-12], [-86], [62], [-32], [50], [-64], [56], [80], [-36], [-26], [-38], [-24], [28], [14], [-50], [28], [-18], [-72], [62], [90], [-18], [34], [-80], [88], [102], [34], [88], [-10], [-58], [36], [-40], [-48], [40], [48], [8], [-58], [96], [-76], [-10], [-24], [-24], [-14], [-76], [-40]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4851_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4851_2_a_o();".
// 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_4851_2_a_o(:prec:=1)
    chi := MakeCharacter_4851_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_4851_2_a_o();".
// 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_4851_2_a_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4851_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>,<5,R![4, 1]>,<13,R![-2, 1]>,<17,R![-2, 1]>],Snew);
    return Vf;
end function;