
// Make newform 4900.2.a.p in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4900_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_4900_2_a_p();".
// 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_4900_2_a_p();".
// 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_4900_a();"
function MakeCharacter_4900_a()
    N := 4900;
    order := 1;
    char_gens := [2451, 1177, 101];
    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_4900_a_Hecke(Kf)
    return MakeCharacter_4900_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 := [[0], [1], [0], [0], [3], [-1], [-3], [-2], [6], [-9], [-8], [10], [0], [-2], [-3], [0], [-12], [-8], [-8], [0], [14], [5], [-12], [-12], [17], [6], [-7], [6], [-19], [6], [-20], [18], [-12], [-2], [18], [-19], [-22], [-2], [9], [-3], [12], [-8], [3], [4], [-12], [-20], [5], [17], [-27], [16], [-24], [9], [10], [-6], [-6], [30], [-18], [16], [22], [-21], [11], [-15], [-13], [-18], [-13], [18], [20], [-2], [-30], [10], [-15], [0], [5], [4], [-4], [36], [9], [5], [-15], [-26], [24], [29], [15], [14], [10], [6], [15], [28], [0], [-8], [15], [-6], [10], [-15], [-25], [-33], [-18], [30], [20], [-25], [-8], [0], [-12], [-6], [-4], [47], [-24], [-33], [-9], [22], [-1], [-2], [-30], [10], [-25], [-30], [41], [48], [6], [-33], [-8], [-20], [15], [-24], [28], [3], [-1], [18], [32], [-13], [11], [-48], [53], [16], [42], [-2], [15], [5], [-3], [9], [34], [-15], [40], [42], [40], [-18], [-22], [6], [4], [-12], [22], [-48], [-20], [-48], [22], [0], [41], [24], [-7], [0], [-36], [-36], [34], [-36], [18], [-33], [-16], [17], [29], [-42], [24], [-14], [27], [-62], [46], [36], [-25], [9], [20], [16], [4], [-60], [10], [42], [36], [54], [-20], [-31], [-31], [-24], [-34], [6], [-7], [12], [60], [39], [-1], [64], [45], [33], [6], [22], [14], [10], [-30], [57], [34], [-12], [-42], [40], [-32], [-18], [64], [21], [-18], [-68], [22], [-24], [42], [-42], [-22], [-2], [-51], [34], [-3], [-10], [57], [-9], [41], [-36], [-68], [-38], [8], [9], [5], [57], [-44], [-24], [57], [-6], [0], [64], [5], [-34], [-9], [-30], [56], [-42], [77], [-42], [22], [42], [-48], [16], [-30], [-57], [-34], [-28], [63], [47], [-26], [6], [16], [11], [-66], [10], [-15], [-78], [40], [30], [-68], [-56], [17], [44], [-22], [29], [36], [2], [17], [6], [33], [77], [48], [-38], [-67], [-9], [-14], [-66], [-32], [0], [-39], [-33], [-12], [0], [22], [-60], [-62], [-9], [60], [-31], [-67], [-42], [-61], [-72], [65], [10], [78], [40], [24], [-68], [0], [-21], [78], [-50], [30], [4], [-90], [39], [5], [-75], [-38], [52], [-18], [-8], [18], [-8], [41], [-76], [-12], [24], [-34], [-48], [94], [51], [-76], [-63], [-43], [-18], [46], [-43], [10], [12], [-90], [35], [6], [-36], [82], [28], [-78], [-51], [-68], [-56], [-15], [-19], [17], [-30], [-30], [-18], [30], [-18], [-80], [-60], [-60], [-9], [-28], [46], [69], [-62], [-43], [15], [-56], [-36], [3], [-62], [-2], [-48], [-69], [-25], [-54], [-37], [-18], [6], [16], [-90], [76], [-39], [-49], [-49], [76], [51], [-31], [33], [21], [92], [-57], [40], [-20], [36], [11], [-78], [-68], [-24], [22], [-87], [-84], [-74], [10], [-57], [44], [12], [17], [18], [42], [-28], [22], [72], [39], [-100], [30], [-15], [-3], [-22], [66], [-60], [11], [45], [-60], [-30], [-20], [24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4900_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4900_2_a_p();".
// 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_4900_2_a_p(:prec:=1)
    chi := MakeCharacter_4900_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_4900_2_a_p();".
// 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_4900_2_a_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4900_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-1, 1]>,<11,R![-3, 1]>,<13,R![1, 1]>,<19,R![2, 1]>,<23,R![-6, 1]>],Snew);
    return Vf;
end function;