
// Make newform 4900.2.a.n in Magma, downloaded from the LMFDB on 19 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_n();".
// 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_n();".
// 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], [2], [3], [1], [-3], [-6], [7], [1], [-6], [4], [-9], [-3], [-9], [1], [7], [0], [-1], [-13], [12], [-15], [-10], [-15], [11], [-15], [-1], [-6], [-8], [-3], [21], [-20], [3], [17], [-13], [-11], [-12], [-9], [21], [10], [-9], [-11], [-18], [7], [-4], [8], [-3], [-11], [21], [-12], [1], [0], [3], [3], [-3], [-11], [13], [30], [29], [6], [-28], [27], [23], [9], [-13], [34], [-9], [-26], [-21], [15], [5], [25], [8], [-33], [15], [-37], [3], [-11], [12], [-22], [-15], [-10], [1], [9], [-18], [-23], [6], [16], [-21], [3], [19], [24], [11], [0], [-3], [-39], [-1], [35], [-8], [33], [9], [-9], [29], [-1], [12], [-21], [-27], [-14], [47], [25], [-6], [31], [-16], [15], [20], [21], [-39], [12], [-11], [-14], [27], [-21], [13], [-18], [-1], [21], [32], [-25], [-19], [48], [-25], [-2], [-3], [34], [-33], [-31], [42], [-33], [52], [27], [-5], [36], [1], [24], [-22], [15], [-23], [-51], [13], [-18], [-44], [39], [-41], [48], [-1], [9], [26], [-27], [45], [18], [16], [-51], [-27], [33], [-19], [59], [-10], [-33], [-39], [-50], [3], [-23], [7], [-6], [32], [3], [-16], [13], [13], [-12], [-38], [15], [27], [21], [-47], [5], [11], [15], [-1], [60], [47], [-27], [-21], [-45], [11], [37], [30], [-21], [-9], [40], [23], [-35], [-36], [3], [61], [-39], [-6], [-41], [37], [30], [-8], [27], [63], [13], [31], [-51], [57], [66], [-61], [-20], [39], [25], [60], [26], [-45], [21], [23], [-45], [61], [-65], [-16], [27], [-4], [-3], [-11], [21], [60], [36], [39], [1], [29], [47], [-30], [45], [8], [-45], [5], [36], [-2], [-15], [-9], [-14], [-21], [-57], [-37], [-67], [-54], [-61], [-5], [60], [25], [26], [-45], [-23], [18], [42], [4], [-63], [61], [-29], [59], [47], [-58], [11], [-3], [-13], [71], [-63], [-63], [65], [-24], [-50], [-49], [15], [1], [30], [67], [-54], [-57], [21], [33], [-24], [-50], [-63], [-23], [-6], [-51], [-40], [23], [-51], [65], [-60], [-25], [-62], [-33], [70], [39], [-35], [-45], [75], [-45], [31], [-72], [-35], [-12], [-63], [14], [6], [67], [-11], [-90], [28], [-87], [-71], [-37], [-13], [-27], [54], [23], [9], [40], [-39], [-19], [-36], [2], [3], [70], [-37], [-23], [-30], [-42], [32], [-63], [-36], [37], [-71], [-12], [-57], [85], [-11], [42], [-19], [-25], [90], [-3], [-21], [-3], [24], [46], [-27], [87], [27], [-55], [-35], [-78], [-89], [-70], [33], [-71], [-39], [-42], [-65], [73], [27], [0], [11], [-87], [14], [-87], [-6], [-56], [57], [-68], [57], [5], [-85], [37], [96], [50], [27], [-87], [-85], [-9], [49], [-29], [42], [-52], [-9], [-23], [-51], [-95], [-33], [-39], [-35], [37], [30], [29], [-99], [71], [9], [36], [-73], [-38], [-15], [15], [41], [-18], [-105], [51], [-1], [-48], [12], [38], [27], [81], [-18], [-65], [-33]];
    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_n();".
// 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_n(: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_n();".
// 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_n( : 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![-2, 1]>,<19,R![-1, 1]>,<23,R![3, 1]>],Snew);
    return Vf;
end function;