
// Make newform 4900.2.a.s in Magma, downloaded from the LMFDB on 05 March 2026.
// 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_s();".
// 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_s();".
// 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 := 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]];
    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], [2], [0], [0], [-1], [2], [-4], [0], [5], [-3], [10], [5], [10], [-5], [4], [10], [10], [-10], [5], [3], [10], [13], [10], [-10], [-6], [10], [-8], [0], [13], [-5], [-15], [0], [10], [-10], [11], [-9], [-10], [20], [-6], [12], [4], [-10], [-24], [15], [15], [10], [-4], [-2], [24], [-20], [25], [-8], [-20], [-20], [-30], [15], [-10], [10], [-10], [29], [-32], [-18], [4], [-20], [28], [15], [-7], [10], [35], [-20], [-18], [19], [26], [15], [29], [-10], [-3], [-16], [1], [-20], [-30], [-1], [24], [20], [10], [-20], [-33], [15], [-30], [-20], [-4], [10], [-5], [-21], [-8], [12], [-30], [0], [20], [11], [-35], [15], [0], [1], [-3], [16], [-10], [28], [-43], [40], [-34], [-35], [45], [30], [1], [33], [12], [-20], [-10], [8], [40], [10], [44], [15], [-10], [34], [-42], [-30], [-40], [-48], [3], [0], [4], [25], [0], [10], [42], [-6], [-4], [-13], [20], [-6], [-55], [35], [20], [-30], [20], [-30], [-40], [-45], [30], [50], [25], [10], [-20], [13], [17], [30], [-54], [50], [-60], [35], [-40], [-20], [45], [-22], [33], [-36], [7], [-20], [-51], [-40], [9], [-30], [10], [0], [-31], [14], [-10], [10], [-5], [0], [25], [-60], [-55], [60], [-25], [28], [-3], [-30], [0], [45], [-9], [0], [55], [18], [-11], [-35], [-44], [42], [-54], [-40], [50], [10], [30], [54], [-20], [-60], [-41], [-50], [25], [-60], [-65], [-64], [-70], [40], [-25], [-40], [-15], [5], [57], [-40], [3], [25], [-14], [46], [-42], [-63], [-64], [27], [-15], [-70], [-28], [-1], [-22], [46], [10], [15], [-33], [-40], [45], [10], [42], [-61], [22], [20], [60], [-40], [28], [0], [-30], [30], [40], [-20], [-20], [28], [-3], [60], [16], [-24], [0], [55], [50], [-42], [10], [0], [-77], [30], [15], [45], [-40], [-20], [-28], [-19], [-20], [-12], [-25], [34], [-41], [40], [-18], [-36], [70], [20], [14], [-41], [-5], [55], [-80], [20], [31], [-66], [25], [10], [-65], [0], [30], [-18], [-50], [-64], [58], [-10], [-7], [75], [-31], [-70], [45], [30], [9], [50], [-50], [-2], [-7], [60], [-15], [20], [-60], [56], [62], [13], [-20], [-10], [60], [-55], [55], [10], [77], [-80], [-35], [-35], [-22], [65], [80], [72], [-52], [36], [57], [30], [70], [34], [10], [45], [10], [69], [65], [-9], [50], [35], [60], [34], [-30], [-65], [66], [-92], [67], [0], [50], [30], [30], [-20], [-45], [-30], [60], [-73], [-40], [-10], [-36], [-5], [-89], [-81], [30], [25], [-78], [10], [75], [40], [11], [88], [-20], [26], [50], [-5], [5], [95], [-10], [72], [11], [34], [35], [-6], [-47], [-2], [23], [-20], [-9], [-15], [40], [70], [44], [17], [30], [75], [65], [26], [-80], [90], [75], [14], [-10], [30], [-58], [5], [60], [73], [5], [-20], [92], [-30], [-40], [-82], [-18], [-50], [20], [60], [-18], [-106], [-25], [74], [-20], [-60]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4900_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4900_2_a_s();".
// 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_s(: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_s();".
// 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_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4900_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 1]>,<11,R![1, 1]>,<13,R![-2, 1]>,<19,R![0, 1]>,<23,R![-5, 1]>],Snew);
    return Vf;
end function;