
// Make newform 4900.2.a.v in Magma, downloaded from the LMFDB on 20 June 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_v();".
// 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_v();".
// 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], [3], [0], [0], [-2], [6], [-2], [0], [9], [3], [2], [-8], [5], [-1], [-8], [-4], [-8], [7], [3], [8], [-14], [4], [1], [13], [10], [-3], [-13], [15], [9], [4], [-16], [-4], [12], [10], [9], [-10], [2], [-8], [9], [16], [-6], [1], [6], [-2], [-14], [20], [4], [-16], [4], [-14], [18], [-26], [10], [30], [-8], [-17], [9], [-24], [-18], [-22], [4], [-4], [-1], [24], [4], [-10], [10], [-28], [-25], [-17], [36], [-10], [-1], [-32], [0], [9], [-10], [-10], [-29], [3], [-24], [-1], [6], [4], [4], [-37], [15], [-28], [-2], [-17], [-5], [-30], [-32], [6], [-6], [-27], [-41], [14], [-4], [33], [15], [2], [13], [30], [24], [-22], [-20], [36], [0], [-6], [-1], [6], [-20], [-10], [-34], [-31], [20], [-1], [34], [4], [31], [-16], [-6], [-29], [-26], [-29], [31], [-6], [-3], [-34], [10], [3], [20], [38], [30], [-2], [18], [11], [-36], [45], [50], [18], [-19], [-23], [-10], [14], [40], [54], [-36], [-37], [-32], [-31], [36], [-53], [37], [54], [32], [-5], [52], [38], [-29], [60], [-37], [-12], [6], [19], [-38], [40], [13], [-28], [20], [17], [-42], [24], [14], [13], [-32], [-42], [-47], [25], [1], [30], [-62], [22], [24], [51], [-52], [-56], [31], [-42], [-24], [56], [-26], [23], [-24], [-42], [37], [26], [12], [40], [-5], [38], [-36], [-51], [-6], [-32], [-4], [-36], [69], [8], [60], [-9], [-1], [-59], [42], [-46], [29], [39], [-40], [-36], [-13], [14], [-63], [-36], [49], [-19], [32], [-10], [-19], [62], [-58], [24], [-74], [-3], [-49], [-45], [31], [-62], [44], [34], [-52], [20], [24], [31], [12], [-30], [-5], [6], [-64], [-8], [56], [51], [63], [-21], [66], [-32], [-19], [-55], [-6], [-12], [51], [-25], [-38], [34], [-72], [74], [-18], [62], [-13], [36], [45], [-36], [52], [-60], [-50], [-8], [28], [23], [21], [-56], [17], [32], [-48], [-63], [13], [0], [20], [-12], [-58], [54], [75], [55], [14], [38], [84], [21], [-80], [-76], [-18], [28], [-42], [-32], [34], [39], [-58], [42], [-44], [63], [-30], [-2], [51], [54], [-41], [67], [9], [65], [0], [48], [-42], [-13], [-38], [50], [-25], [-15], [-6], [17], [-34], [-39], [80], [60], [-46], [72], [28], [-21], [-82], [63], [-62], [90], [-47], [76], [44], [-64], [57], [-64], [-6], [16], [-21], [11], [-66], [0], [-22], [-30], [75], [31], [87], [72], [30], [22], [30], [-63], [16], [62], [61], [64], [89], [-10], [14], [-4], [-23], [-12], [-14], [23], [-90], [-4], [62], [20], [-18], [-14], [66], [-36], [13], [74], [33], [-36], [78], [51], [90], [4], [55], [17], [98], [56], [-38], [60], [-2], [-40], [-44], [85], [36], [-69], [-27], [66], [-25], [48], [25], [64], [52], [-21], [-9], [-90], [-8], [-66], [-12], [40], [-50], [22], [42], [-48], [-42], [67], [85], [88], [3], [-70], [42], [74], [-44], [-21], [-68], [-46]];
    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_v();".
// 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_v(: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_v();".
// 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_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4900_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-3, 1]>,<11,R![2, 1]>,<13,R![-6, 1]>,<19,R![0, 1]>,<23,R![-9, 1]>],Snew);
    return Vf;
end function;