
// Make newform 1050.2.a.n in Magma, downloaded from the LMFDB on 08 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1050_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_1050_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_1050_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_1050_a();"
function MakeCharacter_1050_a()
    N := 1050;
    order := 1;
    char_gens := [701, 127, 451];
    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_1050_a_Hecke( : Kf := Kf)
    return MakeCharacter_1050_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], [-1], [0], [1], [-2], [7], [-7], [8], [-5], [9], [1], [-2], [11], [3], [4], [3], [7], [-5], [-12], [-4], [10], [-6], [-9], [-10], [10], [0], [5], [-2], [4], [-14], [-18], [0], [16], [8], [-3], [-22], [-10], [-15], [2], [-16], [-2], [2], [-1], [10], [-21], [-16], [-13], [9], [-3], [-14], [16], [8], [-30], [-5], [5], [-9], [-6], [12], [2], [-8], [-16], [24], [-22], [2], [10], [23], [25], [-13], [-6], [-7], [30], [19], [-17], [-4], [7], [0], [6], [37], [-12], [10], [-3], [10], [39], [6], [-19], [-4], [-20], [7], [8], [8], [-29], [-12], [6], [-42], [-23], [-26], [-28], [9], [8], [8], [-35], [-14], [27], [18], [31], [38], [39], [18], [-1], [8], [-36], [2], [-38], [-38], [42], [-30], [10], [46], [-34], [34], [-10], [-13], [0], [38], [-14], [-15], [-8], [30], [21], [45], [15], [-39], [4], [26], [-34], [-32], [28], [26], [-30], [-44], [-6], [34], [-14], [-6], [-19], [-36], [-35], [-10], [-16], [48], [32], [-21], [-23], [-14], [-39], [-39], [16], [7], [-6], [24], [12], [-48], [16], [-36], [-18], [32], [-20], [2], [23], [-36], [22], [-43], [-25], [-1], [-52], [7], [-28], [27], [5], [-21], [22], [43], [34], [-13], [16], [-34], [-20], [-10], [-31], [20], [-16], [-40], [-32], [34], [12], [-21], [5], [10], [-18], [36], [6], [63], [-7], [18], [59], [-56], [21], [-18], [-50], [-42], [46], [-30], [46], [-57], [6], [16], [44], [50], [-67], [19], [-64], [21], [30], [28], [-28], [-26], [26], [-24], [28], [42], [4], [56], [-48], [-6], [56], [56], [34], [14], [34], [62], [-60], [-38], [39], [-10], [53], [64], [-67], [12], [4], [48], [-56], [-57], [-51], [26], [24], [42], [38], [18], [30], [2], [52], [28], [-59], [-2], [-35], [66], [81], [3], [23], [39], [1], [37], [36], [-24], [-60], [44], [40], [-10], [7], [-5], [-48], [-40], [32], [19], [20], [63], [-13], [14], [31], [30], [70], [52], [-28], [-24], [-18], [-68], [-49], [-18], [-41], [-62], [-52], [-77], [50], [-4], [-52], [-19], [78], [-15], [-63], [-70], [54], [77], [-78], [0], [87], [18], [36], [67], [52], [60], [-26], [54], [54], [-20], [-10], [-8], [-77], [18], [48], [87], [-18], [15], [-1], [49], [-73], [-36], [-92], [-39], [50], [1], [-38], [-26], [-64], [40], [-59], [-56], [-78], [-71], [-16], [22], [76], [-53], [-61], [31], [78], [-73], [90], [-55], [-42], [-41], [-12], [-70], [72], [8], [-6], [1], [26], [36], [1], [-44], [0], [59], [-13], [41], [8], [-5], [-56], [20], [-47], [-24], [70], [-36], [92], [-28], [-16], [32], [-10], [-2], [4], [-42], [55], [56], [-10], [-98], [-65], [58], [-69], [17], [-52], [42], [67], [-70], [-68], [75], [72], [31], [42], [10], [18], [40], [24], [10], [48], [67], [-45], [28], [-57], [-77], [93], [22], [-17], [3], [-87], [-3], [-16], [60], [-22], [-18], [-32], [-6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1050_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1050_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_1050_2_a_n(:prec:=1)
    chi := MakeCharacter_1050_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_1050_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_1050_2_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1050_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![2, 1]>,<13,R![-7, 1]>,<17,R![7, 1]>,<19,R![-8, 1]>],Snew);
    return Vf;
end function;