
// Make newform 1050.2.a.d in Magma, downloaded from the LMFDB on 03 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_d();".
// 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_d();".
// 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], [-6], [-4], [-6], [8], [6], [-2], [-4], [2], [-4], [-8], [-6], [-8], [-10], [-8], [-6], [14], [-12], [8], [-10], [10], [10], [-8], [-12], [-14], [-6], [-4], [-12], [6], [-14], [10], [-8], [22], [4], [12], [-8], [10], [2], [-18], [8], [-2], [6], [0], [8], [8], [14], [10], [26], [-26], [0], [0], [-16], [18], [10], [-28], [30], [-20], [0], [12], [24], [2], [2], [-8], [8], [-36], [-26], [-20], [6], [32], [24], [20], [-20], [2], [2], [-14], [26], [20], [34], [2], [26], [-18], [12], [-6], [-28], [-18], [-36], [24], [8], [-12], [-30], [-12], [12], [30], [-26], [28], [22], [-40], [42], [4], [-46], [4], [26], [12], [-12], [30], [38], [-8], [-28], [26], [-22], [-4], [14], [-12], [-36], [-34], [-26], [38], [-12], [-40], [-4], [-2], [-34], [-6], [-4], [-24], [-14], [-44], [48], [20], [32], [-6], [30], [0], [-28], [8], [30], [-14], [22], [-12], [20], [18], [-20], [-38], [-24], [-10], [48], [28], [30], [-24], [0], [-24], [6], [60], [-30], [-26], [-10], [12], [10], [-56], [-24], [18], [16], [4], [42], [-46], [0], [-6], [-6], [50], [-8], [-14], [42], [8], [-30], [-16], [26], [8], [16], [40], [-12], [40], [-42], [44], [4], [14], [-20], [-54], [-52], [-44], [-30], [-12], [12], [-30], [48], [24], [-20], [-18], [-18], [-42], [-50], [-16], [24], [18], [-52], [-2], [-50], [4], [54], [16], [0], [52], [-26], [-68], [-14], [-8], [-10], [42], [-50], [54], [-32], [12], [-54], [-40], [-26], [-8], [-22], [-32], [-38], [-16], [30], [-52], [-16], [42], [46], [-30], [60], [36], [2], [-16], [-50], [-40], [60], [0], [60], [8], [8], [-36], [2], [-8], [-62], [-56], [-54], [58], [-20], [16], [-14], [-12], [-28], [22], [-2], [-36], [38], [6], [-10], [8], [6], [-70], [28], [34], [76], [2], [64], [-20], [50], [-10], [48], [24], [44], [-8], [-14], [4], [-10], [12], [10], [-34], [50], [-10], [24], [-30], [-12], [-64], [-66], [-54], [12], [-64], [60], [-50], [-42], [-64], [-36], [-52], [32], [52], [-58], [82], [0], [24], [6], [-30], [-48], [32], [-10], [28], [-26], [50], [-30], [14], [4], [-30], [-16], [6], [62], [-84], [12], [32], [50], [46], [62], [26], [76], [48], [12], [70], [-48], [-26], [-16], [24], [82], [26], [32], [-58], [-18], [-42], [88], [-20], [48], [74], [36], [-18], [24], [-10], [-60], [60], [36], [-46], [-64], [8], [78], [-72], [-10], [-60], [20], [-36], [-24], [54], [-14], [-34], [32], [66], [-6], [-36], [20], [94], [18], [30], [-2], [-78], [58], [-20], [-50], [66], [-84], [36], [-62], [0], [32], [38], [36], [-66], [-20], [10], [-64], [74], [-26], [20], [78], [-14], [-94], [-32], [-24], [-66], [-22], [12], [18], [76], [-48], [-50], [-6], [12], [16], [32], [-46], [-70], [96], [-100], [-48], [-30], [-14], [-88], [-48], [10], [96], [4], [10], [30], [44]];
    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_d();".
// 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_d(: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_d();".
// 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_d( : 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![6, 1]>,<17,R![4, 1]>,<19,R![6, 1]>],Snew);
    return Vf;
end function;