
// Make newform 1050.2.a.i in Magma, downloaded from the LMFDB on 18 April 2024.
// 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_i();".
// 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_i();".
// 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)
    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+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 := [[-1], [1], [0], [1], [-4], [-6], [-2], [-4], [-8], [-2], [0], [10], [-6], [4], [0], [-6], [4], [6], [-4], [8], [-10], [0], [4], [-6], [14], [-2], [-8], [-12], [-2], [14], [0], [-20], [-10], [4], [6], [-8], [10], [-20], [8], [-22], [-12], [-18], [0], [-2], [10], [8], [20], [16], [-12], [-2], [22], [0], [2], [-12], [30], [24], [22], [0], [10], [26], [-4], [-30], [-28], [-8], [-10], [18], [-4], [-18], [-12], [22], [30], [-8], [-32], [-22], [-20], [16], [-26], [-6], [18], [-22], [-36], [6], [0], [-2], [-24], [4], [34], [-10], [22], [32], [-28], [-16], [-8], [12], [-44], [-24], [6], [-6], [-20], [30], [12], [2], [-44], [-6], [12], [-34], [28], [-18], [24], [26], [-48], [42], [22], [-44], [8], [2], [4], [-24], [18], [-28], [-2], [-2], [18], [-12], [-4], [-2], [-10], [0], [8], [-6], [-12], [-24], [48], [-6], [-22], [-14], [2], [36], [-6], [10], [-44], [38], [56], [36], [-26], [56], [-14], [-42], [20], [32], [2], [18], [-20], [24], [-12], [0], [40], [18], [22], [-26], [-4], [-26], [-8], [-12], [-18], [24], [-16], [-14], [50], [-30], [28], [22], [-24], [54], [-48], [-38], [-36], [38], [40], [-10], [0], [-4], [-22], [-10], [-64], [30], [-30], [-44], [-54], [-48], [-34], [-12], [-44], [38], [60], [22], [50], [50], [30], [24], [-66], [48], [34], [-30], [-12], [42], [32], [12], [-54], [-28], [-50], [-50], [-24], [-4], [-24], [26], [32], [2], [-56], [-62], [22], [40], [-30], [-16], [-60], [38], [70], [16], [72], [60], [-30], [-20], [16], [-54], [76], [48], [-62], [-6], [-36], [40], [-68], [-28], [-56], [-18], [-50], [-24], [48], [-52], [-4], [-48], [2], [66], [24], [10], [-38], [20], [-58], [12], [2], [-74], [-64], [44], [46], [26], [78], [28], [30], [-38], [-28], [-6], [6], [28], [6], [-64], [14], [24], [52], [-66], [10], [-36], [0], [24], [-72], [-82], [12], [-64], [-18], [-38], [56], [-30], [30], [-60], [6], [-44], [-14], [6], [80], [-30], [-12], [36], [38], [-78], [64], [-36], [-52], [-2], [-28], [-10], [8], [58], [48], [22], [34], [76], [-40], [-22], [12], [64], [-2], [18], [44], [-58], [54], [-16], [-10], [2], [20], [-4], [-80], [58], [14], [34], [0], [4], [76], [92], [78], [-34], [74], [-16], [10], [6], [-18], [-72], [-30], [-12], [-34], [4], [-48], [18], [92], [-74], [30], [-48], [22], [-26], [16], [-12], [46], [-8], [26], [-22], [32], [-4], [20], [22], [-54], [24], [-38], [-28], [4], [-48], [70], [40], [2], [60], [-16], [-34], [2], [-74], [-90], [86], [-40], [30], [28], [56], [32], [-94], [68], [16], [2], [66], [44], [52], [40], [6], [-32], [26], [12], [-10], [-58], [30], [32], [86], [-18], [-88], [50], [18], [-44], [-4], [-82], [10], [-44], [36], [22], [86], [32], [-40], [30], [8], [-34], [-14], [16], [84], [-26], [-70], [-100], [-102], [-12], [-88]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1050_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1050_2_a_i();".
// 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_i(: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_i();".
// 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_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1050_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![4, 1]>,<13,R![6, 1]>,<17,R![2, 1]>,<19,R![4, 1]>],Snew);
    return Vf;
end function;