
// Make newform 1710.2.a.d in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1710_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_1710_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_1710_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_1710_a();"
function MakeCharacter_1710_a()
    N := 1710;
    order := 1;
    char_gens := [191, 1027, 1351];
    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_1710_a_Hecke(Kf)
    return MakeCharacter_1710_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], [0], [-1], [-1], [0], [-1], [3], [1], [-3], [3], [2], [-10], [-6], [2], [0], [-3], [-3], [8], [-7], [-12], [-13], [14], [-6], [-6], [-10], [-12], [8], [-15], [11], [12], [2], [0], [-9], [8], [12], [-10], [-10], [-22], [-18], [6], [0], [2], [27], [-22], [-6], [-19], [-25], [14], [15], [26], [6], [15], [-4], [0], [-18], [0], [-18], [-7], [8], [-18], [14], [-21], [20], [3], [-1], [-9], [5], [2], [30], [14], [33], [33], [-16], [-31], [35], [24], [24], [8], [-6], [-22], [30], [17], [-24], [-4], [38], [24], [30], [35], [-6], [-16], [-12], [24], [2], [-30], [-10], [-39], [-18], [0], [11], [20], [-4], [12], [12], [-24], [-22], [-7], [-30], [-30], [30], [-34], [-10], [-16], [-6], [44], [32], [12], [50], [33], [0], [-21], [-49], [20], [15], [12], [-10], [-48], [26], [-33], [-13], [-4], [2], [-18], [2], [2], [21], [-55], [-21], [5], [33], [-33], [-7], [12], [-31], [-33], [-13], [-36], [50], [-48], [-16], [12], [-31], [-42], [38], [-24], [35], [36], [-1], [-21], [47], [-45], [-24], [36], [44], [-12], [-6], [-54], [2], [-46], [20], [42], [-45], [-19], [-24], [-19], [-34], [42], [50], [18], [14], [14], [17], [51], [38], [36], [45], [-30], [-43], [-52], [-40], [-21], [-16], [48], [20], [57], [12], [-54], [41], [20], [27], [51], [-66], [-28], [11], [-22], [-6], [60], [-7], [39], [-9], [-1], [59], [0], [32], [-27], [-18], [20], [-49], [-42], [-6], [-6], [-22], [2], [0], [-25], [9], [-58], [-54], [0], [8], [-30], [68], [-25], [68], [30], [26], [-69], [-46], [6], [54], [-6], [36], [20], [5], [-19], [-6], [24], [-16], [24], [-16], [69], [32], [-15], [9], [-28], [54], [12], [38], [44], [30], [2], [-46], [-36], [-46], [11], [-30], [35], [45], [69], [-49], [0], [-31], [11], [-1], [23], [8], [-7], [24], [29], [32], [0], [66], [-67], [-75], [-31], [32], [-36], [-37], [42], [-49], [54], [36], [-72], [-36], [-84], [2], [18], [-46], [-18], [15], [-16], [-22], [3], [41], [-9], [-10], [-10], [-57], [59], [-48], [32], [-24], [-84], [48], [68], [12], [-70], [-42], [0], [-31], [-30], [77], [65], [15], [8], [-9], [-76], [-85], [83], [-78], [-6], [-52], [39], [17], [-42], [2], [-54], [-25], [36], [-73], [-7], [17], [51], [3], [-10], [-51], [48], [38], [-1], [-60], [96], [-4], [32], [12], [8], [86], [24], [27], [-60], [27], [30], [62], [78], [-12], [-27], [-52], [62], [-54], [-40], [-22], [66], [65], [-63], [15], [-16], [20], [15], [-48], [-70], [-33], [-46], [-42], [-6], [-43], [30], [-73], [-42], [47], [86], [-70], [12], [-22], [51], [66], [-88], [30], [14], [50], [12], [20], [30], [14], [-6], [-22], [30], [6], [-7], [-64], [48], [95], [-78], [50], [30], [75], [-52], [-13], [-90], [12], [-82], [63], [0], [-30], [-67], [3], [75], [-88], [-33], [-81], [33], [8], [-45]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1710_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1710_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_1710_2_a_d(:prec:=1)
    chi := MakeCharacter_1710_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_1710_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_1710_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1710_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1, 1]>,<11,R![0, 1]>,<13,R![1, 1]>,<53,R![3, 1]>],Snew);
    return Vf;
end function;