
// Make newform 1573.2.a.a in Magma, downloaded from the LMFDB on 11 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1573_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_1573_2_a_a();".
// 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_1573_2_a_a();".
// 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_1573_a();"
function MakeCharacter_1573_a()
    N := 1573;
    order := 1;
    char_gens := [365, 1211];
    v := [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_1573_a_Hecke(Kf)
    return MakeCharacter_1573_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], [-2], [2], [0], [1], [-7], [6], [-1], [3], [10], [6], [2], [3], [2], [-9], [8], [-15], [2], [-6], [-4], [-11], [0], [-18], [-10], [-5], [-7], [-3], [-4], [-6], [7], [-3], [8], [-19], [-8], [4], [-19], [10], [24], [-21], [9], [19], [8], [14], [-10], [3], [12], [-14], [-18], [-18], [-18], [-30], [8], [19], [-14], [-17], [-26], [10], [2], [12], [-16], [-6], [-2], [21], [17], [4], [-32], [-14], [-28], [0], [24], [14], [-8], [-10], [28], [-10], [-33], [-14], [8], [0], [-28], [4], [-12], [37], [29], [37], [2], [-30], [12], [34], [21], [-2], [-34], [11], [32], [11], [18], [21], [25], [38], [-35], [-10], [-25], [-18], [-23], [-42], [16], [-20], [-32], [19], [-35], [16], [-30], [-16], [10], [-21], [2], [41], [25], [-49], [-50], [-38], [-18], [-4], [50], [-2], [-4], [9], [48], [-36], [4], [-46], [-32], [-25], [30], [16], [12], [-22], [-34], [3], [-10], [-22], [-21], [-8], [-11], [-20], [16], [19], [39], [0], [12], [14], [-44], [-8], [-39], [57], [-13], [54], [37], [48], [24], [11], [50], [-7], [6], [6], [13], [-2], [52], [-42], [-56], [-20], [-29], [-44], [29], [-37], [-26], [2], [43], [11], [8], [-35], [3], [-18], [-42], [-9], [-39], [-6], [4], [46], [2], [-16], [3], [-36], [-12], [-39], [-32], [31], [-42], [1], [60], [27], [-42], [-34], [-28], [58], [0], [12], [30], [28], [-45], [51], [3], [4], [-12], [-40], [55], [33], [6], [-42], [5], [54], [24], [-46], [3], [-62], [-33], [-3], [-16], [-22], [34], [44], [32], [13], [-47], [38], [6], [24], [28], [16], [-2], [28], [-33], [-24], [56], [-48], [-28], [-10], [0], [29], [2], [58], [54], [-37], [-30], [62], [-47], [66], [15], [14], [-7], [-39], [-20], [-51], [66], [49], [54], [0], [16], [-30], [-43], [12], [46], [16], [-17], [26], [12], [44], [-34], [-3], [71], [8], [-23], [-2], [-4], [39], [-9], [-12], [-20], [35], [14], [29], [-54], [42], [-21], [37], [55], [19], [-60], [-38], [-69], [6], [-40], [-35], [32], [-74], [-12], [3], [-54], [33], [-15], [40], [-69], [65], [6], [55], [-16], [32], [28], [-83], [59], [22], [25], [-46], [-28], [-82], [0], [14], [-60], [-1], [-40], [-27], [-56], [-10], [0], [30], [34], [-34], [40], [7], [44], [-27], [-48], [-25], [62], [-76], [13], [-64], [78], [-42], [-92], [60], [-6], [-45], [-13], [-21], [54], [-14], [22], [14], [-26], [54], [72], [98], [-12], [61], [78], [-14], [-78], [69], [27], [-22], [-19], [-7], [10], [32], [-35], [-14], [42], [-99], [46], [84], [64], [58], [-22], [60], [6], [-7], [-80], [68], [-76], [-32], [-28], [-71], [96], [-31], [70], [27], [-11], [-62], [58], [-89], [60], [18], [44], [51], [42], [-58], [-82], [38], [-45], [-22], [-83], [40], [-29], [-69], [-20], [81], [32], [3], [17], [-88], [102], [80], [-26], [96], [69], [56], [58], [-55]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1573_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1573_2_a_a();".
// 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_1573_2_a_a(:prec:=1)
    chi := MakeCharacter_1573_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_1573_2_a_a();".
// 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_1573_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1573_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 1]>,<7,R![-2, 1]>],Snew);
    return Vf;
end function;