
// Make newform 1850.2.a.l in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1850_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_1850_2_a_l();".
// 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_1850_2_a_l();".
// 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_1850_a();"
function MakeCharacter_1850_a()
    N := 1850;
    order := 1;
    char_gens := [1777, 1001];
    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_1850_a_Hecke(Kf)
    return MakeCharacter_1850_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], [0], [1], [-3], [-4], [-3], [0], [-8], [-3], [-7], [-1], [11], [11], [4], [11], [-12], [-15], [-4], [6], [2], [-8], [12], [0], [-1], [14], [0], [-10], [-11], [-1], [16], [-10], [6], [1], [-4], [-14], [-5], [3], [-14], [-13], [10], [-16], [-17], [-14], [2], [16], [-1], [11], [17], [20], [6], [-17], [-14], [20], [6], [31], [-16], [22], [0], [6], [-8], [-15], [-8], [3], [-14], [-21], [2], [-28], [12], [-6], [-3], [-14], [13], [38], [12], [-16], [23], [14], [10], [-4], [20], [-30], [23], [-16], [7], [34], [36], [-37], [3], [-4], [3], [-36], [36], [-36], [-18], [-26], [30], [-7], [4], [26], [35], [-6], [-19], [14], [7], [-2], [15], [-36], [-32], [-47], [-42], [-15], [-42], [-35], [37], [-25], [7], [2], [38], [48], [-25], [0], [-42], [13], [-41], [-22], [29], [20], [-14], [1], [-15], [-13], [-32], [-28], [7], [-44], [-51], [22], [12], [-12], [0], [-14], [40], [-3], [-23], [-12], [-30], [-17], [-20], [-39], [-1], [39], [29], [7], [28], [36], [-16], [-15], [28], [26], [37], [6], [-34], [-1], [33], [-11], [-45], [-44], [5], [-22], [-12], [5], [11], [36], [24], [12], [50], [-12], [-7], [-30], [-24], [18], [24], [47], [-15], [-12], [-47], [-5], [42], [-66], [-54], [6], [10], [14], [4], [30], [28], [-54], [-26], [52], [46], [52], [-57], [-11], [-37], [62], [50], [64], [-20], [-5], [-41], [-63], [-56], [-68], [17], [-13], [-22], [26], [-40], [1], [18], [-32], [-1], [12], [70], [26], [36], [-12], [21], [-48], [59], [-20], [38], [38], [42], [-45], [-66], [72], [-16], [-56], [3], [46], [-64], [31], [44], [63], [44], [-10], [39], [-20], [22], [-2], [-67], [70], [12], [51], [-4], [78], [-57], [-11], [-2], [0], [18], [-41], [38], [-17], [-74], [-40], [-34], [8], [-58], [-19], [-38], [69], [-22], [-9], [24], [-22], [-3], [-52], [64], [-5], [31], [-18], [-16], [40], [19], [31], [-8], [34], [31], [76], [-20], [69], [-13], [-6], [-56], [-54], [76], [-46], [47], [-10], [38], [76], [50], [-7], [57], [-50], [-68], [14], [-24], [66], [-45], [-32], [-18], [-36], [39], [-24], [56], [-60], [-58], [-24], [3], [-45], [-60], [-42], [-13], [1], [-18], [33], [-20], [20], [-59], [84], [14], [-30], [76], [-14], [-38], [-72], [78], [48], [-6], [-67], [-46], [36], [-48], [80], [59], [63], [55], [92], [13], [-46], [-16], [-28], [-27], [-44], [30], [-48], [38], [-22], [-18], [-6], [4], [77], [-42], [80], [30], [66], [-34], [3], [-18], [-57], [67], [-96], [-58], [76], [-36], [-44], [81], [-6], [10], [38], [6], [15], [-40], [12], [-71], [50], [-41], [42], [58], [-63], [-12], [10], [40], [21], [43], [48], [-84], [-61], [-52], [42], [22], [-20], [-2], [-23], [-70], [-3], [-36], [33], [-41], [46], [-51], [-70], [-54], [-49], [30], [-76], [54], [38], [-96], [-35], [18], [-13], [24], [-99], [4], [51]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1850_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1850_2_a_l();".
// 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_1850_2_a_l(:prec:=1)
    chi := MakeCharacter_1850_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_1850_2_a_l();".
// 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_1850_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1850_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<7,R![-1, 1]>],Snew);
    return Vf;
end function;