
// Make newform 1850.2.a.v in Magma, downloaded from the LMFDB on 24 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_v();".
// 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_v();".
// 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
        poly := [-6, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
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], [0, 0], [2, -1], [0, -2], [4, 0], [0, 2], [6, 1], [-4, 2], [0, 0], [-4, -1], [1, 0], [2, 0], [4, 0], [2, -1], [-2, 4], [-6, -1], [12, 0], [-8, -1], [0, -2], [4, 0], [4, -1], [-12, -1], [6, -4], [-2, 0], [-4, -6], [0, -4], [-8, -1], [4, 4], [-8, -2], [-10, -3], [2, 5], [0, -8], [4, 4], [-4, 6], [-8, 0], [2, 4], [-12, 4], [8, 0], [10, -4], [-2, 3], [-4, -6], [4, -3], [14, 0], [-2, 0], [4, -1], [8, 6], [-2, -1], [-20, 2], [-10, 0], [0, -4], [4, 9], [-2, -4], [14, -3], [0, 2], [14, 5], [-4, 6], [-8, 10], [0, -8], [18, -4], [-4, -8], [6, -8], [-4, -7], [0, 3], [2, 8], [18, 0], [2, -5], [16, 8], [12, 4], [12, 2], [-6, 0], [-8, 2], [-10, -3], [-2, 4], [-12, 8], [-20, 6], [-16, 4], [-26, -4], [-14, 4], [-10, 0], [-4, -4], [0, 8], [-28, -5], [-8, -12], [28, 3], [32, 3], [-6, 4], [-2, 0], [-24, 4], [4, 10], [-12, 0], [-12, -7], [0, 12], [-24, -6], [-18, -3], [8, 4], [30, 0], [8, -4], [-4, 2], [20, 8], [4, 6], [-12, 0], [-20, 6], [2, -8], [4, 12], [-16, 6], [-12, 0], [24, 0], [-8, -8], [-8, 0], [12, -6], [30, -4], [0, -8], [-8, 2], [-8, 5], [8, -4], [-28, -2], [-32, 0], [34, 0], [12, -8], [8, 16], [24, 0], [18, 0], [-4, -8], [28, -4], [-4, 4], [-4, -4], [32, -8], [-4, -2], [2, 8], [-36, 4], [-26, -5], [16, -6], [-14, 8], [-14, -16], [-26, 4], [-6, 0], [8, 15], [-24, 8], [6, 16], [36, 4], [-26, -8], [26, -3], [-12, 10], [28, 8], [24, 4], [-18, 8], [-10, -8], [-38, 7], [-6, 15], [-14, -12], [24, 12], [28, -6], [2, 9], [-4, -2], [0, 13], [8, -7], [18, 8], [8, 0], [-4, 14], [-4, -12], [-12, 4], [28, 10], [-40, -2], [-30, -8], [-22, -11], [0, 13], [-22, 0], [8, 8], [10, 16], [18, 13], [-4, -16], [20, -7], [36, -8], [24, -16], [-18, 12], [-10, -17], [-12, 6], [10, -9], [30, 0], [-24, 8], [6, -11], [0, -4], [10, -8], [6, 7], [18, 8], [2, 24], [20, -4], [-30, 0], [24, -8], [-24, 2], [-12, -1], [-2, -9], [-10, 8], [-40, -3], [-12, 4], [10, 8], [-24, -8], [-4, -12], [32, 8], [-8, 12], [-8, -10], [30, -8], [28, -12], [-4, -4], [40, -8], [-16, 14], [-16, -5], [-14, -4], [-8, 10], [8, -10], [-12, 16], [-16, -20], [-4, -7], [8, 3], [20, 8], [16, -12], [26, 4], [4, 6], [14, 0], [60, -2], [16, -4], [-28, -8], [0, -4], [-28, -1], [20, 0], [60, -4], [-24, 16], [-42, 5], [6, 19], [10, 16], [16, 6], [32, 10], [-10, 8], [12, -17], [-30, 17], [-42, 8], [-48, 8], [-10, 5], [16, 19], [12, -22], [-26, 7], [34, 15], [-8, -8], [40, -4], [0, -5], [-20, 2], [50, -7], [-12, 8], [-40, -4], [68, 0], [16, 4], [10, -23], [22, 12], [-18, 8], [0, -10], [-34, -16], [60, 7], [6, 8], [14, 16], [-4, 22], [12, -16], [-18, -8], [2, -12], [40, 2], [-20, -20], [4, -6], [-22, -4], [-20, 21], [40, 16], [-28, 20], [-44, 8], [8, 14], [24, -11], [16, -12], [-6, 25], [-12, -5], [56, -4], [30, 8], [-58, 5], [-46, -15], [-8, -15], [26, -17], [18, -24], [52, 4], [40, -2], [-40, -14], [-22, 20], [-56, 9], [70, 0], [16, 4], [-28, 4], [-28, -12], [-36, 12], [34, 0], [-42, 8], [16, 4], [-54, -8], [22, -3], [40, -3], [34, 0], [-38, 4], [8, -12], [-28, -2], [-58, 5], [-8, -26], [-36, 6], [40, 0], [40, -10], [4, -20], [-6, -25], [36, 6], [24, -8], [-28, 3], [-48, -16], [42, 12], [-72, -2], [-48, 5], [-20, 16], [-18, -8], [14, 17], [-24, -12], [48, -4], [54, 5], [0, -24], [18, -4], [32, 2], [-28, -12], [24, 16], [42, -12], [10, -32], [10, -8], [4, -11], [-36, -10], [50, -7], [12, -1], [-2, 8], [-4, -28], [22, 20], [-30, -3], [30, -4], [-36, 16], [-12, -12], [-28, -5], [10, -24], [78, 3], [-18, 0], [40, 7], [-28, -5], [6, -32], [28, -24], [-56, -4], [-44, 4], [-20, -14], [-4, 6], [40, 16], [-12, 13], [22, 15], [60, 12], [-36, 10], [-8, 4], [-10, 8], [36, 18], [6, 21], [8, 5], [22, 8], [48, -20], [28, -6], [30, 8], [-18, 5], [38, 3], [-30, 21], [-60, -10], [24, -3], [-10, -8], [72, 2], [8, -24], [-4, 32], [42, 12], [-4, -12], [0, 8], [48, -6], [20, 2], [40, 12], [6, 21], [18, 9], [-16, 32], [-30, -8], [20, -14], [16, -12], [70, 0], [6, -8], [58, -7], [-84, -2], [-24, 14], [20, -4], [-20, -15], [-30, 0], [-8, -10], [-18, 0], [-12, -2], [34, 0], [40, -8], [30, -8], [40, 0], [-8, -10], [-10, -8], [46, -16], [-44, 7], [-72, -2], [36, 12], [42, -16], [-4, 12], [-14, -11], [-42, 0], [44, 2], [24, -6], [70, -3], [-36, -24], [-44, -18], [36, -4], [34, -24], [6, -7], [-56, -6], [0, -4], [-14, 28], [60, 1], [54, -16], [16, -6], [36, 11]];
    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_v();".
// 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_v(:prec:=2)
    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_v();".
// 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_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1850_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-6, 0, 1]>,<7,R![-2, -4, 1]>],Snew);
    return Vf;
end function;