
// Make newform 9200.2.a.bv in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9200_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_9200_2_a_bv();".
// 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_9200_2_a_bv();".
// 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 := [-4, -1, 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_9200_a();"
function MakeCharacter_9200_a()
    N := 9200;
    order := 1;
    char_gens := [1151, 6901, 2577, 1201];
    v := [1, 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_9200_a_Hecke(Kf)
    return MakeCharacter_9200_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 := [[0, 0], [0, 1], [0, 0], [1, -1], [-2, 0], [2, -1], [-1, 1], [-6, 0], [1, 0], [3, -2], [-1, 4], [3, -1], [1, -2], [0, 0], [0, -3], [3, -1], [-1, -3], [-4, 2], [-7, 1], [-7, 2], [6, 1], [-8, -2], [-3, 7], [8, -4], [10, -2], [-13, 3], [-16, 0], [5, 1], [6, -6], [-5, 5], [0, -5], [-8, -5], [-10, -2], [5, -4], [0, 6], [0, -9], [9, 3], [6, -7], [-8, 0], [4, 6], [-4, -7], [4, 6], [-8, 6], [-4, -1], [14, -3], [10, 0], [-5, -3], [-2, 10], [20, -4], [-8, 12], [-4, -3], [-9, -4], [6, 0], [-2, -6], [-24, 5], [11, 7], [17, -4], [-11, 7], [2, -3], [-10, -2], [7, -5], [9, -5], [-18, 2], [-4, 3], [-7, -5], [-12, 2], [-13, 10], [2, 10], [4, -8], [-17, 6], [-12, -9], [0, 10], [-21, 1], [14, -4], [24, -8], [11, 5], [-14, -2], [-8, 17], [2, 0], [-11, 16], [22, 2], [22, -2], [4, -4], [-29, 7], [-12, 5], [-4, 5], [-17, 3], [-11, -1], [-30, 7], [-22, 6], [-27, -3], [12, -20], [0, -5], [1, 8], [-7, 10], [7, 7], [-10, 11], [2, 10], [4, 12], [2, 3], [-4, 11], [11, -3], [3, -19], [-2, -12], [-36, 0], [28, -1], [18, 7], [0, 10], [8, -16], [27, -6], [-4, -8], [-10, -10], [1, 1], [0, 8], [-24, 4], [10, 2], [-3, -15], [10, 3], [-10, 19], [-6, 6], [6, 8], [-8, 17], [-19, 5], [28, -9], [-4, 8], [-8, -4], [36, 4], [-5, 9], [-29, -5], [-5, -3], [15, 10], [16, 8], [10, -2], [-11, 1], [-19, 16], [-20, -6], [38, -2], [-35, 3], [29, -15], [-37, 11], [-41, 4], [-22, 8], [-24, 9], [15, -3], [-3, 17], [46, 2], [14, -4], [38, 3], [27, 8], [-6, 1], [-6, -16], [4, -24], [-8, -12], [10, -21], [17, -3], [0, -2], [-28, 12], [-13, 28], [10, -4], [-44, -4], [52, -3], [26, -8], [-52, 3], [32, 6], [-27, 21], [-23, 9], [-33, 13], [-40, 10], [-40, 10], [20, -11], [-44, 10], [-39, -4], [22, 2], [-37, -7], [4, -11], [-4, -12], [-16, -11], [-33, 12], [23, 7], [10, -10], [34, 9], [16, 12], [30, -7], [-14, -20], [11, 15], [10, 2], [6, -20], [7, -9], [-31, -4], [1, 3], [14, 11], [26, 10], [6, -12], [21, 1], [16, 6], [37, -13], [30, 4], [-50, 2], [54, -6], [-22, 1], [16, -12], [-4, 20], [-18, 21], [2, -8], [-24, 12], [-34, -3], [-40, 10], [-14, 3], [17, -26], [8, -15], [22, -21], [2, -23], [-16, 0], [-36, 6], [31, -11], [4, 20], [-14, -1], [17, -20], [-5, 15], [22, -25], [33, -2], [-66, 4], [19, 2], [27, 5], [46, 7], [-21, 20], [-50, 12], [-17, -14], [28, -18], [9, 15], [30, 16], [-10, 6], [30, -8], [31, -9], [-23, -1], [12, -12], [16, -24], [-22, 0], [55, 0], [25, -20], [21, 7], [-17, -8], [-62, -1], [-22, -15], [-4, 14], [24, -17], [36, -7], [34, -12], [30, -16], [9, -11], [14, -26], [-2, 0], [11, -1], [-38, -2], [-42, -8], [-65, 2], [28, 14], [-24, -6], [-6, -11], [6, -11], [-21, -19], [39, -7], [-13, 11], [5, 11], [-20, 14], [-4, -10], [40, -14], [38, 14], [-9, -13], [-12, 23], [-43, 16], [-23, 17], [-53, 17], [50, -6], [8, 3], [18, -5], [16, -4], [29, -31], [0, -22], [-12, 28], [2, -11], [12, -6], [-40, -2], [60, -14], [30, -14], [37, -1], [27, -17], [-46, 0], [32, 0], [2, -20], [6, 12], [19, -19], [16, 6], [-56, 10], [-48, -5], [20, 6], [18, -22], [-14, 15], [-47, 20], [-34, 13], [-43, 23], [43, -1], [-24, 12], [48, 7], [-4, 24], [64, -11], [14, -10], [26, -14], [-48, -10], [4, -14], [12, 13], [18, -4], [-20, 28], [72, -1], [47, 3], [-28, -16], [-5, 22], [7, -4], [27, 9], [-6, 17], [24, 0], [-15, -13], [38, -7], [-34, 10], [-37, -5], [-4, -12], [-28, 2], [-2, 25], [16, 22], [-23, 3], [55, -18], [-6, 3], [17, -4], [-14, 13], [-4, 14], [-20, -5], [-58, 4], [26, -30], [14, -19], [81, -1], [-42, 15], [-62, 6], [26, -4], [48, -6], [69, 1], [-3, -21], [-7, 26], [66, -6], [-46, -8], [-27, 13], [29, 10], [-4, 3], [-17, -21], [63, 7], [-44, -12], [-38, -12], [38, 6], [18, 24], [50, -3], [46, -18], [3, 35], [29, -16], [38, 21], [76, 6], [62, -15], [-2, 13], [64, 5], [37, 1], [-4, -10], [-35, -16], [-61, 5], [34, -33], [32, 4], [-48, -8], [30, -5], [23, 14], [-2, 0], [-73, -1], [-56, 28], [-2, -25], [6, 28], [-66, 12], [-72, 12], [-18, 15], [-4, -16], [-30, 5], [-21, -30], [52, 6], [25, -3], [-79, 11], [-14, 18], [22, 4], [37, -30], [0, 20], [2, -18], [15, -1], [-62, 2], [-16, -20], [4, 0], [-39, 25], [-13, -16], [-34, 33], [-13, -15], [25, 7], [-29, -4], [69, 2], [21, 9], [54, 19], [-28, -8], [4, 8], [8, -19], [-46, 7], [4, -38], [72, -14], [29, -3], [-2, -25], [-24, 8], [16, 28], [-17, 21], [28, -48], [-18, -6], [38, -22], [30, -30], [44, 0], [24, -1], [4, 1], [52, -10], [95, -5], [24, -1], [29, 10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9200_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9200_2_a_bv();".
// 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_9200_2_a_bv(:prec:=2)
    chi := MakeCharacter_9200_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_9200_2_a_bv();".
// 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_9200_2_a_bv( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9200_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-4, -1, 1]>,<7,R![-4, -1, 1]>,<11,R![2, 1]>],Snew);
    return Vf;
end function;