
// Make newform 3200.2.a.z in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3200_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_3200_2_a_z();".
// 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_3200_2_a_z();".
// 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_3200_a();"
function MakeCharacter_3200_a()
    N := 3200;
    order := 1;
    char_gens := [1151, 901, 2177];
    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_3200_a_Hecke(Kf)
    return MakeCharacter_3200_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], [3], [0], [-4], [3], [2], [-3], [7], [-6], [6], [10], [0], [-11], [4], [2], [14], [4], [8], [5], [-2], [-9], [-12], [1], [3], [2], [-4], [6], [-13], [-4], [3], [6], [4], [5], [15], [2], [16], [-2], [13], [10], [-12], [-1], [-22], [6], [-23], [-14], [4], [13], [-20], [-8], [-16], [-2], [-26], [13], [-3], [-18], [22], [0], [2], [-26], [-18], [-11], [16], [-13], [-32], [10], [24], [-1], [-1], [23], [0], [-18], [-16], [-8], [10], [-7], [0], [10], [4], [-35], [25], [31], [-24], [0], [25], [-26], [9], [21], [-39], [-40], [32], [-36], [10], [-10], [0], [28], [-20], [14], [-27], [11], [-4], [-17], [22], [12], [-39], [28], [-17], [29], [7], [12], [-5], [-36], [34], [-38], [16], [14], [-30], [-12], [12], [-8], [-31], [-14], [-18], [-12], [27], [39], [18], [32], [-24], [38], [-6], [-12], [-6], [-18], [4], [-15], [41], [46], [-28], [-42], [-46], [52], [10], [32], [-7], [-6], [-24], [-14], [-39], [15], [18], [8], [54], [-29], [-44], [12], [-16], [34], [-30], [-1], [38], [-36], [-3], [-12], [-37], [25], [0], [4], [36], [41], [-46], [-39], [-32], [-20], [19], [10], [27], [5], [24], [28], [-34], [-32], [-45], [-40], [47], [-34], [-28], [44], [23], [-23], [6], [59], [1], [41], [54], [-9], [14], [15], [66], [23], [-48], [38], [0], [60], [51], [60], [42], [26], [-36], [-6], [-52], [-46], [-68], [-34], [63], [58], [-15], [-14], [-27], [28], [-36], [46], [-58], [25], [-56], [7], [-8], [39], [-34], [-22], [9], [-16], [-43], [28], [23], [-28], [12], [21], [62], [1], [8], [39], [-35], [-10], [-16], [-61], [-52], [-44], [36], [17], [-14], [32], [-9], [-60], [11], [42], [-51], [-14], [-55], [52], [33], [-44], [33], [-50], [-2], [1], [39], [-38], [30], [75], [16], [-62], [45], [19], [-34], [-26], [4], [55], [-68], [22], [51], [32], [54], [-12], [76], [73], [80], [2], [-46], [-24], [14], [16], [-48], [-9], [9], [-40], [14], [-4], [-34], [-36], [-63], [-63], [-60], [44], [-49], [9], [61], [60], [-56], [70], [0], [24], [-90], [-75], [56], [-72], [22], [-57], [-84], [19], [-21], [0], [37], [-2], [74], [34], [42], [25], [-40], [-48], [82], [22], [-16], [-56], [3], [35], [60], [70], [-34], [35], [-84], [-56], [-79], [78], [50], [30], [92], [-2], [-7], [-26], [72], [41], [2], [-60], [-26], [10], [-89], [22], [51], [30], [-24], [-28], [83], [12], [-36], [29], [59], [66], [60], [65], [48], [35], [-36], [-96], [-12], [32], [-17], [80], [-1], [-51], [-74], [22], [-39], [78], [-77], [-24], [64], [-52], [-32], [52], [-42], [83], [-8], [-79], [17], [-34], [-35], [40], [29], [-97], [54], [-66], [65], [-26], [37], [4], [30], [22], [50], [-64], [-15], [55], [72], [1], [-20], [23], [-48], [22], [58], [1], [-52], [102], [20], [46], [29], [45], [-22], [44], [-25], [7], [-66]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3200_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3200_2_a_z();".
// 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_3200_2_a_z(:prec:=1)
    chi := MakeCharacter_3200_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_3200_2_a_z();".
// 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_3200_2_a_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3200_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-3, 1]>,<7,R![4, 1]>,<11,R![-3, 1]>,<13,R![-2, 1]>],Snew);
    return Vf;
end function;