
// Make newform 3200.2.a.i in Magma, downloaded from the LMFDB on 20 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_i();".
// 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_i();".
// 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], [-1], [0], [0], [-1], [-2], [-3], [3], [-6], [2], [2], [4], [-3], [4], [6], [-10], [12], [12], [1], [10], [-1], [16], [-11], [-13], [-6], [0], [-14], [7], [16], [11], [6], [12], [-3], [19], [6], [12], [14], [-23], [14], [-4], [-5], [2], [-18], [9], [18], [4], [17], [28], [0], [8], [14], [26], [-11], [-15], [6], [-18], [16], [10], [-14], [6], [17], [12], [-25], [4], [2], [32], [11], [23], [3], [-20], [-10], [-12], [16], [18], [5], [20], [-22], [-28], [13], [-7], [-21], [32], [-12], [-23], [-22], [13], [5], [9], [-32], [8], [12], [-22], [-26], [-40], [4], [-4], [-34], [37], [39], [20], [-13], [-38], [4], [25], [-4], [39], [-39], [-17], [-40], [11], [-28], [2], [-6], [-24], [14], [10], [36], [28], [-28], [-27], [-10], [-50], [-4], [-9], [35], [30], [-24], [4], [-22], [54], [-12], [26], [34], [-36], [-23], [25], [26], [-36], [38], [26], [20], [-30], [40], [-27], [-10], [-40], [26], [25], [-13], [-34], [-48], [14], [39], [4], [-20], [-24], [50], [18], [47], [-38], [36], [45], [-4], [15], [-7], [36], [0], [36], [9], [-42], [-51], [52], [-40], [-45], [-26], [-21], [41], [-44], [20], [62], [8], [23], [20], [-25], [-54], [-12], [40], [11], [9], [10], [27], [-35], [-27], [30], [67], [30], [-17], [10], [15], [20], [54], [56], [-44], [-37], [4], [66], [34], [52], [-54], [52], [-46], [24], [2], [-45], [18], [-55], [-38], [13], [56], [-24], [22], [34], [1], [-20], [51], [20], [-73], [66], [-2], [-35], [-60], [57], [64], [15], [28], [-40], [-59], [-70], [69], [0], [59], [73], [6], [40], [11], [20], [16], [-28], [-19], [-18], [48], [-57], [32], [-37], [-18], [33], [6], [21], [48], [33], [-64], [45], [10], [2], [-71], [27], [58], [22], [-9], [36], [14], [73], [27], [2], [38], [20], [3], [4], [38], [23], [-56], [-66], [60], [44], [5], [-32], [-46], [-42], [56], [-82], [-76], [-8], [39], [37], [16], [-10], [64], [-6], [28], [-43], [-55], [24], [4], [-37], [53], [29], [-4], [-12], [-10], [-20], [44], [22], [21], [32], [16], [14], [-37], [-60], [91], [3], [-40], [-11], [18], [34], [74], [-62], [-11], [64], [-52], [14], [-38], [-12], [44], [-9], [31], [4], [18], [-66], [-29], [-16], [24], [17], [6], [6], [54], [-68], [2], [85], [-6], [-44], [85], [-70], [8], [82], [34], [-9], [66], [-33], [-2], [32], [-48], [43], [-24], [-76], [-31], [51], [-10], [-44], [-7], [-24], [15], [12], [-88], [-52], [-52], [-69], [-68], [-65], [-43], [54], [6], [-87], [2], [27], [16], [-84], [-92], [16], [76], [-58], [83], [64], [61], [-59], [14], [-59], [-64], [69], [-13], [-26], [-22], [81], [34], [21], [12], [66], [-10], [26], [-56], [-67], [-49], [-40], [69], [-44], [63], [60], [-2], [-62], [-55], [-8], [-78], [-56], [-82], [81], [-75], [-18], [4], [-33], [-37], [-90]];
    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_i();".
// 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_i(: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_i();".
// 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_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3200_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 1]>,<7,R![0, 1]>,<11,R![1, 1]>,<13,R![2, 1]>],Snew);
    return Vf;
end function;