
// Make newform 3200.2.a.g in Magma, downloaded from the LMFDB on 08 June 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_g();".
// 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_g();".
// 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]];
    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], [-2], [0], [0], [2], [2], [-6], [6], [0], [-10], [8], [2], [-6], [2], [-12], [10], [-6], [6], [14], [4], [10], [-8], [-10], [14], [-6], [6], [8], [2], [-2], [-2], [12], [6], [6], [-14], [6], [12], [10], [14], [16], [10], [-2], [14], [24], [-6], [18], [28], [-10], [20], [6], [14], [10], [8], [-2], [18], [-18], [-24], [-2], [-32], [2], [30], [-14], [-6], [22], [28], [-26], [10], [2], [-2], [-30], [22], [-2], [12], [-28], [18], [26], [4], [-34], [-38], [-2], [-22], [30], [-10], [0], [26], [20], [2], [14], [6], [22], [28], [6], [-16], [-16], [-22], [-2], [16], [14], [10], [-6], [-34], [-2], [-22], [-34], [-38], [-22], [-18], [18], [14], [-28], [14], [28], [-14], [-6], [42], [-4], [-2], [6], [-16], [10], [30], [38], [-22], [-38], [18], [-10], [-42], [6], [-8], [-8], [-30], [6], [-8], [-32], [42], [10], [-50], [10], [-18], [-14], [26], [-46], [-18], [-16], [-6], [-10], [-4], [58], [38], [26], [4], [18], [2], [-18], [8], [-22], [48], [20], [-18], [-38], [22], [-42], [6], [-32], [42], [10], [-48], [-24], [42], [-18], [18], [18], [-26], [-28], [-54], [40], [30], [50], [22], [40], [-10], [28], [-34], [34], [58], [12], [30], [26], [46], [-6], [-8], [-66], [2], [30], [-18], [38], [42], [-62], [2], [30], [-8], [-18], [32], [-22], [50], [10], [18], [16], [-34], [30], [-14], [10], [54], [16], [42], [-36], [-22], [68], [-2], [-8], [-54], [22], [-44], [-50], [-52], [54], [-34], [10], [0], [-16], [10], [-6], [6], [-56], [-18], [-46], [52], [-50], [-62], [18], [36], [-26], [10], [72], [46], [-2], [-28], [-52], [38], [50], [12], [-54], [-78], [64], [14], [66], [30], [-18], [18], [18], [-42], [-44], [-42], [-26], [34], [-22], [78], [-26], [-2], [-6], [-78], [14], [-58], [-54], [32], [46], [16], [42], [-2], [-34], [-34], [-28], [-60], [-24], [14], [58], [64], [-38], [42], [20], [50], [-82], [38], [-54], [34], [50], [-10], [-8], [-6], [34], [-50], [-26], [-6], [-8], [-26], [-70], [10], [-62], [30], [-76], [2], [28], [-2], [30], [22], [32], [-34], [-34], [-24], [-34], [66], [-10], [38], [30], [-28], [22], [-2], [-26], [50], [-20], [-14], [22], [66], [80], [-42], [58], [-46], [30], [-66], [-50], [52], [-54], [22], [-18], [-12], [-54], [-26], [38], [-46], [0], [50], [-50], [70], [-58], [-52], [46], [-54], [48], [42], [74], [-80], [90], [-50], [-12], [-70], [22], [42], [-14], [-40], [26], [-90], [42], [-36], [14], [-4], [-38], [30], [-8], [14], [-82], [-42], [-42], [6], [-56], [78], [22], [-24], [-32], [-46], [26], [-68], [-46], [50], [58], [62], [-52], [-58], [-64], [-54], [-22], [-2], [14], [-54], [20], [-42], [-90], [-12], [-62], [62], [38], [62], [46], [34], [-6], [46], [-6], [30], [-80], [-64], [-38], [-16], [6], [-22], [-92], [42], [6], [42], [-82], [30], [-22], [36]];
    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_g();".
// 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_g(: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_g();".
// 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_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3200_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<7,R![0, 1]>,<11,R![-2, 1]>,<13,R![-2, 1]>],Snew);
    return Vf;
end function;