
// Make newform 3536.2.a.b in Magma, downloaded from the LMFDB on 05 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3536_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_3536_2_a_b();".
// 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_3536_2_a_b();".
// 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_3536_a();"
function MakeCharacter_3536_a()
    N := 3536;
    order := 1;
    char_gens := [1327, 885, 3265, 1873];
    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_3536_a_Hecke(Kf)
    return MakeCharacter_3536_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], [2], [-2], [-2], [-1], [1], [4], [2], [2], [2], [2], [2], [0], [-4], [-2], [-12], [-6], [-8], [-6], [2], [10], [12], [14], [-6], [-6], [8], [6], [-14], [-14], [-4], [-10], [14], [6], [-18], [8], [14], [-10], [-14], [-6], [-16], [-14], [-24], [-6], [-6], [-10], [-14], [0], [-26], [-6], [-6], [12], [-6], [12], [22], [-20], [-22], [4], [26], [26], [-6], [14], [8], [14], [10], [-22], [-4], [-22], [-6], [-34], [-14], [32], [-22], [10], [-38], [-16], [-6], [34], [18], [-22], [-6], [10], [-26], [38], [-10], [-20], [-6], [22], [-18], [-12], [-32], [10], [-2], [32], [-34], [42], [22], [-22], [-12], [10], [-34], [-30], [16], [26], [-22], [22], [-12], [-14], [40], [10], [14], [14], [18], [22], [48], [2], [-14], [32], [10], [-12], [-42], [34], [26], [10], [-42], [34], [10], [30], [8], [14], [4], [-6], [38], [-38], [30], [-42], [26], [-22], [22], [-30], [18], [18], [26], [6], [-18], [-42], [34], [18], [40], [-4], [-22], [-30], [-20], [6], [26], [-22], [-56], [-6], [26], [42], [42], [-42], [24], [48], [-30], [18], [38], [-46], [-38], [26], [-24], [2], [34], [-18], [-28], [-30], [-46], [10], [28], [18], [12], [-50], [18], [42], [32], [-22], [10], [-64], [66], [-30], [-54], [6], [-16], [34], [-2], [-14], [-30], [-22], [-30], [0], [34], [-62], [-18], [-10], [16], [66], [12], [24], [-30], [-4], [-38], [26], [-38], [-60], [-18], [-46], [-24], [2], [-34], [50], [54], [58], [66], [18], [12], [-54], [-30], [18], [-52], [6], [-10], [-50], [8], [30], [-12], [24], [-70], [10], [38], [-20], [38], [44], [-48], [-14], [-54], [-18], [42], [-66], [20], [28], [-42], [50], [0], [26], [50], [-8], [58], [58], [18], [38], [54], [76], [-70], [-30], [-14], [-76], [-18], [-22], [14], [-26], [-62], [72], [10], [4], [50], [-8], [52], [2], [-34], [-24], [-24], [42], [-54], [-42], [42], [0], [74], [2], [32], [22], [42], [30], [6], [70], [-86], [-14], [-48], [10], [-6], [-28], [-74], [18], [-22], [6], [6], [82], [-44], [-70], [-60], [-74], [-22], [-54], [-46], [88], [44], [26], [12], [74], [-30], [-30], [-54], [10], [78], [12], [-6], [2], [50], [-66], [-10], [-94], [34], [82], [-58], [-56], [54], [30], [62], [18], [58], [68], [-42], [-54], [66], [8], [18], [-54], [-46], [88], [66], [18], [20], [10], [22], [70], [-38], [-58], [-72], [-54], [-30], [-24], [-6], [66], [48], [-66], [-88], [74], [90], [-28], [-62], [72], [-70], [-58], [-86], [-24], [-46], [-30], [50], [30], [2], [-86], [2], [-6], [2], [82], [-66], [14], [8], [-54], [-22], [40], [34], [-70], [72], [36], [48], [2], [92], [26], [-38], [26], [-70], [30], [20], [-6], [30], [38], [18], [-74], [20], [-78], [-14], [-38], [60], [-50], [-86], [-78], [30], [54], [2], [-56], [6], [-46], [-54], [-88], [-30], [-42], [-74], [-38], [0], [-6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3536_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3536_2_a_b();".
// 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_3536_2_a_b(:prec:=1)
    chi := MakeCharacter_3536_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_3536_2_a_b();".
// 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_3536_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3536_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<5,R![-2, 1]>,<7,R![2, 1]>,<11,R![2, 1]>],Snew);
    return Vf;
end function;