
// Make newform 3536.2.a.k in Magma, downloaded from the LMFDB on 23 May 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_k();".
// 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_k();".
// 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+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], [2], [2], [-2], [-4], [-1], [-1], [-4], [-4], [0], [-2], [2], [-4], [4], [0], [-6], [12], [0], [-12], [-10], [16], [0], [4], [14], [8], [10], [-16], [-14], [-14], [14], [-16], [-18], [6], [18], [-10], [20], [-2], [20], [10], [0], [-12], [16], [8], [4], [-10], [0], [-10], [8], [-8], [6], [-2], [-12], [8], [16], [2], [-8], [-8], [-20], [-24], [-10], [2], [-6], [-4], [4], [-6], [22], [12], [-6], [18], [-30], [-22], [-12], [-12], [-22], [36], [0], [30], [-14], [0], [2], [-30], [-34], [-10], [-14], [-4], [-36], [-24], [10], [6], [-12], [8], [6], [-2], [24], [24], [4], [26], [-6], [4], [-42], [-42], [34], [24], [-26], [2], [-2], [-12], [14], [8], [-42], [-20], [-30], [8], [32], [-12], [14], [20], [-24], [-24], [8], [38], [6], [24], [-12], [-28], [-22], [-26], [36], [-16], [-14], [20], [-26], [16], [-10], [-18], [46], [42], [12], [2], [-30], [16], [-38], [8], [28], [30], [42], [-50], [18], [28], [-32], [38], [-18], [36], [36], [50], [-32], [24], [40], [22], [14], [36], [-6], [4], [48], [-18], [34], [-40], [8], [-40], [-12], [-60], [30], [-4], [6], [-32], [42], [-20], [54], [-40], [22], [8], [10], [-16], [46], [24], [-6], [-40], [4], [-36], [-18], [14], [-52], [16], [-38], [-26], [54], [-52], [-64], [12], [32], [58], [20], [22], [-46], [-44], [30], [20], [-56], [-12], [-8], [2], [62], [-24], [68], [-42], [-68], [-32], [30], [-18], [-18], [62], [26], [46], [62], [72], [-54], [-46], [56], [-16], [32], [-54], [10], [0], [-54], [40], [-20], [-20], [6], [42], [0], [-72], [52], [-56], [-38], [-24], [-36], [-62], [20], [-44], [40], [-34], [-24], [44], [-46], [-18], [52], [44], [-12], [16], [-6], [-68], [-60], [6], [44], [8], [-40], [6], [42], [36], [-42], [-64], [20], [-18], [-64], [-38], [-52], [36], [30], [-58], [-48], [48], [-50], [12], [-74], [-8], [72], [14], [22], [-84], [-66], [-60], [-18], [-22], [8], [4], [-12], [-40], [-6], [-26], [28], [-58], [-66], [24], [78], [-10], [8], [40], [56], [12], [-26], [-72], [70], [-78], [-84], [-76], [22], [12], [30], [-28], [66], [-70], [52], [6], [-20], [-36], [-2], [-44], [24], [-12], [44], [34], [-36], [-16], [60], [56], [-52], [66], [36], [-76], [-40], [62], [-6], [26], [48], [-62], [-94], [52], [60], [90], [64], [-68], [52], [2], [-76], [-34], [38], [80], [44], [58], [-4], [2], [42], [0], [-60], [0], [22], [-82], [24], [70], [-44], [2], [10], [-70], [-40], [52], [84], [-4], [66], [-54], [-74], [54], [-34], [70], [72], [-16], [-70], [-72], [30], [16], [-40], [-12], [-90], [-20], [32], [-32], [6], [16], [-30], [-14], [-54], [74], [26], [8], [-12], [18], [64], [26], [38], [52], [20], [-34], [-98], [12], [74], [-6], [-8], [10], [100], [60], [-48], [-22], [62], [-30], [36], [32], [-78], [-58], [100], [-76], [-24]];
    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_k();".
// 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_k(: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_k();".
// 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_k( : 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![4, 1]>],Snew);
    return Vf;
end function;