
// Make newform 3744.2.a.j in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3744_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_3744_2_a_j();".
// 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_3744_2_a_j();".
// 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_3744_a();"
function MakeCharacter_3744_a()
    N := 3744;
    order := 1;
    char_gens := [703, 2341, 2081, 2017];
    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_3744_a_Hecke(Kf)
    return MakeCharacter_3744_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], [0], [0], [2], [0], [1], [-2], [-2], [-8], [-6], [2], [-6], [0], [4], [-8], [6], [-4], [2], [2], [-4], [-2], [12], [12], [-12], [-18], [2], [12], [4], [-2], [2], [0], [-4], [-16], [-8], [12], [-2], [-14], [-14], [0], [2], [12], [-6], [0], [-6], [0], [24], [-20], [-14], [24], [-10], [-14], [20], [6], [-4], [-14], [24], [-14], [-30], [22], [-8], [8], [12], [22], [-8], [-22], [32], [-30], [-14], [-12], [-14], [16], [0], [-32], [22], [22], [0], [6], [-6], [20], [14], [12], [14], [16], [-2], [20], [-36], [-4], [-18], [-16], [18], [-28], [20], [2], [36], [-6], [-16], [24], [-30], [16], [14], [32], [-12], [-28], [-30], [32], [26], [-28], [4], [-40], [-38], [0], [34], [16], [46], [-34], [22], [-46], [-32], [10], [-12], [-6], [6], [46], [8], [2], [-42], [38], [-24], [40], [38], [-10], [12], [4], [-22], [48], [34], [-28], [38], [6], [-38], [-42], [-20], [-4], [-48], [-6], [-12], [18], [42], [4], [56], [-30], [18], [-48], [24], [-40], [32], [52], [-8], [30], [20], [-52], [-30], [-22], [44], [36], [-52], [-40], [46], [30], [26], [0], [-30], [8], [22], [44], [54], [-26], [4], [28], [14], [-6], [60], [26], [-36], [56], [30], [-30], [-26], [-66], [-8], [46], [60], [40], [-52], [4], [2], [10], [14], [-20], [-56], [-24], [-16], [62], [-34], [52], [-66], [-6], [12], [12], [-24], [-50], [-34], [-28], [24], [-16], [-14], [12], [54], [28], [-36], [-58], [-22], [60], [58], [4], [-26], [6], [-40], [8], [12], [-14], [16], [-10], [-58], [-64], [-60], [22], [-60], [-4], [40], [-36], [-32], [28], [-38], [76], [-16], [34], [-12], [-50], [-56], [-34], [0], [44], [-70], [-6], [-16], [-10], [62], [-42], [2], [32], [-52], [38], [-74], [28], [36], [-56], [-40], [62], [50], [-46], [2], [14], [-20], [74], [-10], [-40], [10], [18], [-12], [16], [-22], [0], [-30], [38], [-48], [-34], [60], [38], [-62], [-66], [-60], [-12], [-4], [34], [-2], [-8], [-18], [-36], [78], [10], [-64], [28], [-28], [-16], [14], [28], [-2], [-36], [-50], [40], [16], [-14], [36], [72], [66], [56], [-80], [-42], [6], [-12], [-70], [-90], [-34], [84], [-34], [26], [42], [56], [-6], [-58], [26], [52], [-48], [-50], [-68], [10], [24], [-26], [28], [74], [42], [-4], [-20], [-48], [12], [-58], [-38], [24], [-18], [26], [-2], [68], [-16], [-22], [34], [-24], [-56], [-22], [40], [26], [-26], [-24], [-24], [-20], [-74], [-24], [38], [14], [-12], [-92], [0], [22], [28], [62], [-12], [80], [-30], [78], [-42], [52], [24], [-78], [-60], [-70], [-36], [-42], [-14], [86], [-32], [-82], [76], [48], [60], [32], [86], [-18], [-42], [-52], [-88], [-78], [-66], [-78], [-52], [0], [-20], [86], [100], [-58], [-64], [66], [-18], [84], [52], [-46], [30], [-76], [28], [12], [24], [90], [34], [44], [92], [74], [-12], [84], [-12], [26], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3744_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3744_2_a_j();".
// 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_3744_2_a_j(:prec:=1)
    chi := MakeCharacter_3744_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_3744_2_a_j();".
// 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_3744_2_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3744_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![-2, 1]>,<11,R![0, 1]>,<29,R![6, 1]>],Snew);
    return Vf;
end function;