
// Make newform 3744.2.a.d in Magma, downloaded from the LMFDB on 19 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_d();".
// 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_d();".
// 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], [-2], [2], [2], [-1], [-6], [2], [0], [-2], [-2], [10], [-2], [-8], [-2], [2], [10], [6], [-14], [-14], [-6], [4], [6], [-10], [10], [-6], [0], [4], [2], [-14], [-8], [-4], [6], [-20], [6], [-6], [2], [-10], [-2], [18], [-24], [-22], [-12], [-6], [-2], [-8], [-4], [14], [-18], [-22], [6], [-30], [-14], [0], [-2], [12], [30], [-6], [-26], [30], [-8], [6], [-18], [24], [-26], [-18], [10], [-10], [28], [34], [-10], [-10], [36], [26], [-22], [34], [6], [10], [-10], [34], [20], [2], [-30], [-14], [8], [12], [-10], [2], [-18], [34], [-16], [-18], [-34], [-16], [-6], [-12], [30], [42], [36], [10], [36], [-2], [24], [-26], [8], [42], [-18], [-18], [28], [22], [-20], [-6], [-18], [6], [10], [26], [42], [24], [-34], [-36], [-22], [-10], [2], [-30], [-22], [30], [34], [-8], [-24], [50], [38], [-6], [-32], [30], [-42], [34], [-18], [6], [30], [-22], [-22], [-2], [32], [-22], [14], [-2], [-30], [-10], [-44], [-14], [10], [30], [40], [12], [56], [12], [-12], [-42], [26], [-10], [18], [54], [-18], [-28], [-2], [6], [52], [30], [-38], [26], [30], [-30], [-16], [10], [56], [38], [-50], [6], [56], [34], [-38], [32], [-50], [-2], [54], [-10], [-30], [-14], [50], [-42], [-2], [-34], [28], [14], [-32], [54], [34], [34], [-2], [-44], [30], [-68], [2], [30], [50], [-54], [-10], [-68], [62], [40], [50], [-30], [60], [-6], [10], [-30], [-68], [-34], [-46], [6], [-34], [58], [30], [-34], [24], [-10], [54], [-60], [-16], [-22], [-14], [-44], [-58], [66], [-20], [-22], [-22], [-66], [0], [-76], [54], [48], [-60], [-30], [-34], [32], [-22], [42], [50], [16], [42], [22], [-18], [-18], [-50], [-30], [-54], [-22], [6], [34], [32], [12], [-22], [-26], [-66], [-52], [38], [70], [-2], [-54], [14], [-46], [26], [-48], [-2], [-18], [-82], [50], [10], [-80], [-60], [58], [-68], [42], [14], [-24], [82], [-26], [-46], [62], [42], [-36], [-42], [-38], [66], [-66], [-4], [-42], [36], [46], [58], [22], [-32], [44], [-76], [-14], [-72], [54], [-66], [26], [12], [-58], [34], [44], [-42], [22], [78], [74], [-14], [-6], [-24], [2], [58], [26], [54], [-14], [-42], [42], [80], [38], [10], [-2], [-20], [-38], [2], [-34], [-30], [14], [2], [-48], [-46], [26], [-58], [-40], [70], [-48], [42], [46], [70], [-74], [-62], [-46], [14], [24], [-42], [-54], [-66], [-66], [-2], [26], [34], [-30], [68], [-10], [32], [-74], [6], [2], [-50], [12], [32], [6], [66], [20], [38], [-18], [8], [26], [-66], [-18], [-66], [46], [-94], [78], [-98], [14], [54], [38], [-62], [44], [-70], [86], [-30], [52], [-34], [-54], [46], [-30], [-76], [-26], [58], [90], [-38], [-66], [14], [20], [2], [-50], [54], [-14], [34], [82], [12], [0], [-38], [-22], [26], [20], [14], [64], [82], [-46], [-38], [36], [-46], [70], [0], [46], [-74], [-36]];
    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_d();".
// 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_d(: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_d();".
// 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_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3744_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<7,R![-2, 1]>,<11,R![-2, 1]>,<29,R![2, 1]>],Snew);
    return Vf;
end function;