
// Make newform 3344.2.a.d in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3344_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_3344_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_3344_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_3344_a();"
function MakeCharacter_3344_a()
    N := 3344;
    order := 1;
    char_gens := [2927, 837, 2433, 705];
    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_3344_a_Hecke(Kf)
    return MakeCharacter_3344_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], [-1], [-3], [4], [-1], [2], [0], [-1], [-3], [-6], [7], [-7], [0], [10], [0], [6], [-3], [-10], [-11], [-15], [8], [16], [0], [9], [-1], [0], [16], [18], [2], [-9], [-2], [6], [21], [-8], [-6], [10], [17], [4], [18], [24], [-15], [-7], [15], [-4], [12], [16], [22], [-5], [12], [-13], [18], [18], [20], [-21], [-6], [30], [-30], [16], [26], [-30], [-14], [18], [-2], [24], [-1], [15], [7], [-4], [18], [-28], [-21], [-12], [-23], [14], [-5], [-33], [-3], [14], [-30], [32], [-12], [26], [6], [-13], [-14], [27], [-27], [-10], [12], [-29], [3], [6], [7], [-30], [16], [-18], [21], [39], [16], [-10], [22], [-30], [6], [6], [46], [-7], [-24], [0], [24], [20], [-8], [2], [-30], [13], [1], [-15], [7], [-21], [27], [18], [-13], [14], [6], [-12], [-17], [30], [-25], [9], [-35], [-22], [-44], [30], [1], [14], [0], [-28], [-6], [-20], [51], [-24], [-20], [36], [13], [42], [11], [-33], [-22], [30], [-5], [24], [-4], [9], [4], [-54], [-8], [0], [4], [30], [-22], [-24], [3], [-36], [22], [15], [-9], [15], [-8], [-40], [-34], [15], [18], [62], [48], [-4], [37], [-15], [34], [51], [46], [-58], [-8], [-42], [-49], [-6], [3], [-54], [38], [-44], [32], [-6], [-49], [-24], [-11], [42], [-12], [9], [50], [-49], [48], [54], [-12], [10], [14], [-16], [-51], [-27], [43], [-12], [-6], [-8], [38], [27], [1], [12], [24], [-7], [-50], [-42], [12], [-9], [56], [4], [-57], [19], [72], [2], [-18], [-24], [-8], [-18], [17], [64], [58], [42], [55], [42], [-67], [-12], [-3], [33], [-21], [16], [-32], [47], [-12], [-36], [-20], [48], [40], [-12], [20], [-30], [15], [26], [-66], [6], [14], [-8], [27], [32], [52], [-6], [56], [74], [-18], [-68], [45], [9], [-20], [6], [-1], [-29], [-46], [-68], [2], [1], [57], [62], [44], [-42], [24], [67], [-48], [-34], [22], [3], [2], [54], [43], [-60], [33], [-36], [36], [6], [62], [-60], [-71], [-9], [-36], [-50], [50], [-78], [58], [-60], [37], [41], [3], [23], [-72], [-34], [-78], [30], [18], [-35], [-48], [-70], [-39], [0], [-22], [12], [-2], [29], [84], [-59], [24], [-55], [7], [79], [66], [-12], [-88], [18], [-62], [42], [34], [63], [-1], [-24], [47], [22], [-1], [-45], [0], [19], [33], [72], [-25], [-17], [48], [-81], [-2], [-43], [42], [-8], [-40], [24], [63], [54], [42], [-51], [-46], [0], [51], [18], [61], [35], [-90], [4], [-52], [-33], [-92], [-36], [84], [16], [-67], [-12], [-90], [92], [24], [62], [-30], [-69], [-20], [48], [-32], [-57], [64], [-31], [34], [27], [11], [21], [-57], [1], [-81], [-76], [-2], [-6], [100], [24], [50], [-99], [-68], [-78], [102], [70], [-10], [54], [4], [45], [-46], [-48], [84], [-2], [104], [-75], [-30], [-53], [54], [0], [-15], [-100], [36], [6], [-10], [-27], [27], [18], [-23], [-12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3344_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3344_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_3344_2_a_d(:prec:=1)
    chi := MakeCharacter_3344_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_3344_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_3344_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3344_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 1]>,<5,R![3, 1]>,<7,R![-4, 1]>],Snew);
    return Vf;
end function;