
// Make newform 1344.2.a.s in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1344_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_1344_2_a_s();".
// 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_1344_2_a_s();".
// 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_1344_a();"
function MakeCharacter_1344_a()
    N := 1344;
    order := 1;
    char_gens := [127, 1093, 449, 577];
    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_1344_a_Hecke(Kf)
    return MakeCharacter_1344_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], [2], [1], [4], [2], [-6], [4], [0], [2], [0], [-6], [2], [-4], [0], [-6], [12], [2], [4], [0], [-6], [16], [-12], [-14], [18], [-14], [-8], [4], [18], [-14], [0], [4], [-6], [12], [-6], [-8], [2], [4], [8], [10], [-4], [26], [8], [2], [-22], [-24], [4], [-16], [-12], [10], [-6], [-24], [2], [-20], [26], [-16], [-6], [-16], [-22], [-22], [-20], [-14], [4], [24], [26], [18], [-4], [-14], [-28], [2], [10], [-32], [0], [10], [12], [0], [-6], [18], [-30], [-22], [-12], [-38], [24], [-14], [24], [36], [-30], [10], [10], [-16], [36], [16], [8], [20], [4], [-24], [10], [18], [-20], [34], [4], [2], [4], [10], [-4], [34], [28], [-6], [-48], [-6], [16], [26], [-6], [-20], [40], [18], [20], [40], [18], [12], [-22], [34], [18], [-12], [20], [-30], [-6], [48], [40], [18], [36], [0], [32], [10], [18], [2], [-14], [-44], [26], [42], [44], [-38], [-24], [-12], [-14], [8], [10], [-14], [44], [24], [-46], [-6], [-28], [-8], [-4], [24], [-8], [26], [42], [-38], [44], [26], [-40], [12], [-30], [-24], [16], [26], [-14], [34], [-12], [-30], [0], [-22], [-32], [2], [28], [26], [24], [-46], [-32], [-60], [-38], [-46], [-24], [34], [2], [4], [10], [32], [-14], [-60], [-28], [-6], [28], [18], [50], [-30], [-6], [-24], [34], [-32], [-6], [-14], [44], [58], [48], [-36], [42], [28], [18], [18], [56], [28], [40], [10], [64], [-38], [32], [-30], [-38], [8], [2], [-32], [20], [10], [-14], [-8], [-8], [-12], [-46], [4], [16], [10], [-36], [-48], [-62], [-38], [36], [-8], [60], [-36], [72], [-46], [-54], [-40], [32], [20], [-20], [-24], [34], [42], [-64], [58], [-22], [44], [10], [12], [18], [26], [-32], [-52], [10], [34], [-22], [-28], [66], [-62], [28], [-70], [50], [68], [10], [16], [-62], [-56], [52], [-30], [10], [68], [64], [24], [40], [-70], [44], [24], [82], [-6], [-56], [-6], [2], [52], [-54], [60], [50], [26], [80], [-14], [-52], [68], [-54], [18], [80], [-20], [60], [66], [-12], [-30], [80], [26], [48], [10], [18], [-12], [16], [26], [-12], [24], [-14], [66], [-28], [-38], [-6], [16], [42], [34], [4], [44], [-56], [-6], [50], [-30], [-32], [-12], [-20], [-20], [34], [-6], [-70], [-16], [74], [-22], [-30], [8], [34], [28], [74], [28], [-48], [34], [52], [-54], [2], [-96], [-22], [2], [0], [12], [-46], [48], [-6], [-62], [24], [-76], [-28], [-86], [58], [-40], [90], [12], [-36], [-72], [90], [-56], [2], [52], [-40], [2], [90], [-54], [-38], [-54], [-40], [2], [-60], [-8], [-32], [-70], [-4], [48], [34], [66], [36], [-28], [-80], [58], [-16], [18], [12], [42], [34], [-94], [80], [-78], [-30], [8], [18], [50], [-76], [84], [50], [-86], [-28], [52], [-54], [-6], [24], [88], [90], [-72], [66], [-70], [-8], [-100], [74], [42], [28], [-54], [-36], [24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1344_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1344_2_a_s();".
// 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_1344_2_a_s(:prec:=1)
    chi := MakeCharacter_1344_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_1344_2_a_s();".
// 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_1344_2_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1344_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<11,R![-4, 1]>,<13,R![-2, 1]>,<19,R![-4, 1]>],Snew);
    return Vf;
end function;