
// Make newform 1323.2.a.r in Magma, downloaded from the LMFDB on 14 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1323_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_1323_2_a_r();".
// 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_1323_2_a_r();".
// 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_1323_a();"
function MakeCharacter_1323_a()
    N := 1323;
    order := 1;
    char_gens := [785, 1081];
    v := [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_1323_a_Hecke(Kf)
    return MakeCharacter_1323_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 := [[2], [0], [-1], [0], [4], [2], [3], [8], [6], [4], [-6], [-3], [1], [11], [9], [-6], [-15], [-4], [-8], [12], [-6], [-1], [-9], [2], [-12], [10], [-2], [-2], [-9], [-2], [-15], [-4], [18], [0], [-12], [5], [8], [-11], [17], [22], [2], [-16], [-2], [-25], [4], [12], [4], [2], [-24], [4], [-6], [18], [10], [-25], [10], [-14], [-21], [2], [13], [-10], [-28], [1], [2], [-3], [16], [18], [-1], [7], [2], [20], [-25], [-34], [30], [-13], [-3], [21], [24], [-6], [0], [-32], [-33], [14], [-24], [2], [-24], [24], [6], [-14], [1], [-23], [12], [-17], [16], [26], [13], [21], [-41], [15], [26], [29], [-29], [4], [-4], [-2], [11], [20], [20], [-33], [-36], [18], [-14], [-26], [6], [44], [-19], [-12], [-26], [-8], [42], [-12], [14], [-2], [30], [-42], [-50], [-18], [15], [33], [4], [0], [12], [-36], [16], [23], [-3], [-44], [49], [2], [2], [42], [46], [8], [-3], [18], [16], [-55], [-8], [17], [-2], [-6], [37], [30], [41], [13], [17], [-12], [-43], [-35], [12], [-17], [-28], [2], [40], [-45], [12], [21], [11], [-40], [1], [-11], [-24], [48], [6], [11], [-56], [31], [28], [52], [-24], [26], [-13], [-21], [-22], [-29], [-12], [61], [-26], [2], [-50], [-25], [14], [36], [17], [-27], [-44], [-21], [26], [39], [18], [-21], [-40], [-32], [-30], [14], [-23], [25], [36], [0], [42], [-22], [21], [-30], [43], [-13], [7], [-28], [-49], [14], [10], [-24], [-49], [2], [-34], [47], [19], [-19], [-67], [50], [-62], [30], [-50], [-34], [-37], [52], [-18], [0], [-16], [-16], [18], [25], [12], [0], [42], [-38], [42], [-31], [-70], [55], [61], [-30], [65], [6], [34], [50], [-19], [56], [-13], [18], [30], [46], [-40], [-58], [-8], [52], [34], [-8], [12], [62], [-11], [34], [38], [68], [-10], [-49], [14], [-8], [-2], [39], [-77], [-17], [-17], [-57], [-23], [68], [22], [60], [-35], [72], [-86], [21], [-38], [-1], [-36], [-45], [-5], [53], [50], [-17], [79], [40], [-30], [60], [40], [10], [-63], [66], [-12], [-30], [10], [-62], [-24], [-46], [-78], [81], [80], [-26], [-75], [54], [44], [48], [10], [61], [-27], [-64], [90], [-34], [43], [-44], [20], [0], [70], [12], [40], [63], [25], [29], [50], [57], [28], [44], [79], [8], [63], [-32], [50], [-8], [80], [79], [75], [22], [26], [-63], [10], [84], [-17], [-17], [-60], [9], [20], [-36], [-33], [-67], [90], [-88], [-32], [-47], [-77], [91], [-45], [24], [-18], [-24], [-6], [70], [22], [-28], [46], [70], [66], [24], [73], [24], [-32], [74], [18], [88], [-100], [62], [41], [60], [-65], [42], [-6], [-62], [68], [46], [-70], [-45], [-12], [-42], [40], [-64], [-11], [-3], [-30], [104], [-75], [38], [-32], [48], [-44], [-22], [50], [-35], [3], [18], [-90], [76], [99], [24], [-54], [32], [98], [49], [-14], [21], [-68], [-48], [18], [24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1323_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1323_2_a_r();".
// 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_1323_2_a_r(:prec:=1)
    chi := MakeCharacter_1323_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_1323_2_a_r();".
// 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_1323_2_a_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1323_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-2, 1]>,<5,R![1, 1]>,<13,R![-2, 1]>],Snew);
    return Vf;
end function;