
// Make newform 1350.2.a.l in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1350_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_1350_2_a_l();".
// 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_1350_2_a_l();".
// 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_1350_a();"
function MakeCharacter_1350_a()
    N := 1350;
    order := 1;
    char_gens := [1001, 1027];
    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_1350_a_Hecke(Kf)
    return MakeCharacter_1350_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 := [[1], [0], [0], [-4], [-5], [3], [1], [-6], [1], [-9], [-5], [2], [-2], [-1], [-13], [0], [-4], [8], [4], [-6], [2], [9], [4], [-14], [10], [17], [10], [6], [-8], [-3], [-4], [-3], [-18], [-8], [-13], [13], [13], [19], [12], [-16], [20], [-22], [-6], [-4], [-22], [3], [0], [-22], [-2], [2], [14], [16], [-23], [15], [21], [-8], [9], [16], [26], [-18], [-20], [-6], [-21], [18], [14], [4], [10], [-16], [-6], [16], [11], [6], [-22], [21], [-28], [-19], [-5], [-13], [-16], [11], [-23], [-26], [-8], [-22], [0], [-30], [0], [-16], [-6], [6], [6], [-2], [12], [24], [-12], [9], [-21], [-22], [-23], [10], [-25], [42], [-4], [-20], [-32], [32], [36], [21], [-42], [-19], [10], [23], [-29], [26], [-4], [-20], [-21], [0], [-26], [20], [44], [-48], [4], [-44], [22], [31], [-24], [-20], [0], [46], [10], [-9], [-7], [-43], [12], [3], [-18], [-13], [-14], [12], [-26], [-46], [-24], [22], [0], [-34], [19], [-6], [2], [39], [1], [-18], [-36], [51], [3], [-36], [33], [-6], [40], [31], [48], [-61], [-2], [-9], [-39], [29], [-17], [-15], [-19], [6], [-27], [48], [-62], [-56], [-41], [-12], [62], [6], [2], [-28], [-4], [-4], [13], [57], [-16], [54], [59], [19], [14], [38], [-6], [-20], [52], [45], [-60], [-30], [-54], [51], [42], [-16], [57], [48], [18], [22], [1], [-6], [15], [46], [-16], [28], [-20], [9], [58], [-18], [26], [19], [-26], [50], [-37], [64], [-36], [-41], [-18], [10], [48], [-12], [49], [38], [46], [-53], [46], [64], [5], [-6], [20], [-8], [-45], [50], [-36], [54], [-30], [8], [-70], [46], [34], [-6], [18], [-63], [62], [73], [57], [64], [8], [10], [14], [-45], [-68], [-59], [-52], [-8], [-36], [-14], [-56], [55], [3], [32], [18], [-68], [-61], [-6], [20], [49], [-44], [-29], [-40], [52], [44], [-40], [-7], [-15], [51], [-1], [-43], [-44], [-29], [-80], [-6], [44], [-37], [-2], [-23], [54], [27], [24], [-46], [42], [-48], [60], [-17], [-72], [-68], [0], [-25], [78], [-52], [-70], [80], [-22], [-76], [-39], [-51], [-42], [-36], [39], [37], [47], [-43], [44], [50], [-66], [38], [4], [-57], [-40], [69], [29], [-24], [-36], [-59], [22], [-44], [22], [-19], [14], [66], [42], [10], [-30], [31], [74], [21], [-67], [70], [-61], [-20], [33], [30], [76], [-4], [-12], [-70], [-54], [-45], [-42], [38], [3], [48], [-27], [55], [13], [62], [60], [-24], [-35], [63], [2], [-66], [54], [-45], [17], [-70], [-53], [6], [-12], [45], [-35], [-82], [-90], [84], [-68], [-33], [-25], [100], [47], [-30], [63], [48], [-38], [39], [-5], [53], [-60], [60], [43], [-10], [14], [80], [-10], [-40], [9], [-8], [7], [-8], [63], [-54], [44], [-39], [-42], [76], [39], [-86], [90], [-6], [76], [-46], [94], [88], [6], [-74], [81], [39], [46], [-47], [-45], [34], [-42], [-22], [74], [-90], [64]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1350_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1350_2_a_l();".
// 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_1350_2_a_l(:prec:=1)
    chi := MakeCharacter_1350_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_1350_2_a_l();".
// 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_1350_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1350_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, 1]>,<11,R![5, 1]>,<13,R![-3, 1]>,<17,R![-1, 1]>],Snew);
    return Vf;
end function;