
// Make newform 1620.2.a.e in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1620_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_1620_2_a_e();".
// 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_1620_2_a_e();".
// 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_1620_a();"
function MakeCharacter_1620_a()
    N := 1620;
    order := 1;
    char_gens := [811, 1541, 1297];
    v := [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_1620_a_Hecke(Kf)
    return MakeCharacter_1620_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], [1], [-1], [0], [-4], [-6], [2], [-3], [3], [-10], [-10], [9], [-4], [9], [-6], [-6], [-1], [11], [12], [-4], [-10], [-9], [9], [-10], [-18], [8], [9], [-19], [-12], [-7], [6], [12], [20], [9], [8], [2], [-4], [-15], [0], [-24], [5], [6], [-4], [12], [2], [-4], [11], [12], [-1], [-30], [0], [-7], [18], [6], [-24], [9], [14], [2], [33], [-1], [-6], [29], [18], [-16], [-12], [26], [8], [24], [23], [12], [-6], [32], [14], [14], [-24], [-15], [-22], [-18], [-10], [12], [-22], [6], [26], [-28], [-27], [30], [-10], [33], [-4], [36], [-30], [-16], [-24], [-4], [3], [-9], [27], [-7], [-7], [-7], [-6], [-39], [6], [32], [38], [-27], [-6], [6], [-22], [-7], [-22], [-12], [26], [-40], [21], [-31], [9], [6], [36], [-10], [-28], [0], [-12], [-10], [-21], [35], [-42], [-19], [14], [2], [33], [-16], [20], [-45], [-31], [-36], [-40], [-24], [-54], [-22], [3], [-49], [27], [-7], [-30], [-4], [-48], [20], [15], [-4], [-15], [29], [-48], [-1], [-6], [-34], [6], [-16], [45], [15], [24], [-1], [42], [24], [9], [2], [-10], [-43], [6], [-12], [5], [-24], [8], [-40], [6], [-16], [-6], [-43], [-31], [-16], [-42], [14], [-36], [-9], [39], [-64], [11], [23], [-30], [20], [21], [38], [-33], [27], [60], [5], [44], [6], [69], [18], [-28], [68], [29], [0], [-24], [20], [39], [-27], [-28], [14], [18], [-49], [36], [-60], [29], [-25], [-42], [-3], [-12], [-49], [32], [45], [29], [63], [26], [6], [-24], [29], [-18], [-58], [-16], [-52], [-3], [-31], [39], [59], [-12], [-6], [-60], [3], [44], [-13], [-1], [-54], [18], [-25], [18], [-46], [-51], [-22], [15], [24], [-13], [60], [-24], [74], [-37], [18], [74], [35], [51], [26], [74], [12], [26], [-30], [-57], [77], [30], [14], [-19], [62], [-34], [8], [11], [-57], [-70], [-31], [36], [72], [-40], [-48], [-67], [-43], [-60], [-16], [-72], [-16], [-9], [-54], [-24], [-6], [42], [-82], [9], [8], [72], [30], [80], [14], [24], [-46], [12], [38], [-64], [-75], [59], [-6], [-52], [36], [-45], [30], [-49], [48], [-55], [18], [-48], [74], [54], [20], [-40], [-54], [-40], [-6], [-91], [38], [-37], [21], [54], [-49], [-6], [92], [-60], [-34], [51], [-43], [12], [29], [-52], [44], [-36], [-63], [-82], [-42], [36], [2], [41], [-78], [-12], [-22], [56], [51], [-91], [-25], [-18], [60], [-72], [-12], [75], [8], [39], [0], [-96], [-52], [-46], [12], [-25], [-70], [54], [56], [69], [9], [32], [-52], [-36], [-72], [56], [15], [-46], [81], [96], [53], [-72], [-94], [24], [26], [-88], [11], [15], [-25], [-12], [18], [20], [-18], [-58], [50], [-6], [38], [-30], [-67], [-6], [44], [66], [-51], [56], [44], [51], [11], [18], [56], [36], [-99], [-40], [-76], [-27], [24], [-28], [-60], [-57], [21], [-28], [-72], [-18], [-34], [12], [-51], [30], [-52], [-42]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1620_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1620_2_a_e();".
// 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_1620_2_a_e(:prec:=1)
    chi := MakeCharacter_1620_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_1620_2_a_e();".
// 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_1620_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1620_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1, 1]>,<11,R![0, 1]>,<17,R![6, 1]>],Snew);
    return Vf;
end function;