
// Make newform 1617.2.a.j in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1617_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_1617_2_a_j();".
// 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_1617_2_a_j();".
// 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_1617_a();"
function MakeCharacter_1617_a()
    N := 1617;
    order := 1;
    char_gens := [1079, 199, 442];
    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_1617_a_Hecke(Kf)
    return MakeCharacter_1617_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], [1], [2], [0], [1], [2], [2], [0], [8], [-6], [8], [6], [2], [0], [-8], [6], [4], [-6], [-4], [0], [14], [-4], [-12], [6], [-2], [-2], [-8], [-12], [-2], [-6], [-4], [12], [2], [8], [-22], [20], [-14], [4], [0], [6], [12], [-22], [8], [-14], [-14], [0], [0], [-16], [-12], [-6], [30], [24], [-10], [-4], [14], [-16], [2], [-20], [-26], [-18], [-16], [6], [-32], [24], [22], [22], [-20], [-22], [4], [-6], [-18], [-8], [32], [-2], [28], [16], [-18], [2], [26], [-18], [4], [-26], [-24], [-34], [20], [28], [2], [18], [30], [16], [12], [-8], [-16], [4], [-4], [32], [-30], [30], [16], [46], [8], [-14], [44], [-42], [-16], [30], [-28], [-38], [-8], [-26], [4], [14], [-30], [-44], [16], [18], [-20], [-8], [-2], [4], [26], [-46], [-18], [20], [28], [50], [38], [-24], [0], [-30], [8], [40], [-8], [-10], [-6], [-2], [-6], [8], [10], [54], [56], [-14], [24], [20], [2], [56], [34], [10], [36], [48], [6], [-26], [-20], [-8], [12], [24], [-20], [6], [-26], [54], [12], [22], [4], [52], [-6], [-24], [40], [-14], [-22], [-6], [-4], [-46], [-40], [2], [16], [-50], [-64], [-10], [-20], [26], [8], [28], [6], [-54], [24], [-54], [-18], [12], [-46], [40], [46], [-44], [12], [42], [-60], [54], [2], [-2], [-30], [-56], [18], [-60], [42], [-10], [-20], [26], [0], [36], [6], [-4], [18], [-6], [48], [-28], [-24], [-26], [20], [-6], [-48], [30], [22], [-36], [-54], [-16], [-36], [-34], [-18], [0], [20], [4], [-50], [-64], [28], [-58], [-4], [-16], [30], [-6], [52], [-40], [-44], [16], [0], [-66], [-22], [24], [40], [-44], [0], [16], [-50], [10], [0], [70], [-18], [28], [-74], [24], [66], [-58], [4], [-60], [-38], [-6], [30], [-28], [-18], [30], [56], [18], [-62], [36], [-10], [-76], [-26], [-80], [-60], [-26], [-38], [-36], [56], [32], [40], [-14], [32], [48], [18], [66], [-16], [26], [-10], [-68], [30], [-60], [-42], [46], [8], [-22], [-36], [16], [6], [18], [28], [4], [-28], [82], [-60], [18], [-24], [-66], [72], [-50], [54], [-4], [0], [-50], [76], [16], [14], [-34], [32], [58], [-2], [56], [-2], [46], [-36], [-20], [72], [-14], [46], [54], [-44], [-28], [-48], [36], [-50], [26], [86], [-44], [38], [42], [54], [-40], [62], [76], [26], [-52], [0], [42], [-32], [42], [30], [84], [-2], [-6], [-72], [76], [14], [-64], [62], [18], [48], [52], [-12], [10], [78], [68], [-54], [-12], [84], [64], [-14], [20], [-2], [4], [-40], [46], [-30], [30], [26], [-54], [68], [-34], [-40], [48], [24], [-86], [32], [-8], [2], [74], [-76], [-28], [-72], [-14], [52], [-90], [76], [50], [-86], [58], [60], [-42], [-10], [12], [-18], [-26], [52], [4], [30], [58], [-92], [-88], [98], [42], [-40], [24], [-98], [0], [62], [-46], [-72], [44], [-10], [18], [-36], [46], [-28], [32]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1617_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1617_2_a_j();".
// 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_1617_2_a_j(:prec:=1)
    chi := MakeCharacter_1617_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_1617_2_a_j();".
// 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_1617_2_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1617_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>,<5,R![-2, 1]>,<13,R![-2, 1]>,<17,R![-2, 1]>],Snew);
    return Vf;
end function;