
// Make newform 1155.2.a.l in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1155_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_1155_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_1155_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_1155_a();"
function MakeCharacter_1155_a()
    N := 1155;
    order := 1;
    char_gens := [386, 232, 661, 211];
    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_1155_a_Hecke(Kf)
    return MakeCharacter_1155_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], [-1], [-1], [1], [-2], [-2], [-4], [-4], [-6], [-4], [-2], [-6], [-4], [8], [2], [-12], [-2], [-4], [16], [2], [-4], [0], [6], [14], [-2], [0], [4], [-2], [-10], [8], [12], [14], [4], [2], [-12], [-22], [4], [-20], [2], [4], [-10], [24], [6], [26], [-28], [8], [-16], [0], [-2], [-10], [-16], [-14], [12], [14], [8], [-22], [-32], [-22], [30], [4], [-22], [-20], [-24], [6], [-6], [4], [14], [20], [-2], [14], [16], [16], [2], [4], [24], [-26], [2], [18], [-38], [20], [-26], [0], [22], [0], [24], [-14], [6], [-2], [-8], [-12], [-8], [16], [-20], [20], [-28], [-6], [-10], [12], [-2], [28], [-30], [-32], [-26], [-32], [-42], [-28], [-10], [40], [-22], [16], [-38], [-10], [-32], [-40], [18], [12], [24], [10], [12], [-10], [14], [-14], [8], [8], [-30], [38], [0], [16], [22], [-24], [-40], [-24], [46], [-6], [18], [-6], [-4], [-30], [-18], [-28], [-38], [8], [-28], [-18], [48], [46], [22], [32], [12], [2], [-18], [44], [-36], [-28], [56], [44], [6], [-38], [-50], [-8], [-42], [8], [20], [-18], [24], [16], [46], [-14], [42], [44], [22], [-40], [46], [52], [-42], [40], [14], [48], [-2], [-32], [12], [-18], [62], [36], [-54], [-22], [28], [34], [0], [46], [36], [4], [26], [52], [6], [10], [-66], [22], [44], [10], [-32], [-62], [50], [44], [-30], [-20], [60], [-50], [56], [-58], [2], [16], [-28], [-48], [50], [48], [-30], [-12], [-6], [-50], [56], [34], [-64], [0], [-66], [6], [32], [48], [-60], [-26], [52], [-44], [30], [52], [-12], [-22], [-54], [20], [48], [56], [-28], [32], [-50], [-18], [0], [-64], [-36], [72], [48], [-14], [-30], [4], [50], [42], [44], [-10], [-44], [42], [18], [-24], [-36], [-42], [-10], [-34], [-48], [54], [-42], [4], [-54], [22], [-44], [-26], [-60], [-38], [-40], [-72], [-50], [2], [-20], [60], [0], [12], [-10], [-36], [-32], [66], [-54], [-60], [-22], [78], [-12], [-42], [28], [-14], [-18], [36], [42], [-44], [-28], [74], [18], [12], [-64], [28], [34], [16], [-26], [80], [18], [-12], [-2], [-10], [-44], [-24], [-54], [-44], [64], [-2], [78], [-76], [42], [-26], [16], [38], [-14], [-60], [-12], [48], [-6], [-10], [-6], [56], [-72], [48], [28], [62], [-66], [-6], [32], [14], [54], [46], [-56], [-38], [-36], [62], [76], [-56], [42], [20], [-62], [-90], [-8], [22], [54], [8], [-52], [78], [60], [42], [-78], [-76], [84], [-52], [-22], [58], [48], [82], [-44], [16], [-48], [42], [-64], [-18], [28], [-8], [58], [-6], [74], [-54], [46], [-40], [-18], [44], [-24], [40], [50], [-92], [48], [-30], [-22], [-76], [-52], [40], [94], [56], [-66], [44], [18], [-34], [-90], [-24], [6], [-18], [-24], [78], [14], [-84], [-44], [82], [42], [-16], [-24], [-58], [-78], [-8], [56], [62], [-76], [-58], [54], [-12], [36], [-58], [-46], [-48], [30], [96], [-40]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1155_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1155_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_1155_2_a_l(:prec:=1)
    chi := MakeCharacter_1155_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_1155_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_1155_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1155_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>,<13,R![2, 1]>,<17,R![2, 1]>],Snew);
    return Vf;
end function;