
// Make newform 1155.2.a.k in Magma, downloaded from the LMFDB on 20 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_k();".
// 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_k();".
// 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], [-6], [4], [-4], [-2], [-4], [-2], [-2], [-12], [0], [-2], [0], [-14], [-8], [-16], [10], [16], [-4], [-6], [2], [10], [8], [4], [-10], [-6], [8], [20], [2], [20], [6], [-16], [-2], [-8], [0], [-10], [20], [14], [-8], [14], [-22], [28], [20], [24], [20], [14], [-26], [-24], [-10], [0], [-14], [0], [-26], [-8], [2], [-6], [-4], [-18], [12], [20], [-6], [-18], [-12], [-26], [-4], [18], [-30], [24], [16], [-6], [-20], [-8], [-26], [-18], [18], [14], [16], [6], [16], [-14], [-32], [-8], [18], [30], [-6], [4], [-28], [-32], [4], [-36], [4], [16], [6], [34], [20], [30], [20], [18], [-12], [34], [-28], [-14], [-36], [-14], [16], [-18], [40], [2], [18], [-48], [8], [18], [-12], [-24], [-42], [-36], [-10], [-26], [6], [-16], [16], [-2], [6], [36], [-8], [-34], [36], [0], [-40], [-42], [30], [-18], [6], [4], [-34], [18], [-28], [-2], [-20], [28], [-34], [-12], [14], [10], [0], [4], [-6], [26], [0], [-8], [0], [0], [56], [-6], [34], [-22], [-48], [-34], [56], [-8], [58], [-56], [-16], [46], [-6], [-26], [-36], [46], [48], [-42], [-4], [-30], [-36], [22], [48], [2], [44], [-28], [-26], [42], [60], [-34], [2], [28], [34], [-8], [18], [-20], [-36], [-18], [60], [26], [50], [-26], [-22], [24], [6], [-8], [-26], [30], [32], [-2], [52], [-36], [-14], [-16], [62], [-50], [-20], [52], [-56], [10], [40], [22], [-4], [22], [-26], [-40], [66], [-44], [12], [-18], [42], [-56], [72], [4], [-18], [52], [40], [-30], [-60], [-8], [10], [66], [12], [-60], [-24], [52], [-48], [14], [-30], [-48], [-56], [-48], [-68], [8], [-38], [54], [-20], [-22], [78], [-60], [-42], [4], [38], [-38], [-64], [-28], [-38], [14], [-14], [0], [-66], [-30], [-60], [-22], [-50], [24], [42], [-48], [26], [-16], [40], [-74], [-62], [-52], [-24], [24], [-48], [-6], [68], [32], [-54], [-46], [-44], [30], [-18], [28], [-58], [-60], [-70], [10], [68], [-42], [20], [4], [34], [-6], [40], [80], [28], [-62], [8], [-18], [72], [26], [-48], [-42], [74], [24], [-8], [-26], [24], [48], [-14], [26], [4], [-30], [-30], [76], [30], [-30], [-4], [76], [72], [34], [-10], [46], [56], [-28], [52], [36], [30], [34], [-70], [-40], [-26], [-22], [90], [-80], [86], [-4], [30], [24], [8], [-58], [-28], [10], [62], [24], [46], [-30], [-12], [60], [54], [-68], [-62], [-42], [-44], [-4], [-52], [-62], [-58], [16], [82], [60], [-8], [8], [-42], [-80], [-50], [-40], [-8], [82], [-66], [42], [-18], [-54], [8], [30], [-76], [56], [16], [-74], [-44], [-72], [-62], [22], [-52], [52], [88], [70], [-72], [90], [52], [-74], [-70], [-6], [-32], [-70], [42], [72], [-82], [-14], [84], [-72], [10], [74], [-8], [-12], [-2], [-2], [32], [-56], [82], [24], [46], [-50], [-60], [36], [10], [-66], [-8], [66], [-40], [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_k();".
// 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_k(: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_k();".
// 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_k( : 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![6, 1]>],Snew);
    return Vf;
end function;