
// Make newform 1155.2.a.h in Magma, downloaded from the LMFDB on 18 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_h();".
// 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_h();".
// 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 := [[0], [1], [-1], [1], [-1], [-4], [3], [-1], [-3], [-9], [-10], [-4], [6], [-1], [-6], [-9], [15], [-13], [2], [-6], [-16], [8], [-3], [-9], [17], [18], [-19], [-6], [2], [15], [-1], [0], [18], [-4], [-18], [2], [17], [-16], [12], [18], [-6], [-10], [12], [-10], [12], [-16], [-22], [23], [21], [-22], [24], [21], [-10], [12], [-24], [0], [3], [29], [-10], [6], [32], [3], [-10], [0], [11], [-18], [-31], [-13], [-12], [-1], [-6], [-27], [17], [23], [-19], [-12], [-12], [2], [6], [14], [-3], [17], [12], [2], [35], [-24], [-18], [-1], [-6], [-22], [-12], [24], [26], [-9], [-13], [27], [-15], [-15], [-4], [-10], [17], [-18], [-12], [-45], [-4], [14], [-12], [-42], [-30], [5], [14], [26], [-18], [-22], [-43], [-6], [17], [24], [-3], [39], [-22], [41], [15], [-12], [-10], [-51], [-31], [51], [23], [-22], [-22], [30], [-19], [-34], [-30], [-37], [6], [8], [-42], [6], [20], [9], [-4], [24], [50], [3], [-46], [18], [50], [-3], [-31], [-45], [-16], [9], [-28], [36], [-34], [-18], [2], [24], [9], [-6], [11], [3], [3], [-6], [-1], [14], [-28], [-36], [-39], [-22], [15], [41], [14], [30], [8], [0], [-22], [26], [8], [48], [26], [-15], [-3], [-21], [65], [35], [14], [42], [-34], [30], [20], [-27], [-36], [-24], [-40], [56], [-9], [60], [-9], [-37], [47], [14], [3], [-60], [50], [54], [-69], [50], [-25], [9], [44], [-48], [24], [-10], [-55], [54], [24], [21], [-4], [-16], [54], [44], [-33], [20], [-6], [54], [-52], [69], [20], [-28], [32], [3], [41], [9], [8], [12], [-18], [-72], [-12], [11], [-40], [-25], [-21], [12], [56], [-51], [-28], [36], [-34], [-6], [-12], [-58], [9], [33], [-55], [-46], [18], [-22], [-61], [12], [53], [26], [-48], [-52], [12], [21], [-37], [-72], [38], [2], [-7], [-10], [-40], [56], [-15], [-76], [2], [-60], [9], [-64], [-60], [11], [44], [12], [-34], [0], [-70], [-18], [-48], [-12], [-72], [60], [-58], [-36], [8], [-36], [12], [-64], [-34], [72], [-4], [9], [5], [-70], [21], [-40], [-18], [59], [0], [30], [57], [2], [-66], [-7], [-60], [-9], [26], [39], [-64], [-46], [30], [20], [60], [-28], [-4], [-61], [78], [24], [14], [-66], [-85], [12], [-22], [12], [47], [6], [68], [62], [-82], [-3], [42], [8], [-3], [9], [20], [-10], [60], [-6], [41], [80], [-48], [-28], [-52], [51], [-72], [60], [-78], [-21], [29], [42], [84], [-63], [-61], [62], [-81], [-7], [-34], [-24], [8], [-48], [15], [5], [56], [60], [-21], [80], [-36], [-28], [-57], [-99], [-16], [18], [-79], [18], [-13], [-49], [-52], [-24], [89], [-42], [-30], [-37], [30], [53], [20], [-30], [-67], [75], [5], [51], [-91], [-36], [-60], [47], [74], [-90], [53], [-87], [56], [-42], [-36], [-52], [62], [69], [-81], [77], [-60], [-48], [-6], [-70], [57], [30], [-1], [96], [81], [-39], [2], [36]];
    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_h();".
// 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_h(: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_h();".
// 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_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1155_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>,<13,R![4, 1]>,<17,R![-3, 1]>],Snew);
    return Vf;
end function;