
// Make newform 1155.2.a.a 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_a();".
// 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_a();".
// 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 := [[-2], [-1], [1], [1], [-1], [-2], [-3], [1], [-7], [1], [8], [-2], [-8], [9], [-12], [-5], [3], [13], [-14], [8], [-8], [-14], [11], [-9], [5], [-14], [-7], [-8], [-16], [-9], [5], [2], [-10], [8], [-6], [14], [-23], [-14], [0], [2], [-4], [2], [4], [26], [-16], [-20], [-22], [15], [11], [-4], [10], [15], [2], [-12], [-14], [12], [-17], [7], [-22], [18], [14], [9], [30], [-4], [15], [30], [-27], [-23], [-28], [-27], [30], [3], [25], [-15], [13], [-14], [-2], [-30], [-12], [26], [1], [9], [16], [-38], [1], [4], [-12], [-3], [6], [-2], [32], [10], [34], [-15], [-41], [25], [-3], [1], [8], [-12], [-25], [-42], [12], [37], [2], [34], [18], [-14], [-14], [-9], [22], [2], [-42], [-6], [5], [30], [-3], [-12], [9], [-3], [32], [19], [9], [44], [-2], [43], [21], [15], [31], [38], [-6], [-18], [17], [6], [-30], [9], [-42], [52], [38], [-6], [-28], [7], [-50], [-20], [38], [-9], [-16], [22], [0], [1], [3], [47], [0], [55], [-6], [-36], [26], [54], [52], [-34], [57], [14], [5], [43], [-41], [22], [47], [-38], [-6], [10], [-33], [-20], [21], [-33], [38], [30], [42], [-20], [-30], [-34], [-4], [20], [-26], [27], [-39], [11], [23], [31], [-32], [52], [42], [-38], [-48], [-3], [-42], [-4], [-40], [-26], [5], [-24], [-63], [-35], [7], [-42], [-49], [-8], [-44], [-18], [13], [44], [-31], [-23], [52], [-14], [-44], [-26], [7], [22], [-28], [-23], [40], [-40], [60], [-2], [-75], [-66], [12], [40], [48], [-29], [54], [-20], [22], [-3], [-63], [-65], [-50], [24], [30], [-36], [60], [-59], [0], [23], [-51], [-30], [-8], [-39], [-14], [8], [22], [48], [-20], [44], [39], [75], [-27], [-20], [-40], [-74], [29], [-52], [-41], [80], [10], [6], [-24], [33], [57], [38], [-14], [66], [-39], [-60], [-22], [-16], [37], [-44], [40], [20], [3], [-60], [60], [-51], [50], [-70], [12], [8], [-2], [60], [0], [34], [-34], [36], [-34], [52], [-40], [-54], [-4], [-20], [34], [-18], [34], [-43], [-23], [28], [77], [-30], [24], [17], [-24], [12], [23], [-48], [-68], [7], [-24], [-3], [-62], [41], [88], [-72], [-30], [34], [-24], [-48], [44], [7], [24], [-86], [44], [34], [-31], [-28], [-62], [-24], [-9], [10], [26], [56], [-56], [29], [0], [-56], [89], [11], [-66], [72], [80], [-52], [11], [16], [38], [-6], [28], [-51], [0], [-54], [-30], [-89], [-65], [-78], [-56], [-25], [-61], [94], [-7], [19], [40], [72], [40], [-100], [57], [-89], [10], [68], [-35], [-20], [-18], [78], [-45], [93], [32], [-36], [35], [-64], [-41], [-5], [16], [84], [61], [-38], [44], [-89], [58], [43], [-24], [42], [73], [-23], [35], [99], [19], [2], [-66], [-75], [-40], [-2], [-3], [-15], [28], [-52], [-44], [-6], [-62], [61], [5], [-11], [-62], [84], [88], [-2], [21], [96], [-5], [66], [13], [99], [14], [-62]];
    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_a();".
// 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_a(: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_a();".
// 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_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1155_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![2, 1]>,<13,R![2, 1]>,<17,R![3, 1]>],Snew);
    return Vf;
end function;