
// Make newform 2205.2.a.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2205_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_2205_2_a_d();".
// 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_2205_2_a_d();".
// 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_2205_a();"
function MakeCharacter_2205_a()
    N := 2205;
    order := 1;
    char_gens := [1226, 442, 1081];
    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_2205_a_Hecke(Kf)
    return MakeCharacter_2205_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], [0], [-1], [0], [0], [1], [6], [-5], [-6], [6], [-5], [-7], [12], [-1], [6], [0], [-6], [-2], [-7], [-12], [-11], [-13], [-12], [6], [10], [0], [1], [-18], [-7], [-18], [11], [-6], [-6], [-5], [24], [-16], [10], [-4], [12], [0], [-18], [7], [6], [5], [-6], [16], [-4], [4], [12], [-29], [-6], [6], [-2], [-18], [-12], [12], [-18], [16], [23], [24], [-11], [12], [7], [12], [-17], [-6], [23], [-13], [24], [-26], [0], [6], [19], [17], [-25], [0], [-12], [-29], [-24], [-5], [-30], [-7], [-6], [-29], [28], [36], [-30], [11], [-6], [-7], [30], [0], [-7], [12], [17], [-6], [-18], [-24], [43], [-19], [8], [-18], [-18], [24], [-7], [7], [-18], [36], [-12], [1], [19], [38], [42], [-35], [-40], [30], [25], [-24], [24], [-36], [13], [-31], [-18], [-18], [-11], [-6], [-10], [6], [7], [13], [-7], [24], [-19], [50], [-42], [13], [-24], [28], [18], [-18], [28], [24], [-40], [6], [19], [-30], [-17], [42], [4], [6], [2], [12], [-31], [12], [-31], [18], [47], [-12], [-29], [-30], [-48], [24], [-55], [-24], [12], [18], [-43], [31], [-7], [6], [12], [-50], [30], [-13], [-53], [60], [-1], [-30], [16], [1], [-1], [30], [23], [24], [54], [48], [-46], [-53], [-31], [24], [13], [-36], [8], [12], [-66], [54], [11], [26], [18], [54], [18], [13], [-50], [-26], [36], [-42], [16], [30], [12], [-8], [-7], [18], [-40], [0], [-66], [25], [8], [-12], [60], [-42], [-31], [-17], [12], [-16], [42], [26], [-24], [30], [19], [-12], [-25], [52], [23], [-36], [-35], [0], [-65], [12], [0], [-24], [24], [-29], [49], [-46], [24], [60], [-29], [24], [20], [-36], [11], [60], [-42], [67], [12], [-6], [29], [-53], [-60], [-38], [65], [12], [19], [7], [-78], [-17], [48], [-66], [68], [12], [-11], [20], [-35], [11], [-38], [40], [12], [-1], [71], [12], [12], [23], [-18], [49], [16], [60], [59], [-60], [43], [12], [72], [12], [-48], [18], [29], [54], [55], [54], [-24], [-5], [58], [-54], [32], [12], [-31], [-25], [30], [-5], [-48], [53], [78], [-24], [12], [-76], [84], [-29], [12], [-6], [-59], [84], [-17], [77], [-18], [17], [42], [-29], [-61], [61], [48], [-54], [77], [-60], [-89], [48], [11], [6], [50], [18], [-77], [73], [89], [6], [-24], [5], [72], [66], [67], [20], [36], [-90], [-92], [-70], [-54], [-8], [-25], [-6], [42], [-36], [0], [-24], [62], [-60], [-30], [-90], [-35], [-22], [-54], [-73], [-34], [12], [43], [-60], [60], [61], [-25], [30], [-12], [70], [18], [37], [-96], [54], [56], [48], [40], [12], [47], [-86], [-19], [-72], [-31], [60], [-6], [-17], [24], [-10], [4], [90], [-13], [-54], [22], [-18], [-16], [-84], [48], [-77], [26], [-36], [-11], [-24], [22], [48], [42], [-76], [-85], [-72], [-24], [97], [-12], [24], [-24], [7], [-6], [-30], [-2], [-60], [54], [-84], [-20], [42]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2205_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2205_2_a_d();".
// 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_2205_2_a_d(:prec:=1)
    chi := MakeCharacter_2205_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_2205_2_a_d();".
// 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_2205_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2205_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>,<11,R![0, 1]>,<13,R![-1, 1]>,<17,R![-6, 1]>],Snew);
    return Vf;
end function;