
// Make newform 105.4.a.f in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_105_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_105_4_a_f();".
// 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_105_4_a_f();".
// 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
        poly := [-16, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_105_a();"
function MakeCharacter_105_a()
    N := 105;
    order := 1;
    char_gens := [71, 22, 31];
    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_105_a_Hecke(Kf)
    return MakeCharacter_105_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 := 4;
    raw_aps := [[0, 1], [3, 0], [5, 0], [-7, 0], [-10, -2], [-12, 2], [66, -16], [58, -14], [140, -20], [-74, -48], [54, 42], [-30, -36], [-138, 100], [-156, -32], [304, -48], [120, 86], [-464, 84], [-114, 24], [-84, 64], [814, 42], [-92, -202], [-392, -104], [356, 216], [290, 8], [104, 314], [-30, 28], [1032, 64], [332, 312], [-230, 156], [92, 342], [-1424, 128], [-1020, -392], [0, -454], [602, -518], [802, -436], [-508, 452], [-1448, -242], [964, -344], [856, 16], [-950, 140], [1246, -594], [458, -508], [-670, 94], [-3038, 288], [-2732, 682], [-1074, -470], [1572, 744], [4012, -188], [1460, -1128], [-2090, -440], [-1716, -434], [1526, 250], [-5346, 116], [-2152, 844], [1214, -1084], [7464, -32], [218, -228], [-3502, -354], [-4238, 1244], [-7338, 52], [1964, 552], [7210, 84], [-144, 2124], [7368, -64], [-2680, 1826], [-1208, 1054], [-1236, -696], [-3886, 1376], [2824, -500], [-826, 896], [-1934, -1232], [-4430, -474], [9884, -348], [1430, -1320], [1456, -716], [-416, -208], [6442, 724], [604, -1190], [6274, 752], [3682, 1640], [-4420, 704], [-10834, -336], [-10538, 698], [3832, -998], [7566, 186], [-5448, 2652], [-4694, 1688], [-6858, -284], [2894, -88], [-2312, -760], [-12916, 1936], [-14632, 1736], [12704, -1400], [-6266, -178], [-17912, -124], [9168, 488], [754, 1172], [2846, -36], [-376, -868], [-6322, 2296], [11636, -616], [-8272, -634], [-13484, -1848], [2762, 4576], [320, 2372], [-3088, 5202], [1060, -5408], [2266, 472], [-1630, 1094], [7414, -4780], [6716, 1044], [-19382, -12], [-12220, -1834], [-2078, 1746], [8392, 3616], [18574, 1908], [3920, 700], [7256, 4528], [-2988, 5378], [-23150, 1306], [6822, 4360], [1382, -7300], [8170, 2708], [-14904, -1764], [-22450, -290], [25462, 176], [12638, 32], [3692, -6908], [5660, 5660], [-3124, -2790], [-184, 916], [8152, 16], [15848, 680], [-16746, -4008], [9226, -5056], [19526, 4924], [23138, -772], [-7272, 7108], [626, -396], [-8442, -5900], [1554, -3710], [6430, -4944], [-1240, 784], [-20788, 6888], [3298, -3644], [-18620, 1876], [-1460, 8130], [-4594, 2492], [70, -5378], [5604, 4268], [14082, -6700], [26274, 80], [12612, -7592], [14112, -9032], [5212, 2280], [43754, 1094], [88, 208], [29154, -6240], [8564, -6010], [-12218, 9232], [39824, -4212], [-24308, -4738], [43632, -2024], [-18576, -924], [28308, -5186], [-13672, -3072], [-224, -4864], [26708, 810]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_105_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_105_4_a_f();".
// 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_105_4_a_f(:prec:=2)
    chi := MakeCharacter_105_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_105_4_a_f();".
// 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_105_4_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_105_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![-16, -1, 1]>],Snew);
    return Vf;
end function;