
// Make newform 98.4.c.c in Magma, downloaded from the LMFDB on 14 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_98_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_98_c_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_98_4_c_c();".
// 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_98_4_c_c();".
// 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 := [1, -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_98_c();"
function MakeCharacter_98_c()
    N := 98;
    order := 3;
    char_gens := [3];
    v := [2];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_98_c_Hecke();"
function MakeCharacter_98_c_Hecke( : Kf := false)
    N := 98;
    order := 3;
    char_gens := [3];
    char_values := [[0, -1]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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]];
    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, -2], [2, -2], [0, 12], [0, 0], [-48, 48], [56, 0], [114, -114], [0, -2], [0, 120], [-54, 0], [-236, 236], [0, -146], [126, 0], [-376, 0], [0, 12], [-174, 174], [-138, 138], [0, -380], [484, -484], [576, 0], [1150, -1150], [0, -776], [378, 0], [0, 390], [-1330, 0], [1500, -1500], [0, -380], [0, -636], [-146, 146], [198, 0], [-376, 0], [0, -2130], [78, -78], [-2338, 0], [0, 1002], [2752, -2752], [520, -520], [0, -1280], [1764, 0], [0, 768], [-1812, 1812], [-448, 0], [0, 2136], [-4430, 4430], [198, 0], [2284, -2284], [4412, 0], [2072, 0], [366, -366], [0, 376], [0, 2262], [2592, 0], [-110, 110], [-1890, 0], [0, -2130], [4992, -4992], [-6816, 6816], [0, -8192], [-2414, 2414], [1962, 0], [-5402, 5402], [-4788, 0], [-574, 0], [8808, -8808], [0, 2770], [0, -7566], [0, 11320], [-4786, 0], [-12648, 12648], [9632, 0], [3390, -3390], [0, 10704], [8584, -8584], [0, 2122], [-4912, 0], [0, -9060], [-8994, 8994], [0, 12976], [0, 3522], [-12710, 12710], [1638, 0], [-12850, 0], [8016, -8016], [2198, 0], [0, 376], [0, -7188], [-14670, 0], [0, 5146], [-1512, 0], [7184, 0], [0, 16518], [-10092, 10092], [-7832, 7832], [-6732, 0], [0, -18668], [-6048, 0], [0, -11328], [4146, -4146], [0, 1006], [0, 14722], [-13480, 0], [-6222, 6222], [-4926, 4926], [0, -22182], [-3296, 3296], [24334, -24334], [1638, 0], [0, 7446], [6504, -6504], [16058, 0], [0, -10208], [14974, -14974], [7254, 0], [-12458, 12458], [28352, 0], [-27390, 27390], [-21490, 0], [-17652, 17652], [0, 4782], [-27144, 0], [11860, -11860], [5546, 0], [0, 14880], [-20964, 20964], [0, -13106], [-4590, 0], [0, 862], [0, 3540], [-4228, 0], [0, -5420], [-1280, 1280], [-35712, 0], [0, -24464], [30242, 0], [0, 2154], [10262, 0], [-9084, 9084], [19798, -19798], [30240, 0], [2346, -2346], [-29806, 0], [0, 1506], [20392, -20392], [36108, 0], [13876, -13876], [23436, 0], [8120, 0], [50010, -50010], [0, -34526], [0, 17256], [0, -8714], [-22806, 0], [40196, 0], [0, -40812], [13588, -13588], [-47304, 0], [0, -1784], [0, 35922], [-26782, 0], [-4044, 4044], [0, 2136], [-15174, 0], [25832, 0], [0, 37686], [54006, -54006], [-33276, 33276], [3760, -3760], [-36524, 36524]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_98_c_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_98_4_c_c();".
// 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_98_4_c_c(:prec:=2)
    chi := MakeCharacter_98_c();
    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_98_4_c_c();".
// 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_98_4_c_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_98_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![4, -2, 1]>],Snew);
    return Vf;
end function;