
// Make newform 110.4.a.g in Magma, downloaded from the LMFDB on 03 January 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_110_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_110_4_a_g();".
// 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_110_4_a_g();".
// 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_110_a();"
function MakeCharacter_110_a()
    N := 110;
    order := 1;
    char_gens := [67, 101];
    v := [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_110_a_Hecke(Kf)
    return MakeCharacter_110_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]];
    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 := [[2], [7], [-5], [11], [11], [2], [-9], [-85], [-138], [45], [227], [-19], [-138], [-88], [-534], [297], [-450], [287], [-304], [777], [962], [290], [1422], [-1455], [116], [402], [1412], [2136], [230], [-108], [1376], [-1593], [756], [1460], [-2145], [2702], [1421], [-2773], [-2109], [-3558], [300], [3122], [-1548], [827], [456], [2825], [-2233], [5402], [2826], [-2410], [687], [1650], [2882], [-7698], [-6354], [-5223], [-4500], [-1708], [5636], [-5058], [-4858], [9402], [-6574], [-8253], [-4858], [711], [8552], [1121], [10986], [-1090], [462], [-60], [-1924], [-6238], [830], [3762], [-5550], [8786], [-3333], [-12820], [13680], [4952], [16032], [692], [-17920], [15852], [6930], [-619], [-10533], [-11698], [1431], [-4740], [-2104], [-12933], [17540], [7632], [2340], [-16338], [7712], [14537], [14096], [2706], [20502], [12960], [-14443], [-10234], [9741], [-16518], [13635], [-13438], [-15199], [21122], [-2304], [-14020], [22457], [8847], [-1213], [-9774], [-14433], [22905], [2942], [-13813], [8436], [10377], [18602], [-31353], [27470], [-10425], [-26854], [-22228], [-12640], [24507], [17087], [7106], [-2238], [-12730], [-22323], [40376], [-21114], [-38550], [-43063], [-2298], [-5428], [-5004], [-25180], [-8760], [35462], [-12489], [10970], [36942], [-48094], [-28878], [6497], [15696], [9551], [1317], [43460], [40995], [-31174], [-41103], [18861], [4857], [-18859], [-20568], [45636], [-33618], [-42928], [55316]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_110_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_110_4_a_g();".
// 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_110_4_a_g(:prec:=1)
    chi := MakeCharacter_110_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_110_4_a_g();".
// 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_110_4_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_110_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![-7, 1]>,<7,R![-11, 1]>],Snew);
    return Vf;
end function;