
// Make newform 210.4.a.i in Magma, downloaded from the LMFDB on 05 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_210_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_210_4_a_i();".
// 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_210_4_a_i();".
// 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_210_a();"
function MakeCharacter_210_a()
    N := 210;
    order := 1;
    char_gens := [71, 127, 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_210_a_Hecke( : Kf := Kf)
    return MakeCharacter_210_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], [-3], [5], [-7], [16], [58], [34], [64], [-16], [62], [60], [150], [474], [-292], [240], [-662], [-324], [-514], [-372], [-412], [-770], [-560], [-852], [1466], [-178], [1278], [-296], [1164], [-778], [-42], [-280], [1180], [-730], [-664], [2974], [-2768], [1890], [-1180], [-1216], [126], [872], [-18], [4420], [-2254], [-750], [3732], [1980], [-6328], [-2596], [4742], [1294], [-2340], [5962], [-1572], [-3910], [-5672], [-1002], [-6140], [70], [-3294], [1852], [-5130], [956], [-3448], [-5850], [3794], [4116], [7506], [-11516], [-11362], [3890], [-1332], [10264], [-7714], [-6020], [-5368], [10526], [15642], [14498], [-13718], [-10484], [-8594], [-420], [-9794], [-1436], [12228], [-6734], [10690], [14902], [17064], [3036], [-13664], [-11744], [-48], [1044], [14432], [-6426], [-11766], [-11900], [-22330], [18396], [-3774], [9412], [11146], [15292], [-22322], [684], [-7806], [-1284], [12706], [9016], [-19386], [14206], [344], [3608], [-18838], [27068], [-18912], [-614], [31248], [-19882], [31866], [19574], [24036], [29496], [-10242], [-13730], [-14840], [-3616], [-24182], [34052], [21192], [-2192], [-39458], [458], [25354], [-25306], [26588], [38862], [6610], [4696], [-20362], [-9888], [-19252], [-12954], [18784], [-4958], [-15326], [49840], [13384], [5006], [14098], [13580], [-14648], [-17996], [-41420], [33640], [-37918], [-5954], [-33066], [28508], [24718], [-52424], [10988], [31446], [33528], [-49856], [9298]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_210_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_210_4_a_i();".
// 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_210_4_a_i(:prec:=1)
    chi := MakeCharacter_210_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_210_4_a_i();".
// 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_210_4_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_210_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<11,R![-16, 1]>,<13,R![-58, 1]>],Snew);
    return Vf;
end function;