
// Make newform 7098.2.a.m in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7098_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_7098_2_a_m();".
// 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_7098_2_a_m();".
// 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_7098_a();"
function MakeCharacter_7098_a()
    N := 7098;
    order := 1;
    char_gens := [4733, 5071, 6931];
    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_7098_a_Hecke(Kf)
    return MakeCharacter_7098_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 := [[-1], [1], [0], [1], [-3], [0], [-3], [5], [-6], [-9], [8], [8], [3], [8], [3], [-3], [-6], [-7], [2], [-6], [-16], [-13], [18], [-15], [-16], [18], [-4], [9], [2], [-12], [-16], [12], [-6], [11], [-6], [-19], [2], [-16], [0], [-6], [-24], [5], [6], [5], [-3], [-4], [-22], [8], [-6], [5], [-24], [-6], [8], [18], [-15], [6], [0], [20], [-10], [-18], [-4], [0], [-13], [-21], [26], [30], [-10], [23], [3], [2], [18], [24], [-10], [-22], [-10], [-21], [-18], [-25], [6], [-22], [-6], [26], [-18], [14], [32], [33], [24], [38], [0], [-31], [-24], [9], [-13], [-12], [32], [-24], [30], [21], [-19], [-34], [-34], [3], [-36], [-18], [26], [-10], [24], [15], [6], [8], [8], [-40], [-42], [-1], [-13], [18], [-31], [33], [21], [-33], [-22], [-1], [18], [12], [8], [-21], [26], [9], [-16], [-37], [2], [-12], [-1], [-34], [42], [8], [42], [35], [-54], [-54], [8], [-15], [32], [-12], [11], [-12], [-1], [-42], [5], [30], [50], [6], [-16], [57], [-22], [-42], [29], [39], [-28], [0], [3], [-36], [-16], [6], [0], [0], [17], [-19], [-34], [-24], [-45], [-13], [30], [-10], [-52], [51], [2], [9], [2], [-55], [56], [12], [-46], [-9], [12], [-24], [38], [11], [38], [-15], [38], [12], [-34], [0], [12], [-21], [5], [26], [42], [-33], [-9], [-10], [-22], [-34], [60], [6], [-40], [-36], [-6], [5], [11], [30], [29], [-36], [33], [32], [65], [33], [54], [57], [32], [-22], [66], [-1], [-12], [20], [-15], [30], [26], [60], [50], [-1], [5], [-60], [53], [-63], [-58], [27], [36], [3], [-45], [5], [-46], [20], [57], [27], [38], [-72], [38], [-6], [68], [-57], [66], [32], [-30], [-21], [44], [-64], [66], [20], [-7], [-33], [-25], [-7], [33], [44], [-21], [30], [50], [-39], [50], [-76], [44], [65], [-16], [-34], [-21], [44], [-13], [-54], [-36], [32], [-15], [53], [-4], [-66], [-61], [15], [-58], [42], [3], [72], [54], [30], [-28], [84], [32], [-42], [54], [-49], [32], [-3], [83], [-36], [-22], [41], [27], [-37], [-54], [-4], [81], [66], [18], [44], [36], [32], [-66], [48], [-52], [-36], [-19], [-13], [0], [47], [12], [-28], [20], [44], [-54], [78], [32], [-30], [-10], [66], [-88], [24], [-4], [-54], [-40], [-58], [-76], [-18], [-6], [80], [27], [-60], [77], [-28], [3], [-96], [-37], [-25], [54], [14], [-52], [-33], [0], [-6], [60], [-12], [62], [81], [-54], [-81], [-37], [89], [-12], [-37], [-7], [-9], [-1], [-78], [-75], [62], [80], [-66], [30], [-94], [-66], [62], [54], [-12], [-61], [-30], [68], [75], [-28], [-10], [-16], [3], [-10], [-24], [33], [-13], [24], [-46], [38], [33], [-40], [-63], [-34], [66], [-61], [3], [0], [50], [-28], [30], [-13], [30], [86], [-42], [36], [80], [-97], [-18], [-72], [26], [69], [-81], [105], [29], [-48], [-78], [-82], [-6], [-9], [24], [-64], [75]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7098_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7098_2_a_m();".
// 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_7098_2_a_m(:prec:=1)
    chi := MakeCharacter_7098_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_7098_2_a_m();".
// 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_7098_2_a_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7098_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<11,R![3, 1]>,<17,R![3, 1]>],Snew);
    return Vf;
end function;