
// Make newform 7098.2.a.bn in Magma, downloaded from the LMFDB on 14 May 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_bn();".
// 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_bn();".
// 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 := [-3, 0, 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_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, 0], [1, 0], [0, 2], [1, 0], [4, 0], [0, 0], [3, -2], [4, -2], [4, -1], [8, 0], [-3, 2], [2, 4], [0, -4], [1, -2], [-4, -2], [-10, -1], [8, 1], [0, 3], [-2, 3], [-5, -4], [-4, 4], [-6, 2], [0, -1], [-10, 1], [2, -6], [0, 0], [2, -5], [12, 2], [-12, 0], [6, 6], [12, 4], [19, 0], [0, -8], [-4, -4], [-15, -2], [-12, 0], [4, 4], [-10, -5], [20, 0], [8, 4], [-10, 4], [4, -12], [-4, 13], [4, -8], [-13, -6], [-10, 7], [-16, -4], [11, 2], [4, 10], [13, -4], [10, -2], [-1, -8], [-2, 8], [-17, -4], [-3, -2], [0, 10], [12, -6], [9, -2], [-10, -10], [8, 6], [-8, 2], [30, -2], [-22, 0], [-4, -10], [18, 10], [13, -2], [4, -10], [6, 0], [-2, -16], [-1, 20], [-18, -5], [-8, 0], [-14, 11], [-2, 14], [20, 6], [-10, -10], [-6, 5], [9, -12], [-6, 6], [-4, -6], [-5, 0], [-12, -6], [9, 8], [-4, -8], [-24, -6], [-12, 6], [-18, 0], [4, 3], [-12, -4], [-16, -14], [-5, 20], [6, 6], [-10, -6], [4, 8], [-2, -1], [0, 4], [10, -14], [-30, 0], [8, -4], [0, 10], [-20, 0], [35, 6], [0, 8], [12, -2], [7, 6], [22, 12], [4, 7], [6, 1], [0, 11], [0, 14], [-10, 3], [-24, 6], [8, 8], [10, -12], [-28, 4], [8, 20], [4, 14], [20, -10], [-22, 9], [12, 6], [23, -16], [5, -20], [8, 4], [-20, 18], [10, -18], [10, 7], [-20, -16], [38, 2], [10, 19], [-9, -8], [6, -1], [19, 12], [12, -10], [6, -8], [0, 4], [-8, 20], [6, -4], [-2, -6], [18, -14], [-12, -16], [16, 8], [-7, 14], [34, 6], [26, -6], [28, 12], [12, 6], [13, -20], [-6, 16], [-2, 8], [36, -4], [-2, -16], [-13, -26], [17, 10], [-2, -8], [-15, 2], [-4, -18], [-26, 0], [18, 7], [-38, 4], [-30, -6], [6, -14], [30, 0], [20, 2], [17, -4], [12, -8], [26, 4], [14, 8], [8, -15], [-4, -7], [-18, 2], [-20, 20], [-6, -28], [24, 18], [0, 5], [32, 6], [31, 10], [-2, -9], [-57, -2], [-38, 3], [-4, -35], [-12, 0], [-9, -28], [20, 18], [-42, 5], [25, -8], [12, 0], [-24, 2], [42, -4], [-12, 0], [6, 22], [-8, -22], [-2, 4], [-19, -6], [28, -22], [46, 6], [6, -16], [-8, -8], [8, -26], [6, -9], [46, 2], [-5, -10], [30, 17], [-34, -16], [6, 30], [-24, -10], [-8, 12], [20, 28], [-8, 2], [-10, -20], [-10, 8], [5, -8], [-14, 20], [-20, 12], [-16, -3], [-16, -12], [16, -4], [32, -2], [-23, 10], [-8, -16], [26, -12], [-40, 0], [-20, 4], [-30, 2], [-56, -8], [-13, 8], [32, -22], [-23, 6], [12, -17], [-8, 18], [-18, 6], [-24, 22], [-2, 14], [-40, -4], [32, 4], [10, -22], [-20, 10], [20, -32], [3, 34], [12, -30], [-12, -24], [22, -22], [-50, 2], [10, 15], [-22, 6], [-6, 13], [-20, 8], [0, 20], [24, 18], [14, 23], [32, -6], [-20, -10], [-4, 0], [-1, -24], [22, 24], [-38, 8], [-6, -38], [-34, 14], [-18, 4], [-8, -22], [16, -32], [8, -4], [-2, 8], [34, 8], [16, -23], [-54, -13], [-18, 28], [-5, -10], [-34, -21], [38, 11], [-6, -11], [-36, 19], [50, 15], [18, -2], [-44, 4], [-22, -2], [-48, 16], [-42, -4], [36, -16], [-4, 13], [1, -8], [48, -12], [-42, 4], [24, -2], [34, -16], [-12, -22], [-8, -27], [-69, 0], [-18, -4], [17, 2], [51, 6], [10, -7], [-40, -20], [20, 6], [-36, 19], [-8, -14], [26, -2], [0, -38], [14, 40], [-3, -8], [-60, -2], [8, -30], [-37, -26], [32, 6], [50, -16], [-11, 2], [0, 41], [12, -28], [20, -11], [12, 32], [22, -18], [6, -30], [47, 2], [-12, 26], [-25, 22], [31, 4], [28, 0], [12, -2], [3, 36], [-74, 2], [32, -30], [-16, 22], [-32, 11], [-14, 2], [-2, 0], [-18, 12], [-22, -10], [8, 2], [-14, -22], [-36, -6], [18, 35], [-14, -18], [24, 40], [-48, 22], [76, 5], [-34, 33], [-12, 10], [-52, 12], [6, -17], [-8, 14], [-38, -19], [20, -12], [-46, 14], [-24, -8], [12, -20], [-47, -14], [12, -24], [-64, 1], [-26, 37], [6, -14], [-4, 6], [16, 24], [-20, 32], [-50, 8], [36, 22], [-88, -4], [-10, -4], [-18, 14], [48, -10], [22, -16], [-5, 28], [14, -4], [-30, 28], [16, -27], [28, -4], [-14, 18], [23, 24], [-76, -2], [0, -18], [5, 12], [-6, -18], [34, -20], [-21, 4], [26, -19], [-34, -13], [12, 0], [-28, 5], [-36, 19], [64, 12], [-18, -24], [-22, 20], [10, -8], [-24, 12], [46, 14], [-66, 6], [-34, 30], [12, 6], [22, -2], [32, 21], [46, -9], [54, 20], [44, -3], [30, -4], [42, -34], [38, -22], [-9, 8], [-3, 16], [4, 20], [-22, 36], [45, -2], [-9, -22], [47, 4], [-66, 12], [-34, -14], [-48, 12], [-14, 32], [-26, 3], [-20, -12], [-36, 30], [32, -34], [84, -11], [44, -16], [6, 21], [56, -2], [-15, -10], [22, 20], [0, 16], [-44, 8], [38, -17], [-70, 9], [-54, -2], [-2, 5], [34, 28], [-31, -4], [65, 4], [46, -28], [-18, -34], [-78, -6], [-44, 14], [22, 12], [-42, 30]];
    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_bn();".
// 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_bn(:prec:=2)
    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_bn();".
// 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_bn( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7098_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-12, 0, 1]>,<11,R![-4, 1]>,<17,R![-3, -6, 1]>],Snew);
    return Vf;
end function;