
// Make newform 9800.2.a.bn in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9800_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_9800_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_9800_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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_9800_a();"
function MakeCharacter_9800_a()
    N := 9800;
    order := 1;
    char_gens := [7351, 4901, 1177, 5001];
    v := [1, 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_9800_a_Hecke(Kf)
    return MakeCharacter_9800_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 := [[0], [2], [0], [0], [4], [2], [0], [-2], [4], [10], [-4], [2], [12], [4], [4], [-2], [-10], [-6], [-4], [-12], [-4], [-4], [14], [-8], [-8], [-14], [20], [4], [10], [18], [12], [22], [18], [2], [-10], [-8], [-14], [-20], [-4], [18], [12], [22], [4], [18], [6], [-20], [-4], [8], [14], [14], [10], [16], [-4], [-14], [-28], [16], [6], [16], [-26], [-10], [-10], [22], [14], [0], [-16], [22], [-20], [-2], [-4], [22], [12], [-16], [-8], [30], [20], [-12], [18], [-22], [18], [20], [-14], [14], [24], [-32], [-16], [-28], [18], [-6], [-42], [16], [-26], [-36], [28], [-36], [4], [-24], [22], [-20], [-18], [30], [44], [30], [2], [-6], [-12], [-12], [18], [-44], [12], [-8], [32], [26], [-42], [30], [-20], [-14], [-30], [12], [-6], [-28], [-14], [26], [-14], [36], [-42], [10], [-6], [-36], [4], [14], [20], [-36], [16], [-6], [12], [4], [46], [-42], [50], [22], [-26], [-2], [8], [-36], [-46], [20], [-14], [32], [-14], [-24], [18], [-8], [-36], [-12], [-28], [0], [-28], [-36], [-12], [10], [-28], [-18], [20], [6], [-6], [36], [36], [6], [30], [34], [-44], [-26], [-28], [-54], [28], [-48], [44], [-6], [8], [-14], [-44], [58], [34], [36], [-12], [-50], [-2], [-26], [6], [-60], [-36], [12], [60], [6], [-60], [36], [-22], [26], [-12], [16], [-18], [20], [-34], [-8], [-18], [-10], [-12], [-36], [30], [10], [46], [-14], [-68], [-42], [-16], [52], [32], [-36], [-16], [42], [66], [-64], [66], [-12], [54], [-70], [36], [-40], [-56], [-68], [-22], [-30], [28], [26], [14], [-48], [-20], [38], [12], [24], [-4], [-10], [-24], [-34], [-4], [60], [0], [-26], [-44], [0], [14], [40], [72], [-16], [-42], [36], [42], [-62], [-18], [-48], [-60], [-36], [34], [54], [0], [-6], [-10], [-48], [-12], [-54], [14], [-44], [12], [-64], [-44], [-24], [-52], [-34], [62], [22], [-40], [-12], [-28], [10], [-26], [-72], [-10], [-58], [-76], [56], [30], [62], [-10], [22], [-50], [-10], [48], [-6], [34], [58], [40], [2], [68], [-60], [68], [26], [-68], [-22], [-56], [46], [-8], [26], [-18], [-60], [40], [56], [-22], [12], [-44], [-18], [74], [-46], [18], [8], [26], [92], [4], [70], [-68], [-6], [-2], [-54], [-28], [50], [-20], [-42], [-6], [-12], [-16], [-12], [6], [22], [54], [84], [-78], [-36], [-26], [68], [-40], [-74], [-54], [14], [14], [20], [-10], [12], [12], [-74], [38], [4], [78], [-64], [44], [36], [38], [54], [26], [-4], [-10], [-20], [-30], [48], [-90], [80], [30], [-90], [48], [44], [-44], [64], [-42], [74], [64], [94], [26], [-80], [-20], [70], [-4], [-76], [-22], [-42], [-60], [82], [40], [-30], [-20], [-44], [4], [2], [-6], [-90], [80], [40], [70], [-12], [-98], [-66], [-46], [-14], [-44], [58], [-60], [4], [58], [-90], [-84], [84], [80], [88], [58], [-34], [-64], [50], [-96], [-42], [36], [-90], [-62], [44], [-76], [-52], [-4], [-96], [86], [68], [-14], [34], [-84], [-44], [58], [10], [50], [36], [-72], [-62], [14], [-36], [12], [2], [52], [96], [60], [52], [34], [-74], [-46], [90], [-74], [42], [4], [32], [-108], [102], [-30], [-94], [-16], [-14], [18], [74], [-44], [-84], [-48], [70]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9800_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9800_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_9800_2_a_bn(:prec:=1)
    chi := MakeCharacter_9800_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(3361) - 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_9800_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_9800_2_a_bn( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 1]>,<11,R![-4, 1]>,<13,R![-2, 1]>,<19,R![2, 1]>,<23,R![-4, 1]>],Snew);
    return Vf;
end function;