
// Make newform 9633.2.a.h in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9633_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_9633_2_a_h();".
// 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_9633_2_a_h();".
// 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_9633_a();"
function MakeCharacter_9633_a()
    N := 9633;
    order := 1;
    char_gens := [3212, 4903, 1522];
    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_9633_a_Hecke(Kf)
    return MakeCharacter_9633_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], [2], [0], [0], [0], [-6], [1], [4], [2], [-8], [10], [2], [-4], [-12], [-6], [12], [-2], [4], [0], [-10], [0], [-16], [2], [-10], [-10], [8], [4], [10], [6], [-8], [8], [-18], [4], [-6], [8], [-2], [4], [-24], [-22], [-4], [14], [-12], [14], [2], [-8], [-4], [-16], [12], [-6], [10], [12], [6], [-24], [14], [12], [-6], [0], [22], [10], [-20], [14], [12], [4], [-22], [6], [-12], [-22], [0], [2], [22], [20], [32], [-10], [-12], [-8], [30], [-14], [-38], [14], [8], [-14], [24], [-14], [-8], [0], [2], [6], [18], [-32], [-32], [-20], [-32], [-32], [-28], [12], [22], [14], [-28], [-14], [4], [-30], [20], [6], [44], [-18], [-8], [30], [-24], [10], [24], [-6], [-2], [28], [32], [38], [-20], [36], [14], [-44], [-6], [-46], [-22], [-36], [-20], [-18], [-38], [-20], [8], [-46], [36], [0], [16], [38], [-50], [-18], [-18], [-44], [-6], [26], [-44], [42], [-32], [28], [-10], [0], [-22], [30], [-28], [40], [34], [-38], [-36], [-40], [4], [-16], [-16], [-34], [-22], [22], [8], [38], [32], [-36], [42], [8], [40], [-58], [6], [54], [-4], [2], [52], [-10], [56], [-38], [-4], [26], [-48], [-18], [-32], [36], [-34], [42], [-12], [-10], [6], [4], [-2], [4], [-38], [-16], [4], [14], [-24], [-2], [46], [-50], [30], [44], [14], [-40], [18], [-22], [0], [30], [-64], [28], [-50], [28], [34], [30], [8], [12], [0], [-26], [-16], [6], [0], [-6], [-50], [8], [42], [72], [52], [-10], [-42], [-24], [32], [-48], [30], [-60], [-40], [-18], [52], [-28], [46], [2], [0], [-48], [-12], [60], [-8], [-14], [34], [28], [8], [-68], [68], [28], [-14], [-2], [-52], [66], [-42], [-72], [22], [-28], [-54], [-42], [32], [-28], [10], [78], [30], [-36], [46], [22], [20], [-2], [-18], [-28], [-10], [24], [42], [-16], [72], [-30], [54], [-48], [24], [-16], [12], [26], [-12], [60], [-46], [-18], [-80], [-50], [46], [48], [26], [60], [14], [54], [-32], [-2], [-84], [52], [26], [30], [16], [-4], [36], [54], [-36], [70], [60], [-26], [4], [34], [38], [-44], [12], [50], [24], [80], [-50], [26], [-4], [38], [-14], [-16], [-18], [42], [28], [44], [24], [-10], [2], [66], [8], [72], [-12], [-8], [-14], [26], [86], [-16], [74], [-62], [6], [-16], [-34], [20], [-42], [-52], [48], [70], [-92], [46], [-6], [-24], [22], [-66], [60], [-40], [90], [-48], [58], [-78], [-24], [12], [76], [74], [-14], [-32], [-6], [80], [-20], [-28], [-6], [-24], [-2], [-52], [-20], [-34], [66], [2], [78], [102], [-40], [78], [-44], [-32], [32], [-10], [76], [96], [38], [54], [-16], [-4], [-32], [-30], [-8], [6], [-12], [82], [66], [2], [64], [-54], [-18], [-88], [-78], [-30], [-84], [8], [34], [62], [60], [68], [10], [30], [-80], [-32], [-106], [-64], [18], [106], [-28], [12], [54], [6], [92], [-42], [-100], [-12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9633_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9633_2_a_h();".
// 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_9633_2_a_h(:prec:=1)
    chi := MakeCharacter_9633_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_9633_2_a_h();".
// 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_9633_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9633_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 1]>,<5,R![-2, 1]>,<7,R![0, 1]>],Snew);
    return Vf;
end function;