
// Make newform 9386.2.a.g in Magma, downloaded from the LMFDB on 02 October 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_9386_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_9386_2_a_g();".
// 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_9386_2_a_g();".
// 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_9386_a();"
function MakeCharacter_9386_a()
    N := 9386;
    order := 1;
    char_gens := [1445, 3251];
    v := [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_9386_a_Hecke( : Kf := Kf)
    return MakeCharacter_9386_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 := 2;
    raw_aps := [[1], [-3], [-4], [-2], [-5], [1], [2], [0], [4], [-8], [-6], [-4], [-7], [-12], [-2], [0], [-3], [-10], [3], [-14], [-7], [10], [-9], [-6], [5], [-4], [-4], [-4], [-2], [9], [-4], [-21], [-9], [-15], [-8], [8], [12], [-11], [-14], [22], [5], [-2], [2], [-2], [18], [0], [16], [4], [-19], [-20], [27], [-26], [17], [27], [-27], [-18], [14], [-10], [-12], [15], [-29], [-14], [7], [-18], [29], [-20], [19], [-3], [-15], [14], [-3], [-4], [-8], [-10], [0], [-18], [16], [-22], [-27], [5], [-24], [26], [8], [-10], [-24], [-5], [25], [-33], [-4], [2], [-21], [6], [-6], [0], [11], [-24], [0], [-21], [-20], [28], [32], [6], [15], [-6], [31], [-7], [-36], [-11], [24], [5], [10], [22], [-27], [-12], [-8], [43], [-23], [26], [6], [-44], [38], [-26], [-26], [12], [20], [26], [22], [30], [46], [26], [43], [-6], [38], [-4], [3], [38], [24], [11], [16], [-39], [28], [-2], [12], [43], [-36], [-30], [-16], [3], [15], [-48], [32], [-5], [-1], [-18], [-3], [34], [-10], [27], [-29], [-4], [-12], [15], [-44], [21], [-15], [56], [28], [6], [14], [-24], [-41], [-2], [-20], [5], [-4], [23], [-45], [-12], [16], [-14], [48], [-52], [-14], [-17], [-54], [-60], [34], [-4], [37], [-58], [45], [12], [-33], [-8], [-21], [-22], [-47], [-32], [9], [30], [54], [-50], [-66], [13], [-43], [36], [-58], [31], [53], [1], [-63], [12], [24], [7], [-46], [-14], [16], [30], [-30], [26], [-74], [-36], [33], [46], [-21], [60], [-27], [6], [22], [-20], [-76], [61], [-36], [25], [-19], [-24], [55], [-46], [-45], [20], [-59], [-25], [-52], [-20], [50], [42], [4], [-57], [-61], [26], [32], [-6], [-46], [17], [-30], [-60], [-56], [-35], [56], [-22], [-10], [63], [50], [38], [-57], [55], [24], [-51], [28], [30], [62], [-28], [-63], [18], [-46], [44], [-43], [32], [-9], [-36], [-60], [4], [-8], [58], [-24], [-26], [37], [78], [-4], [26], [18], [60], [-29], [-12], [2], [-64], [60], [76], [28], [45], [5], [-60], [46], [-33], [5], [41], [55], [-34], [54], [58], [36], [-82], [-57], [23], [18], [38], [65], [18], [29], [30], [57], [11], [-78], [24], [6], [-78], [-45], [19], [56], [-64], [-58], [-72], [-28], [-12], [-55], [36], [72], [-69], [-70], [-56], [-84], [33], [46], [28], [-58], [-39], [10], [-60], [-22], [34], [17], [17], [60], [86], [-6], [-67], [-84], [25], [-43], [-36], [36], [27], [-68], [35], [52], [-86], [48], [-6], [23], [1], [-84], [72], [10], [-40], [2], [28], [-8], [53], [5], [-37], [42], [58], [30], [-55], [-79], [-16], [-20], [-32], [-19], [64], [47], [12], [-65], [31], [102], [35], [-34], [-33], [-89], [-48], [72], [25], [92], [-78], [-28], [-2], [-24], [-35], [76], [-95], [-59], [-8], [-49], [79], [67], [98], [20], [-76], [-18], [54], [-42], [-78], [-4], [-35], [-30], [8], [9], [6], [12], [-76]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9386_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9386_2_a_g();".
// 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_9386_2_a_g(:prec:=1)
    chi := MakeCharacter_9386_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_9386_2_a_g();".
// 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_9386_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9386_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![3, 1]>,<5,R![4, 1]>,<7,R![2, 1]>,<29,R![8, 1]>],Snew);
    return Vf;
end function;