
// Make newform 9350.2.a.bb in Magma, downloaded from the LMFDB on 22 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_9350_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_9350_2_a_bb();".
// 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_9350_2_a_bb();".
// 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_9350_a();"
function MakeCharacter_9350_a()
    N := 9350;
    order := 1;
    char_gens := [8977, 3401, 3301];
    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_9350_a_Hecke( : Kf := Kf)
    return MakeCharacter_9350_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], [0], [0], [0], [-1], [6], [-1], [2], [2], [2], [4], [2], [12], [-8], [-8], [-4], [12], [-10], [-2], [-8], [-2], [10], [16], [6], [0], [4], [8], [-12], [-14], [0], [-16], [16], [6], [8], [24], [8], [16], [24], [12], [-2], [4], [-10], [8], [-10], [-6], [28], [12], [12], [4], [26], [-6], [-12], [4], [-24], [-18], [0], [30], [-28], [-14], [30], [28], [-2], [-12], [20], [16], [-6], [0], [-2], [-36], [-8], [-26], [-8], [22], [26], [-28], [0], [-30], [26], [6], [30], [-20], [14], [-18], [-14], [26], [26], [-38], [-18], [4], [20], [-2], [-2], [2], [-26], [4], [-4], [-6], [14], [24], [22], [-28], [-14], [-24], [-30], [40], [2], [-14], [-18], [-24], [20], [-8], [18], [-32], [4], [0], [-18], [12], [-28], [-50], [-18], [-42], [6], [-38], [0], [-12], [-44], [30], [36], [-8], [-22], [30], [0], [12], [12], [42], [18], [40], [-12], [40], [-40], [20], [38], [-14], [-12], [42], [20], [-26], [-50], [0], [-20], [-18], [-6], [38], [48], [12], [40], [-32], [54], [-38], [38], [24], [-54], [-56], [-24], [-34], [14], [-8], [58], [-16], [-22], [-14], [10], [18], [10], [-32], [-14], [48], [-26], [-40], [-8], [56], [-36], [-22], [-54], [24], [-38], [38], [-14], [52], [-42], [8], [-36], [-28], [22], [36], [32], [52], [-14], [42], [-16], [30], [-34], [-20], [18], [56], [-64], [56], [-48], [4], [4], [-22], [-6], [14], [54], [-34], [22], [32], [-66], [-22], [-48], [-44], [-14], [-10], [-2], [52], [-60], [-40], [-40], [48], [12], [-12], [-20], [-52], [-30], [-46], [8], [38], [54], [12], [48], [44], [30], [-24], [22], [18], [-26], [-42], [28], [-26], [-16], [22], [44], [-44], [2], [42], [34], [-70], [20], [2], [-6], [-44], [4], [30], [10], [-60], [56], [-54], [-42], [12], [14], [-14], [2], [18], [16], [-58], [-36], [10], [-28], [38], [-14], [24], [64], [4], [-28], [-84], [-16], [-32], [58], [8], [-54], [-30], [-36], [54], [68], [34], [-14], [-32], [-56], [-48], [64], [-38], [18], [38], [-12], [44], [-56], [78], [-50], [32], [42], [48], [-58], [-20], [-46], [24], [38], [-52], [-42], [-56], [34], [-72], [48], [0], [-48], [-78], [62], [28], [48], [-44], [30], [-14], [-22], [-66], [-12], [-36], [-36], [-14], [-6], [-58], [-8], [20], [-18], [-42], [-40], [-92], [-76], [10], [-14], [66], [-78], [14], [-12], [-14], [-44], [-16], [-6], [40], [56], [-22], [-24], [-6], [76], [36], [16], [34], [-70], [30], [0], [-36], [-88], [20], [-32], [-12], [-60], [22], [52], [90], [46], [-30], [58], [2], [-66], [-80], [-26], [100], [30], [-8], [-4], [12], [-88], [-26], [2], [28], [-46], [-32], [-94], [36], [-10], [12], [-12], [62], [-26], [80], [32], [-36], [-14], [92], [50], [-26], [-36], [-86], [-50], [2], [56], [82], [-34], [42], [-86], [48], [80], [-42], [-90], [78], [66], [24], [-68], [16], [-68], [-12], [-46], [70], [78], [12], [6], [-58], [-34], [-24], [-74], [-88], [0], [92], [-38], [-32], [-4], [80], [6], [112], [52], [26], [32], [52], [24], [12], [-94], [2], [-90], [-8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9350_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9350_2_a_bb();".
// 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_9350_2_a_bb(:prec:=1)
    chi := MakeCharacter_9350_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(3229) - 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_9350_2_a_bb();".
// 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_9350_2_a_bb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9350_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<7,R![0, 1]>,<13,R![-6, 1]>,<19,R![-2, 1]>],Snew);
    return Vf;
end function;