
// Make newform 9408.2.a.u in Magma, downloaded from the LMFDB on 27 July 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_9408_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_9408_2_a_u();".
// 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_9408_2_a_u();".
// 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_9408_a();"
function MakeCharacter_9408_a()
    N := 9408;
    order := 1;
    char_gens := [1471, 6469, 3137, 4609];
    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_9408_a_Hecke( : Kf := Kf)
    return MakeCharacter_9408_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 := [[0], [-1], [0], [0], [0], [-4], [4], [4], [4], [-2], [8], [6], [12], [-4], [-8], [-6], [-12], [-4], [4], [-12], [8], [-16], [4], [4], [16], [8], [-8], [-8], [14], [2], [8], [4], [22], [12], [-6], [-8], [20], [12], [-8], [-16], [24], [12], [12], [-14], [-6], [-8], [-20], [-16], [20], [-20], [6], [-20], [0], [12], [-12], [12], [24], [16], [10], [22], [4], [24], [-12], [-16], [-8], [18], [-4], [-18], [24], [36], [28], [28], [-16], [10], [12], [24], [-26], [20], [-18], [32], [-20], [-6], [-12], [8], [0], [32], [-14], [10], [16], [16], [20], [40], [8], [0], [-36], [32], [-8], [36], [-28], [-30], [28], [34], [12], [-10], [-12], [32], [-36], [-20], [-12], [16], [0], [-26], [22], [-44], [24], [30], [-4], [-8], [-2], [-40], [-4], [14], [32], [48], [20], [46], [-26], [40], [24], [4], [-20], [-4], [32], [22], [12], [-8], [-24], [52], [-16], [-38], [20], [26], [-8], [8], [20], [-24], [4], [-36], [-36], [20], [14], [12], [44], [-56], [-44], [36], [0], [28], [0], [0], [40], [-6], [16], [-28], [-34], [-56], [-8], [28], [2], [-32], [-24], [4], [20], [6], [56], [-36], [-4], [54], [-64], [-52], [-24], [12], [6], [12], [44], [32], [2], [36], [-58], [-8], [56], [-56], [-52], [-16], [32], [12], [-2], [-34], [-12], [-48], [18], [-56], [52], [-40], [36], [24], [8], [24], [42], [-4], [-18], [-48], [-56], [-28], [48], [8], [-40], [4], [60], [62], [38], [48], [18], [-64], [-12], [-10], [4], [-4], [-64], [40], [14], [-52], [56], [-6], [-12], [0], [64], [-54], [-48], [0], [-32], [-28], [-16], [50], [36], [-8], [16], [36], [20], [-60], [66], [-4], [52], [-56], [-72], [32], [22], [-68], [-72], [-40], [-56], [-80], [-28], [-52], [-28], [-4], [50], [-52], [-20], [54], [-12], [52], [56], [-16], [-8], [32], [-32], [2], [26], [68], [48], [-56], [72], [-52], [20], [20], [-14], [-54], [-8], [-44], [-46], [52], [70], [20], [-78], [-48], [32], [-40], [-36], [-20], [8], [46], [16], [-8], [-60], [-2], [-40], [20], [28], [26], [48], [6], [-82], [4], [12], [-56], [-68], [4], [-56], [-50], [76], [-74], [40], [-8], [-6], [-48], [-20], [-28], [52], [22], [-14], [-18], [-8], [44], [60], [-44], [-34], [20], [8], [56], [74], [54], [88], [-48], [30], [8], [-84], [76], [48], [56], [-36], [26], [18], [-24], [74], [-36], [8], [-12], [-30], [28], [26], [-60], [36], [72], [68], [6], [-24], [16], [10], [8], [-84], [12], [-54], [-80], [-62], [20], [-52], [-16], [52], [-32], [0], [26], [-16], [2], [-20], [-48], [-24], [36], [-20], [-72], [46], [0], [40], [12], [-20], [38], [24], [-12], [-52], [-10], [28], [-46], [32], [12], [56], [-88], [-14], [18], [-4], [12], [-16], [22], [-72], [-52], [-90], [-40], [-12], [56], [-28], [32], [-66], [-52], [28], [-100], [40], [-24], [-16], [54], [12], [24], [-16], [24], [4], [0], [-68], [-4], [22], [-106], [68], [-56], [-68], [-2], [70], [-100], [88], [50], [52], [72], [24], [-12], [28], [80], [24], [-12], [34], [-70], [-34], [-68], [-44], [-38], [-20], [80], [32], [-38], [-20], [14], [-104], [-36], [62], [-68], [72], [96], [32], [-30], [-56], [12], [-50], [80], [-48], [-54], [96], [60], [112], [-40], [-8], [-64], [14], [-36], [4], [96], [-4], [-80], [-70], [-24], [48], [-28], [12], [-102], [40], [-100], [-114]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9408_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9408_2_a_u();".
// 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_9408_2_a_u(:prec:=1)
    chi := MakeCharacter_9408_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(3581) - 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_9408_2_a_u();".
// 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_9408_2_a_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9408_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<11,R![0, 1]>,<13,R![4, 1]>,<17,R![-4, 1]>,<19,R![-4, 1]>,<31,R![-8, 1]>],Snew);
    return Vf;
end function;