
// Make newform 5408.2.a.y in Magma, downloaded from the LMFDB on 26 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_5408_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_5408_2_a_y();".
// 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_5408_2_a_y();".
// 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
        poly := [-3, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-1, 2]];
    Rf_basisdens := [1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_5408_a();"
function MakeCharacter_5408_a()
    N := 5408;
    order := 1;
    char_gens := [2367, 677, 1185];
    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_5408_a_Hecke( : Kf := Kf)
    return MakeCharacter_5408_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, 0], [0, 0], [0, 0], [1, 1], [-3, 1], [0, 0], [0, 2], [-5, -1], [0, 0], [0, 2], [7, -1], [0, 0], [0, 0], [0, 0], [9, 1], [-2, 0], [1, -3], [6, 0], [11, -1], [-5, 3], [0, 0], [0, 0], [7, 3], [0, 0], [0, 0], [-14, 0], [0, 0], [0, 0], [0, 0], [0, -2], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [17, 1], [0, 2], [1, 5], [-3, 5], [-22, 0], [0, 0], [0, 6], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [11, -5], [-21, -1], [0, 0], [-10, 0], [19, 3], [0, 0], [0, 0], [14, 0], [0, 0], [0, 2], [-23, 1], [-30, 0], [0, 0], [0, 0], [0, 0], [17, 5], [0, 0], [0, 6], [0, 0], [-5, 7], [0, 2], [0, 0], [0, 0], [0, 0], [9, -7], [0, 0], [0, -6], [11, 7], [-21, -5], [0, -10], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [15, 7], [30, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [17, -7], [0, 0], [29, -3], [31, -1], [0, 0], [19, 7], [0, 0], [0, 0], [0, -10], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [38, 0], [0, 0], [0, 0], [11, -9], [0, 0], [0, 0], [0, 6], [0, 0], [0, 0], [0, 0], [-35, 1], [25, -7], [0, -14], [31, -5], [0, 0], [0, -14], [0, 0], [0, 0], [0, -14], [-50, 0], [-27, -7], [-37, -1], [0, 14], [0, 0], [0, 0], [0, 0], [0, 0], [29, -7], [-37, 3], [0, 0], [-34, 0], [0, 0], [0, 0], [0, 0], [1, -11], [0, -2], [0, -10], [7, 11], [0, 0], [0, 0], [9, -11], [-38, 0], [25, 9], [0, 0], [-10, 0], [0, 0], [-33, 7], [0, 0], [14, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-54, 0], [0, 0], [29, 9], [22, 0], [19, 11], [0, 0], [0, 0], [-43, -3], [0, 0], [46, 0], [0, 0], [-62, 0], [-45, -1], [0, 0], [0, 0], [0, 0], [0, 0], [58, 0], [-23, -11], [0, 0], [0, 0], [0, 18], [-43, 5], [0, 0], [-50, 0], [0, 0], [-3, 13], [0, -10], [0, -18], [7, -13], [0, 0], [-27, -11], [0, -18], [47, 3], [0, 0], [0, 0], [0, 0], [38, 0], [0, 0], [0, 14], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, -18], [49, -3], [70, 0], [19, -13], [0, 0], [0, 0], [0, 0], [-66, 0], [-2, 0], [0, 0], [-51, 1], [-21, -13], [0, 0], [0, 0], [-46, 0], [-41, -9], [0, 0], [-18, 0], [-35, -11], [0, 0], [47, 7], [0, 0], [-62, 0], [-70, 0], [0, 0], [0, 0], [-43, 9], [0, -18], [0, 0], [-37, 11], [0, 10], [0, 0], [7, 15], [0, 0], [0, 0], [0, 0], [0, 0], [11, 15], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [53, 5], [-55, -3], [31, -13], [0, 0], [0, 0], [0, 0], [17, -15], [0, -22], [0, 2], [-51, -7], [0, 10], [-41, 11], [0, 22], [0, 0], [0, 0], [0, 0], [0, 0], [70, 0], [0, 0], [0, 0], [0, 0], [0, 0], [53, -7], [82, 0], [42, 0], [59, -1], [0, 0], [0, 0], [0, -2], [-37, -13], [-57, -5], [0, 0], [0, 0], [0, 0], [0, 0], [-55, -7], [0, 0], [0, 0], [61, 1], [0, 0], [2, 0], [0, 0], [1, 17], [-82, 0], [86, 0], [0, 0], [0, 0], [53, 9], [0, -14], [0, -2], [0, 0], [0, 22], [0, 0], [49, -11], [22, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [54, 0], [-55, 9], [0, 6], [0, 0], [0, 0], [0, 14], [0, 0], [-51, -11], [0, -10], [-21, -17], [-45, -13], [0, 0], [0, 2], [0, 0], [0, 0], [-58, 0], [-23, 17], [0, 0], [34, 0], [61, -7], [47, -13], [0, 0], [0, 26], [0, 0], [0, -14], [0, 0], [-27, 17], [67, -1], [59, -9], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, -26], [0, 0], [0, 0], [0, 0], [0, 0], [0, 26], [1, -19], [-3, -19], [0, 22], [7, 19], [0, 0], [0, 0], [0, 0], [66, 0], [0, -26], [-51, 13], [-57, 11], [94, 0], [-33, -17], [0, 0], [0, 10], [0, 0], [15, 19], [0, 0], [0, 22], [0, 0], [53, 13], [0, 26], [0, 0], [0, 0], [69, 5], [-2, 0], [0, 0], [6, 0], [71, 3], [0, 0], [0, 0], [98, 0], [0, -6], [0, 0], [0, 0], [61, -11], [0, 0], [25, -19], [49, -15], [-73, -1], [-94, 0], [71, -5], [0, 0], [0, 0], [0, 0], [69, -7], [0, 0], [-27, -19], [58, 0], [-41, -17], [0, -22], [0, 0], [0, 0], [0, 0], [0, -14], [29, -19], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-43, 17], [-75, 1], [62, 0], [-34, 0], [0, 0], [0, 0], [-90, 0], [-106, 0], [-5, -21], [0, 0], [0, 0], [0, 0], [-38, 0], [0, 0], [11, -21], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [77, 1], [0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5408_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5408_2_a_y();".
// 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_5408_2_a_y(:prec:=2)
    chi := MakeCharacter_5408_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_5408_2_a_y();".
// 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_5408_2_a_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5408_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<5,R![0, 1]>,<7,R![-12, -2, 1]>,<37,R![0, 1]>],Snew);
    return Vf;
end function;