
// Make newform 5328.2.a.t in Magma, downloaded from the LMFDB on 01 May 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_5328_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_5328_2_a_t();".
// 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_5328_2_a_t();".
// 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_5328_a();"
function MakeCharacter_5328_a()
    N := 5328;
    order := 1;
    char_gens := [1999, 1333, 2369, 1297];
    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_5328_a_Hecke( : Kf := Kf)
    return MakeCharacter_5328_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], [2], [4], [-4], [-6], [-6], [2], [2], [2], [-2], [-1], [-6], [2], [-4], [-10], [-6], [-14], [4], [-12], [-2], [10], [0], [10], [10], [-18], [-10], [0], [18], [-14], [12], [-2], [10], [20], [10], [16], [-18], [-18], [10], [-14], [-26], [-22], [18], [-6], [26], [-10], [0], [12], [-14], [22], [26], [-6], [10], [6], [-6], [-24], [18], [-20], [10], [18], [-18], [6], [4], [-10], [10], [-30], [26], [-10], [-18], [-30], [2], [12], [-12], [14], [-16], [18], [-30], [-14], [2], [-14], [8], [-30], [30], [14], [-10], [36], [-14], [10], [2], [-14], [26], [18], [2], [-32], [30], [-30], [18], [-38], [-38], [10], [-2], [42], [42], [18], [16], [2], [-2], [-34], [12], [-6], [22], [26], [-2], [28], [46], [-18], [-34], [22], [2], [-20], [-22], [-14], [-6], [2], [-20], [-38], [26], [-20], [22], [-46], [-4], [-16], [40], [-6], [-18], [-22], [-6], [-8], [26], [-22], [-40], [6], [12], [-38], [-14], [56], [2], [-6], [50], [44], [-2], [6], [-34], [48], [-22], [34], [-10], [18], [-26], [-2], [10], [46], [18], [-48], [42], [-32], [-2], [-30], [-18], [-30], [-50], [26], [-18], [-54], [8], [18], [-42], [-34], [-56], [-14], [-18], [-34], [-6], [-22], [16], [-6], [2], [10], [10], [48], [-22], [8], [38], [-34], [-60], [-38], [10], [2], [-2], [62], [-6], [52], [22], [-26], [-8], [26], [32], [44], [50], [8], [-46], [10], [18], [24], [-18], [42], [30], [34], [-38], [18], [38], [-40], [18], [2], [28], [34], [30], [-60], [0], [26], [-2], [-32], [-40], [10], [28], [-28], [46], [10], [-30], [-10], [42], [30], [12], [-14], [6], [-6], [-22], [66], [52], [6], [2], [18], [-4], [42], [-62], [-44], [-66], [-76], [22], [2], [-30], [-18], [-34], [-26], [42], [56], [-34], [50], [36], [50], [74], [-22], [66], [-34], [-50], [52], [20], [18], [-34], [42], [36], [-2], [44], [-2], [-30], [-64], [-38], [78], [-18], [18], [42], [30], [58], [-72], [-14], [-10], [-56], [54], [-42], [72], [-78], [18], [-76], [18], [50], [-78], [14], [82], [64], [-6], [-72], [-10], [-46], [4], [-90], [-22], [36], [14], [2], [50], [-46], [-82], [-54], [-52], [-54], [18], [56], [70], [62], [-50], [-14], [26], [10], [-54], [46], [0], [2], [62], [82], [-48], [62], [-70], [18], [10], [-30], [-6], [-6], [8], [14], [-22], [20], [58], [10], [46], [-54], [-50], [6], [98], [-18], [-18], [34], [-34], [-86], [34], [28], [26], [34], [46], [50], [-26], [-38], [-32], [-22], [82], [38], [48], [-28], [-14], [-14], [58], [-30], [42], [38], [-26], [-26], [-40], [-20], [-86], [-54], [18], [54], [10], [-30], [18], [36], [-58], [94], [-18], [12], [30], [-26], [-78], [38], [-102], [10], [12], [42], [-2], [4], [16], [-58], [-98], [-42], [-82], [-6], [-66], [-64], [60], [6], [-86], [-38], [-22], [40], [-72], [14], [18], [-20], [10], [-76], [46]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5328_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5328_2_a_t();".
// 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_5328_2_a_t(:prec:=1)
    chi := MakeCharacter_5328_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_5328_2_a_t();".
// 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_5328_2_a_t( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5328_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<7,R![-4, 1]>,<11,R![4, 1]>,<13,R![6, 1]>],Snew);
    return Vf;
end function;