
// Make newform 5328.2.a.n 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_n();".
// 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_n();".
// 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], [-1], [-5], [3], [1], [-7], [3], [4], [4], [-1], [-10], [4], [0], [9], [-6], [-2], [0], [-8], [-11], [6], [-9], [-3], [-8], [2], [-14], [5], [-11], [-2], [5], [-18], [8], [-20], [6], [-3], [-14], [19], [21], [15], [-10], [-14], [15], [-16], [23], [-16], [-12], [-16], [6], [-20], [8], [-8], [18], [-10], [-3], [16], [3], [-24], [21], [5], [19], [-27], [10], [12], [-24], [22], [-8], [9], [-26], [-2], [-6], [8], [7], [4], [20], [21], [-6], [38], [1], [-30], [29], [26], [-35], [-21], [-34], [20], [-26], [-36], [-18], [4], [-6], [-15], [8], [15], [-13], [-8], [-15], [-16], [24], [-33], [-1], [-2], [14], [-13], [12], [20], [0], [18], [48], [-29], [-32], [2], [-42], [4], [-2], [38], [1], [15], [-22], [-24], [3], [29], [-9], [30], [2], [12], [51], [10], [8], [4], [-6], [28], [12], [49], [-30], [20], [-25], [-24], [42], [-47], [30], [37], [9], [-48], [29], [-42], [-13], [43], [47], [6], [-22], [-36], [-41], [14], [35], [0], [38], [36], [-22], [58], [38], [-26], [20], [28], [41], [18], [10], [-39], [50], [30], [-48], [-46], [-44], [17], [1], [42], [40], [-3], [-44], [6], [38], [-26], [-21], [-38], [24], [7], [20], [29], [2], [-16], [-34], [-36], [-39], [11], [44], [-32], [10], [2], [6], [21], [-64], [-11], [62], [30], [28], [60], [-43], [69], [57], [30], [22], [58], [16], [-23], [-45], [41], [-2], [42], [-1], [-21], [34], [1], [2], [-72], [36], [-29], [2], [-46], [-8], [-42], [4], [52], [35], [38], [40], [4], [-35], [26], [-60], [-12], [6], [17], [1], [39], [-44], [-9], [10], [-16], [6], [-19], [-37], [18], [10], [-40], [-12], [44], [76], [-60], [-23], [-58], [-54], [6], [46], [-16], [-3], [-26], [-33], [-26], [-46], [-60], [3], [3], [10], [40], [-49], [29], [-81], [2], [-45], [-16], [48], [-6], [-40], [10], [-40], [-68], [56], [-66], [-50], [65], [-2], [-68], [-72], [-11], [-44], [-26], [41], [85], [0], [30], [-60], [-70], [72], [32], [67], [-52], [-68], [34], [30], [3], [60], [-23], [42], [-34], [-27], [0], [-75], [57], [14], [-29], [-33], [-19], [82], [43], [-26], [16], [64], [-4], [29], [-34], [66], [30], [22], [58], [19], [52], [28], [10], [28], [72], [88], [-64], [-84], [10], [-84], [42], [-48], [-76], [-48], [58], [-80], [79], [10], [-50], [-26], [-52], [3], [52], [-24], [84], [-22], [-58], [21], [4], [-10], [46], [54], [-50], [-10], [40], [4], [34], [-9], [-52], [10], [-93], [-82], [61], [-101], [-87], [-4], [-51], [-38], [-82], [-28], [-34], [-43], [86], [88], [-23], [1], [58], [-10], [11], [68], [-91], [-10], [-4], [-100], [69], [-78], [-83], [-76], [-98], [-24], [77], [-39], [-26], [-4], [88], [-89], [57], [88], [91], [-20], [17], [72], [1], [102], [-36], [-28], [15], [-44], [-33], [81], [-38], [-27], [-62], [-74], [-72]];
    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_n();".
// 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_n(: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_n();".
// 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_n( : 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![1, 1]>,<11,R![5, 1]>,<13,R![-3, 1]>],Snew);
    return Vf;
end function;