
// Make newform 5328.2.a.u 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_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_5328_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_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], [-2], [6], [6], [8], [-6], [-2], [-1], [0], [10], [-12], [-4], [-4], [10], [4], [-12], [-10], [-10], [0], [2], [-2], [0], [-6], [-12], [-10], [-6], [8], [-12], [-12], [-4], [8], [20], [2], [6], [0], [-12], [12], [2], [8], [-14], [-24], [-2], [4], [-16], [-4], [10], [0], [0], [14], [-20], [-6], [-20], [24], [-20], [-18], [-30], [26], [-28], [32], [0], [34], [20], [-10], [26], [20], [6], [10], [0], [16], [-22], [24], [32], [14], [-2], [-18], [-34], [16], [30], [24], [-30], [14], [16], [-6], [30], [30], [10], [4], [8], [30], [-16], [6], [-32], [28], [-24], [-14], [-34], [22], [30], [-36], [-6], [8], [18], [28], [-16], [-20], [14], [-22], [-6], [4], [16], [-38], [40], [10], [-8], [-18], [12], [26], [6], [-4], [-12], [20], [6], [-26], [-24], [-34], [-50], [24], [36], [8], [-2], [20], [2], [24], [44], [2], [22], [16], [-28], [16], [-44], [-10], [-52], [-46], [-6], [-22], [-24], [2], [-28], [-6], [52], [26], [-16], [-38], [-24], [22], [28], [12], [20], [2], [16], [38], [-36], [-42], [-6], [-62], [-30], [28], [-14], [-16], [-42], [32], [-50], [-14], [-64], [-52], [18], [18], [-12], [30], [-6], [-44], [-28], [30], [-26], [14], [56], [-18], [12], [-2], [60], [-24], [44], [26], [6], [-40], [-24], [-6], [16], [42], [-34], [68], [30], [-16], [0], [18], [40], [10], [6], [-26], [24], [32], [62], [-22], [-66], [-16], [-4], [-42], [28], [-56], [34], [-28], [66], [12], [40], [-24], [52], [14], [4], [4], [-4], [48], [72], [50], [42], [60], [-72], [4], [-34], [28], [10], [52], [-24], [-38], [-12], [-4], [16], [-50], [-20], [24], [-10], [18], [-12], [-10], [-28], [-68], [-62], [-38], [44], [66], [-26], [6], [-56], [60], [70], [44], [10], [26], [-10], [-26], [-22], [82], [24], [-40], [-22], [-66], [4], [0], [-78], [-24], [30], [22], [-20], [-10], [-16], [34], [-6], [-30], [20], [-36], [-72], [22], [-16], [-32], [12], [-76], [8], [50], [-20], [12], [36], [74], [46], [-76], [6], [48], [58], [16], [4], [68], [-12], [72], [-18], [44], [-40], [14], [54], [50], [78], [-26], [-36], [-44], [6], [28], [-10], [24], [0], [-70], [-10], [-2], [-12], [-58], [20], [-74], [48], [-58], [72], [-18], [88], [66], [14], [38], [-36], [34], [56], [0], [-8], [56], [70], [-6], [46], [74], [-12], [0], [84], [-56], [32], [34], [36], [72], [-60], [64], [34], [-78], [-46], [-78], [-68], [-10], [4], [48], [6], [-98], [-52], [12], [46], [98], [22], [-30], [-58], [-2], [28], [-58], [-12], [8], [-42], [46], [-48], [-2], [-34], [-36], [-42], [-12], [86], [-74], [-36], [-8], [56], [66], [-50], [94], [-78], [94], [88], [38], [96], [-4], [-64], [66], [0], [-20], [2], [-50], [-92], [-24], [-52], [-60], [64], [26], [-6], [-24], [76], [-18], [-72], [-44], [-12], [-16], [-24]];
    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_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_5328_2_a_u(: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_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_5328_2_a_u( : 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![2, 1]>],Snew);
    return Vf;
end function;