
// Make newform 1320.2.a.b in Magma, downloaded from the LMFDB on 19 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1320_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_1320_2_a_b();".
// 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_1320_2_a_b();".
// 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_1320_a();"
function MakeCharacter_1320_a()
    N := 1320;
    order := 1;
    char_gens := [991, 661, 881, 1057, 1201];
    v := [1, 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_1320_a_Hecke( : Kf := Kf)
    return MakeCharacter_1320_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], [-1], [-2], [1], [4], [0], [-4], [4], [-2], [-8], [-2], [-6], [-6], [-4], [6], [-4], [-6], [0], [-16], [-4], [-8], [6], [14], [6], [-6], [-8], [6], [-2], [-2], [-6], [0], [-2], [-8], [18], [-16], [-22], [4], [-18], [4], [20], [-14], [0], [4], [4], [-16], [20], [20], [-14], [22], [12], [12], [-26], [28], [2], [-14], [-2], [0], [28], [-22], [10], [-8], [-6], [24], [2], [2], [20], [-12], [14], [10], [-14], [-20], [4], [32], [-28], [32], [-18], [-30], [30], [-14], [28], [-38], [-16], [38], [-20], [4], [14], [-8], [-38], [4], [8], [4], [-12], [-4], [4], [-42], [-2], [-10], [-2], [-18], [6], [-36], [42], [6], [40], [6], [-16], [-32], [-8], [-2], [-22], [32], [22], [20], [40], [-10], [-28], [-8], [-18], [4], [-2], [20], [0], [-8], [36], [-42], [-22], [8], [32], [-16], [16], [10], [24], [26], [-10], [-50], [-42], [-38], [2], [-22], [24], [-42], [-16], [-34], [18], [0], [-8], [-48], [52], [24], [40], [34], [36], [-42], [-24], [48], [-4], [26], [32], [-6], [32], [20], [50], [-36], [46], [0], [-16], [32], [46], [-2], [-36], [18], [-12], [-48], [0], [42], [-48], [38], [-54], [-10], [24], [-28], [26], [36], [36], [2], [-32], [0], [2], [-8], [30], [-34], [28], [30], [-6], [-46], [-26], [46], [36], [-6], [46], [60], [30], [-50], [60], [42], [-64], [-6], [50], [20], [0], [18], [16], [-56], [-44], [26], [14], [46], [48], [-58], [18], [-40], [66], [8], [14], [30], [22], [-24], [34], [0], [-30], [-56], [52], [42], [-64], [-54], [-50], [32], [28], [-48], [-44], [-20], [24], [-14], [36], [-20], [-20], [28], [56], [-38], [-76], [78], [24], [-54], [-52], [12], [58], [46], [-38], [24], [-10], [18], [10], [-4], [-78], [-20], [-54], [-30], [-54], [32], [-18], [-32], [14], [52], [-72], [-76], [56], [-46], [6], [0], [66], [-16], [-54], [-74], [6], [0], [46], [48], [-8], [-66], [-18], [-44], [36], [-8], [44], [-30], [-8], [-66], [20], [-14], [-72], [60], [-44], [-64], [52], [-62], [-48], [6], [-48], [24], [46], [-74], [-34], [-32], [78], [-34], [68], [12], [-50], [54], [-68], [-50], [54], [-84], [-4], [78], [-52], [-28], [-6], [52], [-6], [-54], [16], [-30], [-20], [-92], [-34], [68], [-58], [-34], [10], [42], [10], [64], [-66], [-24], [38], [28], [-56], [-10], [-32], [-70], [42], [74], [-78], [40], [-16], [-24], [-84], [96], [-76], [-34], [36], [-48], [52], [-42], [12], [62], [-74], [-12], [52], [26], [-38], [68], [66], [60], [12], [56], [54], [48], [-10], [-14], [2], [-48], [-60], [40], [-32], [-22], [14], [48], [30], [22], [44], [-24], [40], [56], [-36], [-6], [20], [-82], [14], [78], [86], [42], [-46], [-56], [-62], [62], [-32], [-52], [8], [72], [-12], [-80], [36], [82], [24], [8], [-70], [6], [-46], [28], [48], [-24], [-94], [22], [12], [-42], [100], [-44]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1320_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1320_2_a_b();".
// 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_1320_2_a_b(:prec:=1)
    chi := MakeCharacter_1320_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_1320_2_a_b();".
// 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_1320_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1320_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<13,R![-4, 1]>,<17,R![0, 1]>,<19,R![4, 1]>],Snew);
    return Vf;
end function;