
// Make newform 1560.2.a.f in Magma, downloaded from the LMFDB on 19 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1560_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_1560_2_a_f();".
// 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_1560_2_a_f();".
// 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_1560_a();"
function MakeCharacter_1560_a()
    N := 1560;
    order := 1;
    char_gens := [391, 781, 521, 937, 1081];
    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_1560_a_Hecke( : Kf := Kf)
    return MakeCharacter_1560_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], [4], [4], [1], [6], [0], [-4], [-6], [-8], [-2], [10], [-4], [8], [-2], [4], [14], [-12], [-8], [-10], [0], [-4], [10], [-2], [10], [-8], [-12], [2], [14], [16], [0], [18], [-4], [-18], [16], [-2], [-4], [-8], [6], [24], [-2], [0], [6], [-26], [-8], [-4], [-28], [-12], [-22], [14], [24], [-30], [0], [14], [-12], [18], [8], [-10], [-6], [-4], [-26], [-28], [-24], [-14], [-2], [24], [-22], [-20], [34], [26], [24], [0], [-10], [32], [-16], [10], [-26], [18], [-14], [0], [-30], [24], [-6], [0], [-4], [18], [38], [30], [36], [-20], [-40], [4], [-8], [8], [-12], [14], [2], [-44], [-38], [44], [14], [4], [-30], [28], [22], [-20], [2], [8], [26], [16], [14], [2], [0], [-40], [-30], [20], [20], [-18], [0], [-14], [18], [-42], [-12], [48], [-46], [-38], [-24], [48], [22], [-40], [8], [8], [38], [10], [2], [-42], [-20], [22], [-54], [-40], [30], [8], [-12], [-34], [0], [14], [-42], [36], [-16], [-2], [10], [-20], [-36], [-20], [-24], [-16], [34], [-38], [-26], [28], [46], [-12], [-48], [-46], [-8], [-32], [-26], [18], [46], [44], [50], [16], [-18], [-32], [34], [40], [30], [48], [-26], [28], [0], [22], [-22], [-32], [-30], [-2], [-28], [2], [56], [34], [60], [-20], [-10], [36], [6], [58], [-34], [-30], [-60], [-50], [40], [22], [-46], [36], [-2], [-40], [-44], [-54], [36], [58], [2], [-64], [-12], [-48], [-6], [24], [-6], [8], [-66], [-2], [-8], [66], [-20], [-12], [-50], [30], [-40], [-16], [-12], [14], [-52], [56], [2], [60], [-48], [10], [-26], [0], [16], [60], [52], [32], [-38], [-14], [-56], [-28], [-12], [-40], [12], [38], [-14], [-32], [-54], [-42], [4], [38], [4], [30], [22], [-72], [60], [26], [-2], [18], [-44], [78], [42], [28], [54], [70], [-20], [62], [-56], [34], [-76], [76], [10], [34], [-24], [12], [-40], [-12], [26], [44], [-24], [-54], [-42], [-56], [34], [26], [-4], [74], [44], [-2], [-46], [48], [38], [40], [60], [-62], [-18], [16], [20], [4], [-74], [-12], [22], [48], [-58], [36], [-10], [58], [28], [-16], [42], [60], [48], [46], [2], [68], [-90], [82], [52], [42], [-62], [72], [76], [76], [-34], [-22], [-18], [-40], [-44], [-24], [28], [10], [-86], [-14], [64], [54], [-26], [-2], [24], [30], [32], [54], [-68], [64], [54], [-40], [-2], [54], [-8], [14], [14], [72], [4], [70], [-32], [-26], [42], [-52], [44], [-12], [34], [46], [60], [-38], [-32], [-36], [16], [-14], [88], [78], [52], [80], [-74], [10], [-78], [-58], [34], [44], [-46], [56], [0], [-32], [22], [-4], [-36], [-62], [102], [36], [-36], [-88], [-46], [88], [-54], [-44], [-90], [26], [-66], [52], [-22], [38], [-64], [38], [-30], [-20], [-60], [10], [-50], [76], [68], [-6], [10], [24], [88], [66], [-76], [-38], [-26], [-24], [80], [-106], [14], [28], [-54], [64], [24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1560_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1560_2_a_f();".
// 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_1560_2_a_f(:prec:=1)
    chi := MakeCharacter_1560_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_1560_2_a_f();".
// 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_1560_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1560_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-4, 1]>,<11,R![-4, 1]>,<17,R![-6, 1]>],Snew);
    return Vf;
end function;