
// Make newform 5760.2.a.f in Magma, downloaded from the LMFDB on 31 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_5760_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_5760_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_5760_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_5760_a();"
function MakeCharacter_5760_a()
    N := 5760;
    order := 1;
    char_gens := [2431, 901, 641, 3457];
    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_5760_a_Hecke( : Kf := Kf)
    return MakeCharacter_5760_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], [-1], [-2], [2], [-6], [6], [6], [-2], [2], [-4], [-10], [2], [-8], [6], [6], [-10], [14], [8], [8], [2], [0], [-12], [10], [-6], [2], [-14], [-12], [-2], [14], [10], [-18], [-18], [-14], [2], [-16], [-2], [-20], [-14], [-14], [-6], [-10], [12], [-14], [-18], [12], [-6], [14], [8], [-10], [14], [16], [2], [-14], [14], [-26], [-14], [-4], [22], [-10], [-32], [26], [0], [-24], [-14], [14], [-26], [10], [-36], [-2], [14], [4], [-14], [22], [2], [-14], [-30], [-14], [-30], [38], [26], [22], [-12], [-38], [-12], [-24], [14], [26], [-30], [14], [-8], [-8], [-18], [18], [6], [-18], [2], [22], [-40], [14], [-8], [-14], [36], [42], [-10], [-14], [28], [6], [20], [-38], [26], [26], [30], [10], [-32], [-18], [-4], [42], [38], [-22], [-50], [18], [-18], [24], [50], [18], [6], [32], [30], [-34], [22], [30], [4], [38], [-10], [54], [6], [8], [-18], [-42], [22], [10], [-14], [44], [-34], [-52], [-22], [-42], [50], [-54], [-50], [-30], [-36], [10], [-12], [28], [-52], [30], [2], [42], [-48], [-34], [-26], [10], [6], [38], [28], [18], [-26], [22], [14], [14], [-48], [-38], [-40], [10], [14], [42], [-54], [-34], [-22], [54], [-6], [22], [10], [26], [-2], [28], [6], [36], [18], [-48], [10], [-30], [16], [38], [-66], [42], [22], [22], [42], [44], [18], [-14], [54], [66], [48], [-12], [-6], [-10], [-38], [18], [34], [20], [12], [-50], [-38], [2], [-30], [6], [-50], [12], [26], [-2], [-8], [6], [-50], [-48], [14], [42], [26], [62], [-28], [22], [-16], [-18], [18], [-58], [38], [32], [36], [-2], [-62], [-10], [-34], [60], [18], [-50], [26], [-38], [18], [-18], [-78], [74], [2], [-6], [-50], [-28], [10], [-22], [6], [8], [-50], [14], [-2], [-26], [-70], [34], [-40], [-6], [-26], [-40], [-22], [-16], [-14], [50], [-12], [14], [10], [30], [42], [-80], [-22], [22], [4], [-36], [-30], [46], [44], [42], [26], [-72], [-34], [42], [78], [-30], [28], [10], [22], [64], [42], [-46], [80], [28], [-50], [34], [-36], [-2], [12], [6], [-70], [-30], [82], [-12], [-6], [-10], [-6], [-12], [-22], [10], [-62], [42], [34], [-26], [-26], [2], [78], [24], [-18], [-42], [-74], [-30], [8], [12], [54], [28], [-34], [54], [74], [-78], [54], [62], [10], [8], [-14], [-54], [-66], [28], [-84], [-18], [-22], [18], [10], [-66], [-58], [6], [-40], [10], [54], [-66], [-62], [66], [6], [-18], [0], [-14], [-66], [90], [-18], [-74], [34], [-54], [-22], [0], [-30], [-38], [-92], [66], [14], [-46], [10], [6], [22], [-10], [-50], [102], [-36], [-26], [-24], [38], [70], [90], [14], [40], [48], [10], [-8], [-42], [-50], [34], [-50], [14], [10], [30], [26], [40], [18], [-46], [76], [-62], [-86], [42], [32], [82], [-46], [106], [-48], [78], [-18], [-90], [-86], [98], [-50], [-42], [10], [82], [-20], [54], [54], [60]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5760_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5760_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_5760_2_a_f(:prec:=1)
    chi := MakeCharacter_5760_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_5760_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_5760_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5760_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<11,R![-2, 1]>,<13,R![6, 1]>,<17,R![-6, 1]>,<29,R![-2, 1]>],Snew);
    return Vf;
end function;