
// Make newform 5760.2.a.x in Magma, downloaded from the LMFDB on 10 February 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_x();".
// 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_x();".
// 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)
    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], [4], [6], [-4], [-4], [8], [0], [-2], [2], [4], [-6], [12], [0], [14], [-6], [-6], [4], [-8], [-2], [-6], [12], [-6], [-14], [10], [16], [-8], [-18], [12], [-16], [-2], [8], [8], [-18], [6], [0], [12], [0], [-6], [18], [-18], [12], [14], [-2], [2], [8], [16], [8], [-22], [8], [-12], [-18], [18], [-4], [16], [30], [-22], [8], [-6], [4], [-18], [4], [12], [6], [14], [12], [30], [24], [2], [16], [20], [-4], [32], [-12], [8], [-6], [4], [-2], [-18], [34], [10], [28], [-18], [10], [0], [-42], [-22], [-10], [-36], [12], [0], [16], [-26], [4], [-32], [26], [14], [-20], [-10], [20], [-30], [-32], [-18], [-28], [-6], [-16], [4], [12], [10], [-28], [-4], [24], [24], [-38], [18], [12], [0], [-14], [-10], [-38], [-18], [-6], [-20], [16], [-38], [-38], [-4], [32], [40], [44], [8], [2], [48], [-2], [10], [-18], [-12], [18], [-6], [-52], [54], [8], [20], [-2], [-36], [-28], [-4], [28], [-48], [-32], [58], [-20], [-32], [44], [8], [50], [-34], [-46], [-14], [-52], [8], [16], [-30], [-4], [56], [30], [-24], [2], [-30], [-6], [62], [0], [22], [-46], [-14], [-64], [-18], [-20], [-42], [-24], [-38], [32], [-52], [16], [58], [-28], [-36], [-2], [52], [-34], [-12], [-32], [-18], [40], [-36], [-18], [60], [-60], [8], [-30], [54], [-36], [-50], [14], [-42], [22], [48], [18], [48], [-18], [-26], [-16], [-12], [32], [-14], [12], [-18], [-32], [-6], [50], [-22], [-6], [20], [-4], [-74], [-36], [20], [-12], [18], [-44], [52], [-6], [-30], [4], [0], [6], [-6], [-2], [-48], [-8], [20], [24], [-38], [48], [16], [-28], [-54], [4], [64], [48], [62], [-8], [-10], [14], [-42], [6], [-52], [-2], [-26], [-8], [-44], [50], [28], [-48], [28], [-78], [6], [-76], [-62], [62], [28], [38], [10], [-58], [24], [-56], [-22], [-74], [46], [-72], [-38], [56], [-2], [-28], [68], [58], [-54], [14], [-18], [-78], [-56], [-24], [-14], [4], [-26], [46], [-42], [-18], [-44], [6], [58], [54], [48], [32], [-78], [-52], [42], [-72], [0], [-48], [-10], [22], [-52], [-48], [-30], [66], [-8], [22], [-74], [-20], [38], [42], [16], [28], [50], [28], [20], [24], [34], [2], [18], [-66], [-72], [-32], [4], [-2], [72], [50], [0], [-32], [12], [-6], [50], [6], [38], [-38], [-12], [-80], [22], [-92], [42], [10], [16], [10], [-48], [-72], [58], [72], [88], [-32], [78], [-24], [54], [-60], [-38], [62], [60], [-10], [38], [-4], [56], [86], [-66], [8], [14], [16], [50], [-50], [10], [54], [24], [-20], [-80], [-68], [-40], [10], [8], [60], [16], [-14], [-78], [26], [-28], [-88], [58], [2], [-30], [48], [42], [94], [-72], [-32], [-16], [6], [10], [104], [-50], [20], [42], [26], [10], [4], [56], [-26], [94], [-88], [68], [72], [64], [6], [48], [-40], [-6], [22], [-18], [-60], [94], [80], [-60]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5760_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5760_2_a_x();".
// 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_x(: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_x();".
// 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_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5760_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-4, 1]>,<11,R![-6, 1]>,<13,R![4, 1]>,<17,R![4, 1]>,<29,R![2, 1]>],Snew);
    return Vf;
end function;