
// Make newform 7728.2.a.z in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7728_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_7728_2_a_z();".
// 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_7728_2_a_z();".
// 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
        poly := [-10, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_7728_a();"
function MakeCharacter_7728_a()
    N := 7728;
    order := 1;
    char_gens := [4831, 5797, 5153, 6625, 6721];
    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_7728_a_Hecke(Kf)
    return MakeCharacter_7728_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+1)]];
    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, 0], [0, -1], [1, 0], [0, 0], [2, 1], [0, 0], [-2, 2], [1, 0], [4, -1], [-6, 0], [0, 1], [4, 1], [-2, -1], [0, -1], [2, 2], [8, -2], [10, 0], [-4, 0], [0, -2], [2, -2], [0, 0], [10, -2], [-8, -2], [10, -3], [12, -2], [6, 3], [0, -4], [12, 1], [-8, 1], [-6, -3], [-8, -2], [12, -3], [10, 3], [-10, -2], [2, -3], [-6, -2], [12, 0], [14, 2], [4, -2], [-2, -1], [2, 4], [-8, -4], [4, -3], [-12, 5], [10, -3], [12, 0], [6, 4], [8, 5], [6, 6], [14, 4], [4, 4], [-6, 3], [12, -7], [-6, 8], [-2, 5], [4, -2], [-10, -2], [-6, -2], [8, -1], [10, 0], [-18, 4], [6, -3], [-6, 6], [-12, 2], [4, 1], [-12, 8], [10, 2], [-2, -5], [12, 6], [0, -9], [-6, -1], [10, -7], [10, -4], [14, 1], [8, -4], [22, 4], [-16, 6], [14, 4], [-10, -6], [10, -2], [12, 1], [6, -3], [6, 3], [-18, 4], [-6, -3], [-2, 8], [6, -6], [20, 2], [2, 3], [12, -1], [0, -2], [2, 3], [-12, 8], [0, -10], [4, 2], [0, -4], [8, 2], [-6, 12], [2, -6], [-8, 6], [14, 2], [-4, -11], [0, 3], [12, 0], [6, -8], [16, -4], [-16, -7], [0, 8], [18, 2], [2, 0], [8, -9], [-42, 0], [30, 0], [-24, 4], [-8, -1], [-18, 4], [-18, 6], [36, 3], [4, -8], [-6, -4], [-12, 1], [-18, 4], [-36, 0], [38, 1], [22, -2], [-30, 4], [4, -5], [44, 0], [-6, 10], [-16, -6], [-8, 8], [-8, 0], [22, -4], [-14, 8], [-18, -3], [4, 11], [2, -14], [-28, -1], [-14, -2], [22, -9], [6, 0], [-6, -11], [-4, 10], [-8, -6], [12, 2], [30, 3], [28, -7], [-22, 7], [-4, 4], [-26, -6], [12, -6], [-24, 2], [30, 6], [14, -3], [2, -1], [-16, 4], [20, 1], [14, 9], [24, -9], [-22, 5], [-6, -4], [24, -12], [26, 2], [-44, 1], [4, 14], [-20, 0], [56, -2], [18, 4], [-4, -4], [-20, -10], [54, 1], [-18, 17], [-10, 12], [-22, 6], [32, 2], [-56, 2], [2, 8], [-42, -3], [54, 0], [44, 0], [-24, -1], [14, -18], [-48, -1], [-22, 11], [12, -9], [6, -12], [26, 10], [4, -3], [-24, 7], [-46, 4], [-30, 9], [-24, 12], [-16, 2], [0, 6], [4, 4], [-26, 12], [38, 4], [-12, 18], [8, 1], [-2, 8], [38, -6], [26, -3], [60, -2], [44, -1], [-28, -8], [6, -11], [4, 8], [-14, 4], [-8, 20], [32, 1], [4, -14], [-16, -12], [62, -2], [-20, 5], [-44, -6], [-16, 12], [0, -3], [6, 9], [20, -1], [-42, 6], [-8, 8], [18, -8], [72, 0], [4, 4], [-6, -12], [-12, -12], [-4, 16], [-10, 3], [-36, 12], [-6, -12], [6, -16], [40, 0], [-14, 10], [-50, 6], [12, 12], [2, 7], [-26, -8], [26, -7], [-22, 14], [24, 10], [-36, -8], [54, 2], [30, -4], [-4, 11], [-28, -7], [18, -8], [-20, 13], [-10, -9], [62, 1], [0, 19], [-32, 10], [38, -11], [-30, 15], [24, 0], [-42, -9], [20, -15], [-18, 12], [8, -10], [62, -6], [8, 4], [48, 6], [-6, 21], [34, 8], [-42, 12], [-2, 12], [-30, 0], [4, -4], [22, -1], [-14, 8], [6, -15], [-8, -12], [-30, 5], [20, -2], [-2, -12], [-6, 0], [-10, 7], [6, 16], [2, -8], [0, 9], [-8, -1], [12, 6], [-12, 6], [-50, -6], [-22, 3], [40, -10], [-66, 6], [58, -8], [-18, 8], [-36, 1], [-50, -2], [28, -17], [14, -22], [24, -17], [-26, 20], [-26, -8], [-58, 9], [20, 16], [-20, 10], [38, 3], [-50, -1], [-30, 2], [-20, 6], [-14, 13], [30, -9], [12, -13], [46, 7], [30, 6], [10, -1], [-10, 0], [-24, -15], [-42, 16], [30, 8], [36, -12], [24, -12], [-64, 6], [0, -20], [-12, -12], [-6, 9], [6, -2], [-14, -12], [14, -12], [-28, -10], [6, -15], [8, -7], [-4, -6], [-12, 12], [-14, -21], [34, -1], [-26, -14], [10, 14], [52, -1], [18, 12], [44, 4], [14, 15], [16, -20], [-32, 5], [64, 2], [-22, -8], [-6, 3], [30, -12], [-8, -17], [16, -14], [0, 0], [42, -18], [-42, 3], [10, -7], [-48, 10], [0, -12], [28, 13], [4, -16], [20, 7], [4, -11], [8, 0], [-48, 17], [48, 11], [4, -2], [-10, 2], [18, 10], [12, -6], [22, 0], [-12, -15], [-56, 4], [60, -4], [-2, -21], [36, -3], [-12, 0], [24, 16], [-2, 2], [-20, 16], [-66, 1], [-28, -10], [-50, -2], [20, -12], [66, 6], [20, 10], [-38, 11], [46, -3], [-40, 2], [10, 2], [10, 8], [18, -8], [4, -12], [66, -8], [-14, 14], [12, -3], [46, -13], [-94, 1], [18, -15], [-52, 10], [6, 4], [64, 2], [30, -1], [-14, 3], [-72, 4], [-66, 4], [66, -1], [-12, 0], [-54, -1], [12, -1], [6, 17], [-2, -20], [-12, 14], [12, 16], [8, 10], [14, -24], [28, -9], [-6, 6], [38, 10], [-24, 12], [18, -4], [4, 11], [-68, -4], [48, 4], [102, 0], [0, -12], [-20, 2], [-18, 24], [72, 6], [12, 6], [42, -20], [-22, 10], [48, 0], [-52, -4], [6, 18], [48, -8], [-64, 8], [-34, 14], [46, -15], [-8, -1], [-6, 21], [-8, 2], [-20, 20], [4, -4], [78, -3], [48, -18], [-40, 3], [-90, -2], [12, -12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7728_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7728_2_a_z();".
// 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_7728_2_a_z(:prec:=2)
    chi := MakeCharacter_7728_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(3067) - 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_7728_2_a_z();".
// 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_7728_2_a_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7728_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-10, 1, 1]>,<11,R![0, 1]>,<13,R![-4, -5, 1]>,<17,R![0, 1]>],Snew);
    return Vf;
end function;