
// Make newform 7728.2.a.bh in Magma, downloaded from the LMFDB on 25 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_bh();".
// 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_bh();".
// 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 := [-8, -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], [-1, -1], [1, 0], [-4, 0], [-1, -1], [0, 2], [-4, 0], [1, 0], [7, -1], [-2, 2], [5, 1], [-1, 3], [1, 3], [-7, -1], [0, -2], [-2, -2], [-2, 0], [4, 0], [-10, 2], [-8, 2], [4, -4], [-4, 0], [-2, -4], [-7, 1], [-6, 4], [3, -3], [-4, 0], [1, -3], [-9, 3], [9, -1], [-14, 2], [-1, 3], [-11, -1], [4, -6], [1, -1], [-16, -2], [-4, 0], [-4, -4], [2, 4], [9, -5], [-2, 8], [-12, -4], [7, 3], [-19, 1], [5, -5], [12, 0], [14, 2], [1, 3], [4, 2], [14, -4], [-4, 4], [-5, -1], [9, 3], [10, -8], [-3, 3], [-2, 8], [8, -8], [24, -2], [-15, 1], [18, 2], [-10, 0], [-5, 1], [4, 4], [-6, 0], [9, -3], [0, 4], [8, -6], [-19, -1], [26, -4], [29, -3], [-17, 1], [9, -1], [-22, 4], [-13, 1], [-16, 0], [2, -4], [-10, 0], [-6, 8], [-4, -2], [-4, 0], [-15, 5], [-3, -5], [-1, -5], [-26, 2], [11, 1], [10, -8], [4, 6], [-22, -4], [-5, -3], [-17, 5], [2, -2], [-13, -3], [12, -8], [2, 10], [6, 2], [12, -6], [-22, 0], [26, -6], [4, -6], [26, 2], [0, 6], [-27, -1], [-27, -3], [4, -8], [14, -12], [-16, -4], [-5, -1], [20, -12], [16, -6], [-18, 2], [35, 3], [-26, 4], [-30, 2], [16, -8], [1, 9], [22, 6], [8, 0], [-5, 11], [-4, 8], [-6, -4], [3, 7], [14, -8], [-12, 0], [-1, 13], [-20, -6], [-14, -4], [9, -1], [-20, -4], [4, -6], [10, 2], [-8, 16], [-4, 4], [-50, 0], [-14, -8], [-35, -3], [-25, 7], [-16, -4], [-5, -5], [-8, 2], [-3, 7], [-14, 4], [25, -9], [2, -6], [34, 4], [-38, -2], [5, -15], [-17, 3], [-15, 3], [-4, 4], [-8, 6], [-22, 8], [26, 2], [8, 0], [15, -11], [-9, 17], [8, -16], [23, 3], [-31, 1], [-57, -1], [27, 9], [14, -4], [44, -4], [0, -12], [-13, -9], [18, 6], [4, -4], [-50, 0], [10, -8], [16, -2], [-18, 14], [35, -5], [9, -9], [-22, 0], [-16, 0], [6, -12], [-34, 6], [-42, 0], [41, -9], [-6, 4], [20, -4], [-5, -7], [16, 6], [5, -3], [35, 5], [43, 3], [2, -4], [0, -4], [31, -13], [-15, -9], [14, -12], [-1, 13], [12, -8], [38, 0], [-2, -10], [-12, 6], [54, -4], [-2, 8], [10, -8], [23, -7], [22, 0], [-52, 4], [-5, 11], [-10, 12], [23, -11], [0, -10], [-7, 15], [-28, 8], [26, -16], [-36, 0], [31, -5], [22, -8], [-40, 0], [-24, -4], [-1, 1], [-30, 8], [12, 4], [-17, -5], [-47, -1], [23, -1], [36, 2], [16, -8], [-30, 0], [-16, 16], [0, 4], [10, -4], [8, 2], [-20, -12], [51, -3], [-16, 4], [2, -20], [-10, -10], [-12, 12], [-8, 2], [-44, 0], [-24, -8], [7, -13], [-18, 16], [7, -11], [-52, -4], [-22, -6], [12, -16], [24, 0], [-42, -8], [37, 5], [-5, 21], [-22, -10], [5, 15], [1, 3], [25, -1], [23, -9], [-34, 4], [-19, 19], [-1, -13], [60, -6], [17, -13], [-31, 5], [2, 18], [-34, 16], [-28, -10], [28, -12], [-10, 14], [-37, -5], [-46, 12], [-6, 24], [-26, -10], [-18, 8], [40, 2], [-55, -5], [30, 8], [-5, 19], [-12, 0], [-39, -7], [30, 2], [50, 0], [46, -6], [41, -13], [-6, 4], [2, 8], [53, -1], [1, 7], [46, -6], [2, -10], [16, 6], [-43, -1], [14, 2], [28, -10], [42, -12], [22, 2], [39, 3], [-4, 26], [-25, 13], [48, -6], [-21, 1], [-22, -12], [66, -4], [-19, 11], [48, -2], [-14, 10], [-47, 11], [-21, 7], [-36, -6], [26, -10], [-21, 9], [-23, -5], [9, -15], [41, -13], [16, -18], [-9, -7], [-14, -12], [39, 1], [42, -12], [14, -28], [-4, -16], [12, -4], [-14, -8], [84, 0], [36, 12], [-35, -3], [-8, 10], [14, -2], [22, -12], [-30, -12], [-15, -9], [-1, 3], [34, 16], [40, -4], [53, -1], [-25, 1], [12, 2], [40, 6], [23, -1], [34, -18], [-4, -8], [-79, 3], [-48, 12], [-15, 13], [22, -20], [22, 20], [23, 1], [22, 0], [-7, 9], [6, -24], [0, 8], [8, -2], [-3, 7], [-33, -9], [-6, -22], [12, 20], [17, 13], [-28, 0], [5, -3], [45, 9], [32, 16], [21, 1], [41, 9], [-38, 6], [-20, -8], [20, 14], [30, -6], [-54, 4], [55, -5], [52, -4], [-48, 12], [-23, 3], [23, -5], [-16, -10], [8, 16], [36, -10], [44, 0], [-49, 5], [30, 18], [-44, -6], [64, 8], [20, -6], [-26, -2], [-45, 5], [61, -3], [-34, 20], [-76, 6], [-2, 16], [-22, 16], [-16, 8], [-18, 20], [40, -12], [7, 9], [15, 9], [-1, 7], [-25, -3], [-30, -18], [70, -4], [-46, -4], [-15, 11], [53, -9], [28, -20], [-22, 0], [-21, 29], [40, 18], [-83, 7], [-81, -1], [3, -5], [-46, 16], [-42, 26], [0, -22], [-2, 16], [18, 6], [15, -17], [-36, 6], [-48, 12], [64, 4], [-10, -4], [-5, 19], [-36, 0], [68, 0], [-26, 4], [-52, 10], [-22, -10], [-14, 6], [-50, 4], [66, 6], [18, -28], [52, -14], [-40, -8], [8, -12], [-36, 14], [-24, -18], [-36, -8], [-24, 34], [7, 13], [-55, 7], [-47, 1], [58, 10], [80, -12], [44, -12], [-25, -9], [-50, 20], [29, -27], [-28, -14], [-44, 0]];
    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_bh();".
// 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_bh(: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_bh();".
// 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_bh( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7728_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-6, 3, 1]>,<11,R![4, 1]>,<13,R![-6, 3, 1]>,<17,R![-32, -2, 1]>],Snew);
    return Vf;
end function;