
// Make newform 7800.2.a.bx in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7800_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_7800_2_a_bx();".
// 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_7800_2_a_bx();".
// 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 := [6, -12, -14, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [4, -8, -3, 1], [-3, -11, -2, 1]];
    Rf_basisdens := [1, 1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_7800_a();"
function MakeCharacter_7800_a()
    N := 7800;
    order := 1;
    char_gens := [1951, 3901, 5201, 7177, 4201];
    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_7800_a_Hecke(Kf)
    return MakeCharacter_7800_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, 0, 0], [1, 0, 0, 0], [0, 0, 0, 0], [1, -1, 0, 0], [1, 1, -1, 0], [1, 0, 0, 0], [2, 1, 0, 1], [1, 1, -1, 1], [2, 0, 1, 0], [1, -2, 0, 0], [1, 1, 1, 0], [-1, 1, 1, -1], [1, 1, -1, -1], [2, -2, -1, 0], [0, -2, 2, -1], [3, -2, -1, 2], [-3, -1, 0, -2], [4, 1, 0, -1], [0, -2, -1, 1], [5, 1, -1, 1], [-2, -2, -1, -2], [-3, 1, 1, -1], [5, 1, 0, 0], [4, -2, 2, 0], [-2, 2, 0, 0], [4, -1, -1, 1], [4, 2, 3, 0], [1, 1, 2, -1], [-5, 1, 1, -3], [0, 0, 2, -2], [1, 3, 1, -1], [7, -1, 3, -1], [-1, 1, -2, -1], [-4, 2, 0, 2], [6, -2, 0, 0], [3, -1, 2, -2], [-10, 1, -1, 1], [6, 2, -3, 2], [1, -5, -1, 1], [1, -4, 5, -2], [-4, 6, -1, 0], [10, -3, 1, -1], [14, -4, 0, 0], [6, 4, 1, 4], [-4, 0, 3, 2], [-9, 1, 0, 1], [6, 0, 1, 0], [-4, -4, 5, 0], [3, 5, -3, 4], [3, 5, -2, 1], [6, -2, 2, 0], [-3, 5, 0, 2], [6, 0, -1, 0], [11, 1, -1, -1], [4, 3, 1, 3], [6, 0, 1, -2], [11, -2, -2, 2], [11, 3, -4, 2], [-12, 0, -4, 2], [3, -3, 0, 1], [4, -6, -2, -2], [0, 6, -4, 2], [-9, -3, 1, -1], [6, 0, -2, -4], [5, 6, -3, -2], [-18, -2, 3, -6], [10, -2, 1, -2], [-12, -3, 5, -1], [-3, 3, -2, -1], [2, -4, -3, -2], [-5, -5, -3, 1], [10, -2, -2, 1], [7, -1, -2, 3], [-12, 5, -1, 3], [-7, -1, 3, -2], [11, 3, -1, 3], [15, -1, 0, -3], [-15, 1, -4, 3], [-18, -4, 3, -2], [12, -8, 3, -2], [3, 5, -3, 3], [6, 2, 2, -2], [11, -1, 1, 3], [-1, -5, 2, 1], [-7, -5, 3, -1], [15, -1, -3, -3], [3, -1, 0, -5], [-12, -4, -1, -2], [-16, 2, -2, 2], [1, -3, -3, 2], [-13, -3, 4, -5], [10, -8, 3, -1], [5, -5, 3, 0], [8, 6, 4, 2], [-8, 8, -6, 3], [-10, -2, -1, -2], [-2, -8, 3, 0], [-12, 2, 0, -2], [-4, -8, 8, -2], [12, 0, 3, 0], [-6, 8, 3, -2], [8, 6, 1, 2], [23, -3, -1, 1], [10, 3, -5, -3], [14, -4, 3, -4], [-14, 2, 0, 4], [-27, 7, -3, 2], [9, 3, 4, -1], [2, 6, -5, -2], [-9, -4, -5, 0], [-1, -5, 4, -3], [-26, -4, 4, 0], [1, -7, -4, -1], [12, -6, 3, 0], [-4, -6, 5, -4], [-16, 7, 0, -1], [-13, -1, 5, -1], [18, 4, -6, 2], [0, 7, -4, 3], [-7, -7, 3, -5], [1, -7, 1, 3], [17, 6, -1, 4], [-26, 0, -2, 0], [19, 1, 3, 2], [12, -2, 0, -5], [-18, 13, -6, 3], [-6, -6, 1, 4], [4, 6, 5, -4], [12, 2, -5, 4], [-21, -1, 0, -1], [12, 8, -7, 3], [2, 0, 1, 1], [-3, 3, 3, 3], [2, 1, 6, -3], [-13, -9, 3, -1], [-6, 8, -4, -2], [-22, -6, 4, 0], [-3, -7, 1, -8], [-18, 5, 0, -5], [-4, -8, 8, -6], [-5, 3, -4, 2], [6, -4, 3, -2], [-5, 5, 4, -1], [-1, -3, 1, 6], [-6, 13, -2, 3], [26, -2, -1, 6], [-31, 5, 2, -3], [2, 8, 0, 0], [10, -10, 1, 0], [-20, 4, -3, -1], [-17, -3, -2, -1], [-4, 7, -5, 3], [10, 8, -8, 2], [8, 0, 5, -2], [22, -2, 0, 2], [4, -10, 3, -2], [-31, -3, 0, -5], [5, 7, -8, -1], [-18, 5, 4, -1], [-14, 6, -1, -2], [-1, -7, 9, -6], [18, -1, 3, 1], [15, 9, 1, 6], [25, 1, -4, -5], [2, -2, 10, -6], [19, 3, -3, 0], [-27, -1, -1, 3], [-2, 7, -6, 5], [0, -6, -2, -4], [0, -5, -1, -3], [-12, 12, 3, -4], [11, 5, 2, -1], [4, 4, -7, 4], [-16, 2, -5, 0], [-9, 13, -2, -3], [-4, 3, -3, 5], [1, 5, -5, 8], [4, 0, 2, 6], [3, -1, 2, 1], [-14, 9, -5, 1], [0, 12, -5, 0], [18, 4, -3, 8], [0, -8, -4, 6], [11, 5, 0, 3], [9, 7, 6, -4], [7, -5, -4, 5], [8, -7, -1, 5], [-54, 2, -2, 1], [0, -10, 5, 4], [11, 11, 7, -4], [-27, -1, -10, 3], [15, -1, 10, -6], [6, 12, -7, 0], [-32, -4, 1, 0], [-5, -11, -4, -3], [16, -1, 5, 3], [-12, 6, 3, 0], [14, -5, -5, 3], [2, 4, 2, 0], [-4, -12, 0, -2], [-38, 6, -1, 2], [17, 5, -2, -3], [2, -8, -2, 0], [31, 2, 1, 0], [19, -7, -3, 2], [31, 1, -4, 5], [-43, 3, 2, 2], [1, 7, 1, 5], [-6, 2, -2, 6], [-18, -6, -1, -6], [-19, -2, 4, 0], [3, -5, -4, -3], [2, 12, -4, 10], [11, -7, -2, 8], [4, 4, -7, -5], [8, -10, 6, -4], [16, 6, -2, -2], [-28, -1, 1, -7], [-9, 5, 0, 0], [-2, 6, -6, 6], [-8, 0, 4, -2], [6, 14, 3, -2], [-21, -1, 4, -1], [-17, -7, -10, 0], [39, 3, -3, -3], [-12, 9, 0, -3], [18, 4, -8, 0], [8, 2, -8, 6], [-9, -1, 1, -9], [1, -1, -4, 2], [-16, -2, 2, -6], [-4, -6, -4, -4], [13, -11, -1, 4], [-36, 0, 4, -2], [-14, 4, 3, 4], [6, 2, 5, -6], [-10, 8, 4, 8], [-2, -4, -3, 2], [1, -17, -4, 5], [-46, 2, 2, 2], [-10, 0, -11, 2], [-22, -2, 1, 0], [-23, -5, 3, 5], [-21, 7, 5, -1], [-14, -4, 7, -2], [32, 0, -3, 4], [-18, 10, 1, 2], [4, -6, -9, 0], [-12, -6, 11, -11], [14, -4, 3, 2], [13, 9, 6, -3], [-26, -4, -1, 6], [21, -11, -1, 1], [-15, 6, 7, -8], [-5, -5, 10, -7], [7, -11, 2, 4], [8, -6, -6, 6], [-2, 14, 3, -9], [1, 10, -6, -2], [18, 6, -2, 0], [25, -1, -4, -5], [-11, 5, -3, -1], [-3, -13, 0, 5], [-10, -14, 6, -12], [40, -14, 6, -4], [-34, 0, 3, 4], [-38, 2, -1, 0], [-3, -7, -5, 1], [-3, -7, -4, 10], [2, -1, -3, 3], [-4, 16, -4, -2], [6, 14, 8, -3], [4, -4, 4, -6], [-13, 15, 1, 7], [-33, -2, 5, -4], [-27, -11, 7, -4], [-8, 12, 3, -4], [-23, -5, 1, 1], [-24, 8, -12, 4], [-7, -9, 16, -5], [10, 6, -6, -4], [-40, -8, 11, -8], [-8, -6, -4, 0], [48, 2, -1, -2], [-12, 4, 0, 5], [0, -10, 7, -6], [-4, -15, 2, -7], [4, -16, -5, -2], [-31, -5, 1, -6], [0, 3, 7, -9], [-58, 1, 3, -3], [1, -7, -7, 5], [9, 15, -8, -1], [5, -5, -11, 0], [-28, 1, -5, -1], [14, 1, 0, -1], [-6, -8, 0, -10], [-28, 8, -2, 6], [8, 12, -5, -2], [11, 9, 0, 12], [27, 6, -6, -6], [14, -2, -1, 4], [-30, -10, -7, 2], [47, 1, 4, -1], [24, -18, 5, 0], [46, 2, -5, 6], [-55, -1, 4, -7], [-4, -4, 6, 2], [35, 7, 5, -1], [8, 9, -1, 13], [-36, 16, -9, 6], [-18, -10, -8, -2], [12, 1, -2, 5], [-6, -10, 1, -4], [-52, -2, -1, -3], [-19, 19, -6, 3], [-17, -11, -8, -2], [-2, -2, 3, 2], [-6, -10, 0, 2], [34, -7, 8, -3], [12, -12, -1, 12], [-14, 4, 3, 8], [3, 17, 6, 1], [43, -5, 0, -2], [-3, 1, 5, -3], [-7, 16, 0, 2], [0, -14, -8, 9], [-22, 4, 6, 9], [22, 10, 5, 2], [27, 8, 6, -2], [5, 11, -2, -1], [-4, -4, 4, -2], [10, 2, 5, -6], [-11, -1, 8, -10], [-6, -14, 7, 3], [-17, -7, -6, 2], [-38, 0, 4, -4], [9, 7, -4, -11], [-34, -6, -5, -8], [-11, -1, -2, -9], [-29, -1, 5, -3], [-18, -2, 6, -12], [-44, 12, 1, 6], [-42, 12, -3, 0], [-44, -2, 5, 4], [-16, 4, 4, -12], [34, 7, -3, 1], [-39, 1, 0, -2], [0, -20, 3, -7], [-7, 10, 10, -2], [2, -2, 1, 3], [8, 2, 2, -12], [10, 2, 5, -4], [27, 3, -1, 13], [2, -18, 4, -8], [22, 9, -8, 15], [8, -6, 16, -9], [1, 5, 2, 0], [40, 1, 1, 3], [-24, 12, -4, 9], [21, 7, 0, 5], [-12, -18, -8, 10], [0, -8, 10, 4], [-20, 8, -3, 8], [38, 4, 0, 0], [48, 0, 2, 6], [-22, 8, -13, 4], [5, 7, -4, 4], [-5, -18, 9, -6], [11, -5, 15, -5], [-16, -14, 17, -12], [-4, -18, 5, -4], [-11, -18, -1, -4], [-39, -9, -2, 3], [-4, 9, -14, 7], [-27, -11, -9, 2], [-2, 2, 2, 0], [-22, 6, -2, -2], [-2, 19, 3, 3], [3, -13, 0, 5], [-6, -10, 7, -12], [-25, 5, 4, 3], [-37, -5, 1, -4], [4, -6, -1, 10], [-21, -9, 3, -9], [25, -11, -3, 9], [17, -3, -1, 8], [-10, -9, 14, -1], [-67, 9, 1, 1], [-30, 24, -6, 4], [-6, -10, -2, -10], [-30, -14, 2, 2], [7, -9, 0, 12], [32, -14, -6, 2], [17, 7, -8, 12], [1, -8, 12, -10], [23, 1, 9, -6], [16, -9, 0, -9], [-24, -8, 12, 2], [6, 4, -3, 12], [17, 1, -3, -3], [-14, -7, 14, -15], [1, 1, -13, -2], [9, -23, 8, -9], [-20, 4, 2, -10], [22, 6, 0, -4], [16, 4, 6, -2], [-44, -2, -11, 2], [-22, 8, 0, -5], [24, -16, -7, 8], [22, -5, 2, 1], [-6, -3, 5, -11], [-6, -10, 3, -14], [-40, -12, -2, -2], [-7, 15, -14, 7], [23, -10, 7, 2], [-9, 9, 9, 3], [1, -5, -6, 13], [8, 6, 11, -6], [-64, -2, 6, -10], [51, 0, -9, 16], [-3, -1, -7, 1], [-62, 4, -2, -1], [3, 3, 4, 11], [-28, -18, 11, 4], [28, 12, -3, 12], [18, 0, -8, 10], [-9, 1, -15, 5], [-11, 13, 2, -6], [32, -2, 5, 0], [-38, -14, -5, 4], [3, -3, 15, -10], [72, -8, 0, 2], [1, 11, 11, 6], [-12, 0, 7, -8], [-44, 9, 5, -1], [-26, -6, 2, -4], [29, 3, -14, 13], [-18, -8, -7, -10], [5, -15, 6, 4], [-31, 13, 4, 0], [-5, 1, 12, -9], [-27, -3, 10, -5], [-6, 6, -2, -6], [29, -9, -4, -5], [20, 2, 4, 2], [16, -18, 1, 4], [6, 8, -9, -1], [37, -22, -1, 0], [18, -8, 18, -4], [-6, -2, -10, 5], [30, -10, 3, -7], [-42, -2, -1, -2], [-42, 12, -3, -2], [52, 0, 3, -4], [32, 1, 0, -11], [16, -22, 1, 2], [-41, 21, -4, -1], [36, 15, -17, 3], [27, 15, -1, -1], [-32, 2, -6, 4], [-33, -1, 4, -6], [-19, 11, -2, 5], [-24, 1, 2, 5], [22, -22, -2, -7], [72, 0, 8, -4], [18, -16, 0, 10], [-38, 18, -1, -2], [6, -17, -3, 3], [-26, 0, -10, -4], [7, 9, -16, 8], [-17, 3, -16, 6], [-16, -18, 7, -16], [-24, 8, 7, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7800_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7800_2_a_bx();".
// 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_7800_2_a_bx(:prec:=4)
    chi := MakeCharacter_7800_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(3361) - 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_7800_2_a_bx();".
// 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_7800_2_a_bx( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-20, 39, -11, -3, 1]>,<11,R![-72, 159, -29, -5, 1]>,<17,R![216, 171, -29, -7, 1]>,<19,R![36, 144, -42, -3, 1]>],Snew);
    return Vf;
end function;