
// Make newform 7800.2.a.bw 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_bw();".
// 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_bw();".
// 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, 8, -7, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, -4, -1, 1], [-2, 8, 1, -1], [-4, -1, 1, 0]];
    Rf_basisdens := [1, 2, 2, 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], [0, 0, 0, 1], [-2, -1, 0, 0], [-1, 0, 0, 0], [1, 1, 1, -1], [-3, 1, -1, -1], [1, -1, -1, -1], [2, 0, 0, 1], [-4, 0, 0, -2], [-2, 2, 0, 0], [-1, 0, 1, 0], [1, -1, 1, 2], [-3, 2, 1, 0], [0, 0, 2, 0], [-2, -1, 0, 0], [3, 1, 1, -2], [-6, 0, -2, 0], [-3, -2, 1, 0], [-6, 2, 0, 1], [-3, -1, -3, 0], [-3, 2, -3, -2], [-1, 0, -3, -2], [-2, -2, 0, 1], [-2, 0, 0, -5], [0, -4, -2, 0], [-6, 0, 2, 2], [1, 1, 1, -1], [-1, -3, -1, 1], [-3, 3, 3, 2], [-2, -2, 2, 3], [-7, 2, -1, 0], [2, -2, 2, 2], [-4, -1, 0, -2], [2, -2, 2, 4], [-3, -1, -5, 0], [-4, 0, 0, 4], [-9, -2, 3, -2], [-2, 0, 0, -6], [-2, 4, 2, 1], [12, 2, -2, 2], [-10, -2, -2, 2], [-6, -4, -4, 3], [-3, -2, 3, 4], [-3, -1, -3, 0], [11, 1, -1, 2], [6, 0, 2, 3], [-15, -2, 1, 0], [7, -5, 3, 3], [-3, -1, -3, 3], [-17, 0, -1, 0], [0, 0, -2, 2], [4, 0, -4, 1], [-13, 1, 3, 1], [-1, 5, 1, -3], [12, -2, -2, 3], [2, -4, 2, 4], [5, 3, -1, -4], [-7, -4, -1, -2], [-4, 4, 0, -4], [-13, 0, 1, 4], [0, -6, 4, 2], [6, 0, -2, -2], [-9, 3, -1, -4], [-3, -2, 3, 0], [7, 5, 1, -5], [1, -3, 1, 8], [2, 2, 2, 6], [17, 3, 1, -5], [3, 4, -3, 0], [5, 4, 5, 0], [-6, 2, -8, -4], [-12, 6, 2, 2], [-3, 1, -1, 5], [-19, 4, -3, -2], [8, -4, -2, 3], [-8, 0, -2, 6], [11, -2, 1, -4], [-8, 4, 2, 0], [-4, -4, 0, -1], [3, -5, -5, 1], [-11, 6, 1, 0], [1, 5, 1, -4], [8, 4, 0, 4], [-4, -6, 4, -2], [3, -2, 1, -2], [-12, -4, 6, 1], [12, 3, -4, 0], [0, -4, 0, -5], [-18, 0, -2, -2], [-1, 0, -1, 4], [4, 6, 0, -3], [6, -2, 2, 11], [-3, -1, 3, -7], [5, -1, 3, 5], [10, 5, 6, -4], [-20, 0, -2, -4], [-23, -1, 1, -2], [15, -1, -5, -3], [5, 7, 1, -2], [1, 2, 3, 8], [0, 4, 8, 2], [16, 8, 2, -8], [11, 5, -1, -4], [-16, -2, 6, -1], [-1, 6, -1, -4], [5, 2, -1, 2], [-12, -10, -4, 4], [21, 3, -1, -6], [-2, -2, -8, -4], [4, -4, -10, 0], [-5, -2, 5, 4], [-13, -3, -3, -3], [-4, 0, 4, 4], [0, -2, -6, -6], [6, -4, 2, 8], [-11, -3, 3, -7], [0, -2, 6, 0], [-12, -2, -8, -5], [31, 1, 3, 1], [-2, -6, -6, -4], [16, -4, -2, 0], [3, 6, 11, 2], [-3, -1, 7, 7], [8, 4, 2, 3], [19, -1, -1, -9], [-12, -6, 4, -4], [25, -1, 7, 4], [6, -4, -4, -2], [-9, 9, -3, 1], [-5, 4, -5, -2], [20, -4, -4, 4], [-22, -4, 4, 0], [-19, 4, 3, -4], [10, 0, 0, 8], [-19, -8, -1, 4], [4, 6, -8, 0], [24, 4, -6, 2], [20, -4, -10, -2], [-3, -7, 3, -1], [2, -1, -6, 2], [-16, 0, 2, -8], [1, 2, -7, 6], [7, 1, -11, -2], [-11, -2, 9, 0], [8, 0, -2, -6], [13, 11, 1, -3], [1, -1, -3, -14], [-11, 2, 1, -6], [-2, -6, 0, 2], [-4, 4, 10, -2], [10, 6, -6, -4], [-7, 13, -1, 3], [-6, 2, 2, -8], [-14, 10, 2, 8], [-11, 3, 5, -2], [1, -8, 7, 2], [7, 3, -1, -10], [-12, -5, 0, 14], [-5, -2, 3, -2], [27, 1, -1, -9], [-4, 6, 4, 5], [-32, 2, 0, 1], [7, -4, -3, -6], [-5, -12, 3, 2], [24, 8, 0, -2], [-13, 9, 1, 4], [-14, 6, 0, -6], [-4, -6, 6, 2], [-8, 1, -2, 10], [11, 9, 7, -9], [-2, 4, -10, -4], [14, 6, -8, -1], [4, -4, -4, 2], [-12, -2, -6, 6], [17, -9, -9, -1], [-12, -1, 8, -6], [18, 6, -8, 2], [25, 3, 7, 4], [26, -2, -2, -7], [6, 0, 10, -1], [10, 12, 12, -4], [25, 4, 3, 2], [1, 8, 1, 2], [10, -14, -4, 1], [5, 3, 3, -6], [-12, -4, 0, -4], [14, -2, 8, 10], [3, 8, -5, 4], [-14, -6, -10, 0], [-5, -14, 3, 8], [20, -8, 0, -4], [20, -5, -8, -2], [6, -10, 2, 12], [-1, -15, -1, 1], [4, -2, 6, 0], [2, -12, 8, 6], [41, 4, 3, -6], [-17, 3, 5, 9], [0, -9, 8, 8], [-8, 4, -8, -4], [10, -8, -4, -6], [0, -6, 6, -4], [10, 3, 4, -12], [18, 10, -8, 2], [-12, 4, 0, -10], [-6, 0, -6, 6], [-5, -2, 9, 0], [7, 5, -1, 10], [-3, 1, 5, 4], [28, 4, -10, 5], [3, 1, 5, 12], [-17, 2, -9, -14], [-9, 12, -1, 0], [4, 2, -6, -10], [27, 5, -11, 4], [-50, 0, 0, 4], [3, -8, 3, 6], [9, -2, -5, -8], [26, -4, -8, 2], [-10, 16, 2, -2], [-19, -10, 7, 0], [26, -8, -2, 5], [-20, 2, 0, 0], [2, -4, 8, 14], [25, -7, 5, 11], [-40, 6, 0, -4], [-19, 3, 3, 0], [-4, 7, 14, 4], [9, 11, -1, 4], [-24, 0, 4, 8], [-16, -14, 0, 4], [-42, -4, -4, 2], [19, 9, -9, -6], [-19, -2, 1, -6], [6, -6, -16, 0], [13, 0, -1, -16], [-18, -4, 10, 5], [8, -8, 0, -2], [9, 2, 1, -6], [19, -3, -13, -2], [-11, 3, -5, 4], [-11, 11, 9, 1], [-25, 6, 5, 2], [46, -4, -2, -6], [22, -6, 10, 11], [26, 5, 8, -2], [29, -3, -1, -13], [-9, -3, -7, 5], [-4, -2, -6, -2], [-9, -6, 1, -6], [-13, 8, 3, 0], [9, -1, 3, -6], [-24, 14, -6, -4], [-14, 3, 12, 6], [-29, -11, 1, 10], [10, -2, -2, 8], [24, 0, 2, 0], [-42, 0, -4, -3], [19, 1, 5, 8], [-20, 0, 0, -12], [-17, -5, -5, -1], [-7, -1, -13, -10], [17, -2, 3, 14], [1, 3, 13, 12], [-16, 13, 8, 4], [-25, 14, 1, 2], [-30, 0, 6, -4], [34, 0, 0, 12], [37, 7, 3, -4], [-10, -4, 2, -10], [-12, 4, -2, 15], [0, 0, 0, -4], [15, 3, 3, 10], [30, -6, -2, -3], [23, -2, -1, -8], [35, -3, -1, -11], [8, 10, 10, -12], [-2, -10, 6, 5], [19, 3, 5, 1], [28, 2, 8, -6], [7, 7, 5, -7], [-23, -1, -3, 3], [28, -2, 0, -10], [14, 10, -2, -14], [37, -3, 9, 8], [9, 10, -5, -16], [-6, 6, -2, -6], [8, 0, 14, 0], [58, -2, -4, 1], [12, 4, 4, -18], [-17, 8, -3, -6], [12, 1, -6, 10], [3, -3, 13, 14], [2, -2, -8, 3], [0, 8, 16, 2], [38, 2, 4, 4], [32, -2, 0, 11], [32, -6, -4, -6], [-14, -6, 2, 6], [-1, 12, 9, 4], [12, 0, -4, -6], [36, 0, 0, -16], [6, -2, -10, 0], [-8, 12, -10, -1], [26, 2, 2, 2], [22, 0, -4, -2], [-7, -4, 1, 4], [0, 2, -2, 8], [-9, -11, 5, -9], [10, -5, 6, 4], [-12, 0, -6, 6], [-12, -8, 8, 4], [19, -6, 15, 8], [-21, -11, -3, 2], [38, -1, 0, -8], [-7, -2, -3, 12], [-30, -2, 8, 1], [-44, 6, -10, -10], [-16, -16, -4, 10], [30, -4, -4, 1], [-32, 4, 10, 7], [26, -8, 2, -5], [31, -12, -3, -6], [-5, -11, 9, 14], [19, -1, 5, -3], [14, 0, 2, -18], [-33, 3, 5, 9], [-34, -4, -8, -2], [45, -7, 13, 13], [36, -2, 10, 10], [-13, -7, 11, -2], [-7, 6, -15, -8], [-43, 5, 3, 7], [-37, 2, 3, 16], [-9, 5, -9, -3], [-25, 4, 1, -4], [-36, -8, -10, 2], [26, 2, 0, -8], [0, -4, -14, 2], [-13, 9, -1, -7], [10, 3, -6, 6], [24, 0, 8, 2], [-15, 20, -9, -12], [4, 6, -4, -9], [13, -5, -17, -2], [-6, -10, -10, -2], [13, 10, 1, 8], [20, 0, 10, 0], [-19, 11, -9, -7], [-10, -2, 12, -1], [-10, -11, -18, -6], [-41, -3, 9, -12], [7, 5, 17, 16], [23, -3, -1, 11], [-39, 10, 5, -4], [2, 3, 4, 12], [3, -7, 7, 3], [7, -4, -9, 16], [-10, 6, 8, -6], [30, -12, 4, 4], [-27, 7, -5, -11], [2, 5, -8, -14], [-13, -11, -1, 10], [24, 12, -4, -14], [-11, -10, 3, -8], [12, -6, -16, 7], [20, 10, 10, -4], [-8, 0, 16, 3], [6, 6, -10, -2], [-45, 4, 3, 14], [-6, 2, 8, 5], [11, 1, -5, 0], [-49, -7, 1, 2], [-16, 21, -2, 6], [-28, -4, -4, 4], [-4, -2, -6, -25], [-4, -10, -18, 8], [-20, -8, -8, 2], [0, 3, 12, 10], [37, -4, -5, 8], [-28, -8, 0, 3], [7, -8, -11, 2], [-31, 9, -5, 9], [-27, -4, -3, 8], [0, 10, 0, -8], [-10, 0, -8, -8], [-42, -2, 10, 2], [17, -1, -5, -3], [16, -8, -2, 12], [-31, 18, -5, 0], [24, -17, -6, 0], [-10, -6, 6, 8], [9, -4, -15, 0], [-23, 1, 5, -16], [2, 4, 2, 20], [2, 0, 8, -16], [-1, 9, -1, -18], [12, -5, 12, -4], [27, 5, 11, 13], [-9, 7, 7, 17], [60, 4, 0, 3], [33, -10, 3, -10], [-18, 9, -14, -16], [-23, -5, 1, -2], [16, -10, 2, 4], [-9, 6, -11, -14], [10, 8, -2, -16], [40, -1, 2, 6], [-36, 0, -8, 2], [-32, -8, 10, -16], [22, -2, -6, -4], [40, -12, -4, -2], [10, 10, -2, 0], [32, -12, -2, -7], [29, -10, 9, 20], [-7, -9, 15, 14], [-51, -4, -9, 2], [7, 1, 1, 29], [-8, -10, -6, -3], [-36, -8, 6, -2], [-25, -10, -13, -4], [16, 4, 8, -15], [-4, 18, 10, 1], [-7, -4, 3, 0], [12, -10, -4, 2], [-27, 2, 7, -14], [-5, 7, 17, -11], [42, -10, -6, -4], [36, 8, -18, 6], [-40, -11, 6, 2], [49, -5, 1, 4], [19, 18, 7, -2], [-18, 6, -4, -22], [10, 10, 4, -14], [-36, 4, -2, -8], [-5, -19, -21, 3], [-21, -7, -17, -6], [40, 4, -4, -2], [-3, -2, -3, -16], [-5, -4, 9, 16], [-35, -21, 5, 9], [-8, -10, 16, -2], [-24, 10, -2, -8], [1, -19, 5, 5], [-23, 11, -3, 2], [-35, 0, -3, -4], [-27, -1, -13, -6], [-56, 6, 2, -4], [26, 4, -2, -4], [23, -2, -13, -12], [-9, 2, -9, -2], [-35, -10, 3, 6], [-24, 4, 14, 5], [-14, 2, -12, -7], [-31, -11, -7, -1], [4, 4, 4, -27], [-9, 9, -11, -8], [-15, -8, 3, -10], [-28, 16, 8, 10], [18, -8, -6, 8], [-6, -6, 6, 9], [-7, 19, -17, -12], [-48, 6, -12, -6], [-72, 0, -2, -7], [17, 11, 17, 4], [21, -14, 13, 16], [-4, 10, -6, -4], [15, -3, -17, 10], [24, -6, 8, 1], [35, -10, 11, 6], [35, -8, 11, 16], [-60, 12, 14, 6]];
    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_bw();".
// 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_bw(: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_bw();".
// 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_bw( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-8, 0, 1]>,<11,R![8, -24, 6, 8, 1]>,<17,R![-224, 224, -44, -4, 1]>,<19,R![-448, -224, 12, 12, 1]>],Snew);
    return Vf;
end function;