
// Make newform 3630.2.a.br in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3630_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_3630_2_a_br();".
// 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_3630_2_a_br();".
// 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 := [100, 10, -21, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, -11, -1, 1], [-11, -1, 1, 0]];
    Rf_basisdens := [1, 1, 10, 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_3630_a();"
function MakeCharacter_3630_a()
    N := 3630;
    order := 1;
    char_gens := [1211, 727, 3511];
    v := [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_3630_a_Hecke(Kf)
    return MakeCharacter_3630_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 := [[-1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0], [0, -1, 0, 0], [0, 0, 0, 0], [-2, 0, -1, -1], [1, 1, -1, 1], [-1, -1, 2, -1], [3, 0, 3, -1], [-1, 1, -1, 0], [1, -1, -3, 0], [4, 0, -3, 0], [0, 1, 0, 2], [3, 1, 5, 1], [2, -2, 1, 1], [-2, 1, -6, 0], [4, -2, 1, -2], [4, -2, 0, 0], [11, -1, 5, 2], [-2, -2, 0, -2], [6, 0, 0, 0], [-5, 1, -9, 1], [8, 0, 4, 0], [4, 1, 4, 2], [4, 2, -6, 0], [-3, 1, -1, 1], [5, 1, -2, 1], [2, 2, 2, -2], [12, 0, 4, 0], [-1, 3, 3, 0], [7, 3, -2, 1], [-13, 1, -11, 0], [-9, 1, -3, 2], [4, 3, -4, 0], [11, -1, 1, -1], [2, -4, 4, -2], [1, 0, 1, -4], [1, -3, -9, -1], [2, 2, 7, -1], [-2, -1, -2, -2], [5, -2, 13, 0], [-8, 0, 0, 0], [-2, 4, 6, 4], [10, 2, 0, -2], [-10, 1, -18, 0], [1, 1, 3, -1], [-14, -2, -6, 0], [0, -3, 4, 0], [0, 2, 8, 4], [-4, 6, -2, 0], [7, -3, -1, 0], [4, 0, 18, 0], [-4, -3, -2, -2], [-7, 6, -7, 1], [14, 0, 12, 0], [-6, -4, -13, 0], [-11, 1, -1, -2], [9, -3, 9, -1], [11, 0, 23, 1], [2, 0, 8, 0], [-11, 3, -11, 0], [2, 1, -2, 2], [1, -5, 3, -2], [4, 2, 6, -2], [-4, 2, -4, -2], [3, -7, 8, -3], [-1, 3, -18, 3], [-2, -2, -10, -2], [8, -6, -4, -6], [10, 4, 4, -2], [-9, -1, 3, 3], [4, 0, -2, 4], [14, -2, 10, 0], [5, 5, 8, 1], [2, -3, 6, -6], [-9, 2, -9, 4], [-5, 1, -9, 3], [23, 2, 7, -3], [3, 5, 2, 5], [10, 0, -5, -1], [-10, 4, -1, 0], [2, -4, 16, -2], [-10, -2, -8, -4], [0, 6, -12, 4], [5, 5, 15, 2], [-4, -2, 2, -4], [-17, 1, 2, 3], [-2, -2, -10, 0], [-1, 5, -3, 3], [7, -9, 12, -1], [6, -2, 14, -6], [-2, 4, 10, -2], [8, 4, -16, 4], [-8, 4, -3, -1], [-18, 1, -10, -4], [8, 0, -1, 1], [5, 1, 1, 4], [-10, -5, 6, -4], [0, -6, 12, 0], [-2, 2, 2, 6], [13, 1, -17, 2], [-19, -1, 2, -3], [16, 4, 4, 2], [6, -3, 14, 0], [-6, 5, 2, 6], [6, -6, 0, -4], [2, -6, 22, -2], [9, 1, -3, 0], [-4, 0, 2, 6], [-3, 5, -12, 9], [20, -4, 12, -2], [-6, -4, -8, 4], [0, -4, 0, 2], [-21, 3, -4, 1], [-23, 1, -9, 2], [-15, -1, -30, -1], [9, 7, 7, 2], [9, 1, 3, -3], [-13, -3, -22, 1], [12, -3, 6, -4], [8, -2, 0, 0], [-4, 0, 10, 2], [-4, 6, -6, 8], [-20, 8, -10, 0], [-12, 9, -4, -2], [-26, 2, -16, -8], [-10, 6, -4, 4], [-20, 2, 2, 2], [12, 5, -12, 2], [-11, -1, -15, 7], [-16, -3, -20, -4], [11, 0, 15, 4], [-1, -1, 13, 1], [-10, 8, -13, 3], [14, 0, -8, -4], [16, -4, 23, 0], [-8, -1, 0, 4], [-17, -5, 7, -3], [1, -5, 6, -5], [7, -3, -14, -3], [6, -3, 18, 2], [-32, -2, -12, 2], [-9, 3, 2, 1], [10, -8, -2, 2], [-24, -2, -10, 4], [-30, 0, -10, -4], [-36, 5, -14, -2], [-23, -1, -1, -3], [-4, 3, 20, 2], [-6, 0, 25, 0], [-8, 4, 9, -4], [-14, 3, 6, -2], [-15, 1, 3, -5], [1, -2, 21, 1], [37, 1, 19, 3], [0, 2, 8, 2], [-21, -3, 5, -7], [-10, -9, -14, -2], [22, -2, 18, 2], [13, -5, -5, -2], [-30, 4, -32, -2], [-1, 9, -15, 3], [-21, 11, 0, 5], [4, 7, -14, 8], [14, 8, 0, 8], [-32, 1, -26, 0], [-13, -5, 5, -7], [-35, 7, -27, -5], [2, 0, 29, 1], [22, -4, 12, 6], [28, -8, 13, -1], [-2, -4, 8, 0], [-26, 2, 4, 4], [6, -12, -2, 0], [15, -5, -5, 4], [25, -1, 24, -1], [-41, 3, -20, -7], [1, -3, 15, -3], [-28, -5, -8, 2], [18, 6, 2, 4], [34, -2, 10, -2], [13, -1, 4, 3], [-5, 9, -29, 1], [7, 5, 5, 2], [15, -7, 20, 1], [-15, 3, 13, 1], [-5, -1, 15, -2], [-13, 5, -19, 6], [15, 7, -13, 6], [-14, 2, 8, 4], [-18, -2, -6, 0], [4, -8, -12, -8], [-37, -1, -4, -5], [25, -7, -11, -7], [-20, 8, -16, 6], [-17, 5, -17, -4], [-7, 9, 4, 5], [30, 0, 12, -8], [-9, -1, 17, -6], [-15, -4, -11, 5], [11, 5, 7, -1], [-33, 1, -31, -6], [11, 11, -8, 3], [-9, 5, -13, 9], [15, 5, -15, 0], [-3, 3, -24, 9], [-13, -1, 13, 5], [10, 0, -2, -4], [18, -11, 10, -8], [4, 8, -8, 10], [16, 8, -4, 8], [-9, 1, -41, 0], [16, -3, -4, 2], [32, 2, 10, 4], [2, -2, 12, -8], [-4, 2, -15, 11], [-20, -8, 8, 0], [-13, -3, -20, 9], [9, 7, 19, -4], [2, 4, 20, 0], [14, -6, -4, 2], [-9, 3, -25, 1], [2, 5, 14, 4], [-16, -7, -32, -6], [6, -14, -16, -4], [-18, 6, -30, -2], [13, -3, 5, -7], [-4, -6, -4, -8], [25, 1, 12, 3], [15, -2, 7, 0], [7, -1, 20, -5], [2, 7, 26, 4], [5, 3, 27, 2], [48, -5, 20, 6], [41, 3, 7, 3], [-33, 5, 5, 0], [26, -10, 37, 2], [-15, -1, -8, 7], [-3, -5, -1, 4], [32, 4, 24, -2], [-20, -6, -4, 4], [-38, 5, -22, -6], [0, -3, -16, 6], [-44, 2, -16, -2], [23, -9, 5, -1], [-2, -2, -16, 6], [11, 11, 2, -1], [28, -12, 20, 0], [6, 10, 2, 8], [-15, 8, -19, 12], [-1, 2, 11, -8], [9, -1, -24, -1], [-2, -2, -17, 3], [25, 0, 5, 11], [9, -13, 8, 5], [-31, -1, -28, -5], [-4, 6, -36, 2], [-43, -3, -3, 3], [49, 3, 10, 1], [12, 2, -8, 4], [12, -12, 20, 4], [24, 2, 16, -2], [-6, 6, -16, 12], [-19, 10, -15, 7], [5, -15, 9, -8], [53, 5, 16, -7], [5, -7, 3, -18], [19, -7, -8, 1], [-13, -13, 19, -4], [11, 13, -32, 5], [-8, -4, 6, 6], [23, -3, -13, -3], [24, 8, 38, -2], [7, -15, 9, -3], [8, -2, 20, -4], [-3, 13, 14, 13], [18, -10, 10, -2], [12, -6, 28, -10], [40, 0, 29, -4], [-3, -3, -5, -3], [-12, 5, -6, -8], [-47, 3, -15, -9], [-21, -3, -4, -11], [20, -6, 48, -6], [-39, 1, -21, 11], [12, 6, 6, 12], [-14, 2, -4, -8], [-11, -1, -2, 9], [27, 11, -25, 4], [4, 1, 4, 2], [43, -3, 39, 4], [-46, -4, -2, -8], [16, -10, 0, -10], [1, -2, -11, -3], [5, 0, -7, 8], [39, 7, 5, 3], [23, 1, 23, 0], [26, 11, -10, 2], [18, 0, 21, 0], [3, -3, 49, -2], [14, 8, -26, 0], [-36, -5, 4, -2], [-37, 13, -1, 0], [30, 10, 22, -4], [3, -1, 6, 1], [-2, 0, -10, 10], [8, 0, 30, 4], [64, -2, 26, 8], [6, -18, -6, -10], [-32, -1, -22, -10], [-19, -3, 27, -7], [-5, 1, -29, -1], [-36, -6, -14, 4], [39, -11, 39, 1], [-44, 6, -21, 2], [-17, 7, -17, -6], [31, -6, 11, 0], [2, 10, 8, 4], [20, -4, -32, -2], [-23, 3, -48, 1], [2, 7, 10, 2], [24, 4, 26, 8], [-12, -8, 12, -14], [-36, 7, -8, -4], [12, 14, -4, 10], [-3, 0, 5, 11], [-12, 5, -4, -2], [-21, 1, -39, -7], [12, -5, 30, -8], [-7, 15, 4, 7], [28, 6, -14, 2], [19, 13, -10, 1], [-3, -9, -5, -18], [-30, 4, 12, 8], [9, 1, 8, 13], [-20, 0, -26, -10], [12, -2, 32, -4], [41, -9, -9, -6], [-30, -4, -3, -11], [22, -2, 18, 0], [-12, -3, 10, 2], [-23, -7, 11, -2], [9, 5, -7, 0], [-29, -1, -13, -8], [44, 3, 8, 8], [-3, -5, -36, -1], [-8, 20, -20, 4], [-11, 13, -15, 12], [2, -4, 14, -4], [13, 3, 48, 7], [-16, -4, -32, -6], [42, -14, 24, 8], [-7, -5, -13, 5], [-25, 11, 14, 5], [6, 0, -8, -2], [4, -2, -16, 6], [-2, 2, 8, -14], [-7, -2, 13, 0], [-9, 5, 53, 3], [37, 1, 27, 9], [44, -6, 13, 7], [25, 7, 16, -3], [11, 5, 21, -4], [9, 4, -23, 5], [-45, -5, -21, 5], [8, 6, 4, 2], [-32, 1, 8, 0], [-23, 1, 8, -7], [1, 0, -19, -8], [40, -13, -2, -8], [17, -7, -38, -5], [-34, -12, 23, -8], [2, -14, 20, -8], [-4, -8, 8, 0], [-27, 8, -7, 3], [11, -5, 17, 3], [-46, 4, -4, -4], [-8, -6, -14, -8], [-27, 11, 2, 3], [22, 2, 12, 2], [-19, -3, -15, -12], [43, 3, 11, -1], [44, -3, -4, -6], [-37, -7, -25, 8], [-10, -3, -18, 4], [21, 14, -27, 12], [23, -3, 59, 1], [-46, 4, -29, -1], [-1, -15, 41, -10], [-14, 0, 1, -12], [5, 0, 5, -9], [-9, -11, 18, -9], [13, 2, -27, 5], [-35, -15, 29, -13], [30, 10, 16, -6], [38, 10, 4, 4], [-21, 5, -7, 10], [29, 11, -26, 3], [-30, 14, 10, 0], [15, -5, 39, 3], [18, -8, -2, -6], [-5, -13, 9, -10], [4, 5, 24, 2], [-4, 26, -28, 6], [-83, -3, 1, 0], [-23, 11, -1, -5], [8, -9, -14, 2], [8, 7, -28, 6], [1, -13, 11, -6], [-29, 0, 7, -9], [-4, -4, 16, -14], [-53, 1, 6, -7], [-18, 8, 4, -8], [-39, -3, 28, -5], [-16, -6, 28, 2], [25, -21, 35, -2], [32, -4, 20, 2], [36, -8, 28, -6], [-19, -15, 11, -7], [-11, 8, 29, -4], [-50, 10, -20, -4], [5, 7, -47, 3], [-7, 16, 1, 12], [66, 0, 47, 0], [-16, 8, 8, -4], [-36, 1, 8, 10], [-16, 10, -30, -2], [28, 1, 32, 4], [-41, 7, -24, 7], [-30, 2, -48, 10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3630_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3630_2_a_br();".
// 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_3630_2_a_br(:prec:=4)
    chi := MakeCharacter_3630_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_3630_2_a_br();".
// 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_3630_2_a_br( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3630_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![100, -10, -21, 1, 1]>,<13,R![-11, -112, -27, 4, 1]>],Snew);
    return Vf;
end function;