
// Make newform 3630.2.a.bs in Magma, downloaded from the LMFDB on 25 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_bs();".
// 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_bs();".
// 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 := [121, 19, -24, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-27, -14, 2, 1], [-125, -42, 10, 3]];
    Rf_basisdens := [1, 1, 4, 4];
    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, 0, -1, 1], [0, 0, 0, 0], [1, -1, 0, 1], [2, 1, -1, 0], [1, 1, 2, 0], [0, -1, 0, -1], [0, 0, -4, -1], [4, 0, 0, -1], [1, 2, -1, 0], [0, -2, 5, -1], [2, -1, 3, 0], [3, 1, 2, -1], [4, 0, 3, -1], [-1, 0, -7, 0], [-4, 0, -2, -2], [2, -2, -2, 1], [2, 2, 0, 0], [-6, 0, -8, 0], [-4, 3, -3, 0], [8, 0, 0, 0], [-8, 2, -3, -1], [-2, 0, -4, 2], [6, 1, 11, 0], [9, -3, 2, 0], [-12, 2, -2, 0], [4, 0, 8, 0], [0, 0, -2, 1], [7, -3, 4, 2], [-6, 0, 6, -1], [-2, -2, 0, 1], [-8, 0, -11, -1], [10, -1, 5, 0], [-10, 2, -2, -2], [4, 0, -7, 2], [-2, 1, -9, 2], [1, -1, 6, 1], [0, 2, -5, -1], [-4, 0, 9, 0], [16, 0, 4, 0], [4, 0, -2, -4], [-2, 2, 8, 0], [8, -4, 3, 3], [6, -3, 3, -2], [0, -4, 4, -2], [4, 0, 1, -1], [-8, 0, -6, 2], [2, 0, 0, -2], [-4, 0, 4, -1], [2, -4, -2, 4], [-10, 2, 3, -1], [4, 1, 8, -3], [-2, 0, 0, 4], [5, 4, 7, 0], [-2, 2, -2, -3], [4, 1, 5, -2], [-8, -1, 4, -3], [-6, 0, -8, 4], [0, -4, 0, -1], [-12, 2, -3, 1], [14, -2, -4, -3], [-6, 2, 6, -4], [8, -2, 4, 4], [-3, 1, 16, 0], [13, 1, 2, -4], [8, 2, 6, -4], [-8, 2, -4, -4], [6, 2, -6, -2], [4, -1, 3, 0], [-6, 0, 10, 0], [12, 0, 16, -2], [-9, 1, -8, 0], [12, -2, 19, -1], [8, 0, 5, 2], [-12, -1, -19, -2], [-20, 3, -14, -1], [-11, -1, -18, 0], [-1, -7, -4, 1], [3, 2, -5, 4], [6, 2, 2, 2], [-22, 0, -10, 2], [0, 0, 6, -6], [-6, 2, -2, -1], [10, -4, 8, 2], [1, 1, 8, -2], [-4, 0, 20, -2], [-10, -1, 3, -2], [-11, -1, -12, 0], [0, 2, 6, 4], [16, 2, 16, -2], [-16, 0, -4, -4], [-25, 1, 2, 3], [16, 4, 1, -3], [11, 1, 2, -3], [12, -4, 22, -1], [8, -4, -1, 3], [-16, 4, -6, 2], [16, 6, 10, 0], [6, 2, 2, -5], [5, 5, 12, -2], [0, -2, -14, -2], [8, 0, -7, -7], [-8, -2, -1, 3], [-14, 0, 14, -2], [8, 2, 6, 0], [8, -8, 10, -1], [-6, -2, -16, 2], [-15, -1, 0, -4], [16, 2, -2, -6], [14, 0, 20, 4], [-20, -2, -2, 2], [11, -9, 6, 2], [-10, 2, 6, 7], [-13, 5, 6, -4], [-6, 10, -12, 1], [-22, -3, -1, 0], [-9, -3, 2, 0], [14, -4, 9, 1], [0, -4, 2, 6], [-6, -6, -12, -2], [-10, -4, -20, 2], [2, -4, -6, 4], [-8, -2, -15, -5], [-2, 4, -10, 2], [10, 4, -10, 2], [14, 2, -18, 4], [-8, -2, 11, 1], [-8, 5, -5, 0], [12, 0, -13, 1], [-12, 0, -29, 4], [30, -1, 21, 2], [-27, -5, -8, 5], [-10, -4, 4, 0], [1, 0, 13, -4], [-8, 12, -5, -3], [28, 3, 3, -2], [-11, -1, -22, 4], [-7, 3, 2, 0], [20, -2, 7, 3], [4, -6, -4, 0], [11, -3, -8, 2], [32, -2, 16, 2], [-22, 0, -16, 2], [4, 4, 10, 4], [-10, 2, 1, 1], [-22, 5, -11, 2], [0, 10, -5, 1], [-21, -4, 1, 4], [-23, -2, -13, 0], [-12, 2, 9, -3], [10, -3, 15, -2], [-4, -7, 6, -5], [-46, 1, -1, -2], [-24, 2, -4, -4], [22, -5, 7, 0], [12, -2, -3, 1], [-20, -2, 6, 0], [-2, 6, 8, -1], [-6, -2, 2, 6], [10, -1, 11, 6], [29, 1, 2, 2], [10, -8, 7, 7], [-26, 4, -4, -4], [-2, -8, 11, 3], [6, 3, -3, 2], [12, -7, -13, -4], [-3, 7, -12, 7], [6, 6, 2, 2], [3, 1, -2, -1], [30, -4, 8, 4], [-34, 8, -18, -6], [-28, -4, 14, -4], [0, -8, -10, 3], [-11, 5, -8, 0], [7, 3, -6, -10], [-26, 9, -1, 4], [12, 2, 9, -5], [4, -4, 0, -2], [-16, 6, -30, 0], [-31, 5, -24, -4], [-20, -5, -15, 0], [-30, 2, 12, -3], [-15, -3, -4, 0], [4, 5, -1, 2], [-16, 2, 0, 1], [14, -2, 12, 3], [-28, -6, -20, 1], [18, 4, 6, -2], [-8, -8, -20, -2], [8, -8, 8, -4], [-5, 1, 0, 8], [-24, -7, 1, 0], [-8, 6, -14, -2], [-8, 0, -4, 7], [5, 11, 0, -4], [-2, -4, 0, 12], [-30, -2, -2, 7], [-12, 5, -40, 1], [20, -5, 1, 10], [6, -2, 20, -1], [9, -5, 20, 4], [20, -1, 33, 4], [18, 8, 6, 3], [27, 1, -14, 2], [-22, 11, -15, -2], [0, 0, 6, 0], [0, 0, 5, 5], [28, 2, 6, -6], [24, -4, 4, -4], [12, -12, 0, 3], [4, 2, -9, 5], [6, -4, 0, 10], [-18, 0, -26, -2], [-5, 5, -24, -1], [12, 4, -12, -4], [19, -1, 28, 0], [-6, -8, 2, -7], [34, -8, 32, 0], [-6, 2, -28, 4], [12, -1, 23, 6], [28, 0, 1, 5], [0, 2, 23, 9], [6, 4, 22, 2], [16, -2, 18, 4], [-4, -7, -23, 4], [8, -12, 2, 6], [27, 7, -18, 6], [4, -8, -1, 12], [9, 3, 12, -8], [40, 4, 15, -1], [-2, 2, 16, -7], [-12, 2, -5, 1], [30, -3, 5, 4], [14, 4, 30, -1], [13, 6, 7, 4], [15, -5, -20, 4], [2, -4, -18, -7], [56, 2, 10, -2], [4, 0, 22, -2], [32, -2, -9, -1], [8, 6, -17, 5], [-12, -2, 16, -4], [-26, 3, -23, 4], [-22, 2, -24, 0], [-17, -5, -22, 4], [-16, 0, -8, -4], [-12, 8, -20, -2], [-12, 0, -39, 4], [-16, 4, 17, 8], [5, 5, 8, -12], [-3, 3, 14, -15], [28, 5, -2, -3], [39, 5, 14, 2], [37, -3, 16, -4], [20, -2, -12, 8], [52, -3, 33, 2], [33, 1, 36, -2], [-28, 0, 2, 2], [0, -12, 4, 0], [-24, -10, -24, 4], [2, 0, -14, -6], [-48, 1, 6, 1], [0, 8, -26, 7], [-15, 7, -8, -8], [-10, -2, -10, -7], [-1, -13, 0, 0], [28, -4, 22, -5], [-27, 1, -20, 4], [34, -6, 32, 2], [-4, -1, 29, 0], [10, -2, -8, -2], [6, 7, -13, -8], [-28, -12, -6, 2], [-7, -3, -26, 4], [24, 6, -6, 8], [-12, 10, -8, 0], [-29, -4, 23, -4], [-18, -3, -21, -8], [38, 8, 27, -1], [-4, 1, -17, 4], [-19, -7, -12, 8], [12, 6, -12, 8], [14, -7, -9, 6], [-22, 0, -48, 2], [-6, 4, -14, 2], [-7, -7, 4, -10], [20, -4, 34, 11], [16, -2, 1, -1], [40, -8, 20, 7], [0, -4, -26, 8], [48, -10, 40, 8], [0, -3, 8, 1], [-48, 4, -11, -2], [-2, -1, -3, 4], [24, -8, -12, -7], [12, -6, -9, 7], [25, 6, -3, -8], [-18, -6, -36, 7], [-28, 0, 2, 4], [8, 10, -3, -9], [-32, 4, -36, -7], [4, 8, -12, -10], [-15, -5, 4, 10], [4, -2, 24, -10], [30, -8, 6, 8], [26, -4, -4, 2], [48, -2, 34, 0], [-38, 2, -5, -9], [-10, 1, 7, 0], [12, -1, 1, 6], [-14, -4, -8, 10], [4, -1, -11, 4], [3, -2, 31, 0], [0, 6, 4, -3], [40, 0, 23, -8], [6, 0, -2, -6], [20, -10, 34, 2], [-35, -9, -38, 2], [8, 10, -23, 9], [-2, 0, -6, -4], [16, 2, 6, -2], [12, 0, 23, -7], [24, 2, 8, -12], [8, 5, 6, -3], [-48, 14, -19, -5], [-22, 11, -13, -12], [26, 8, -11, 1], [19, -1, 12, -8], [6, -10, -18, 8], [-29, -11, -18, 0], [38, -2, 16, 1], [-6, 4, -28, -4], [53, -1, 44, 0], [42, 2, 12, 2], [0, 4, -30, 10], [-22, 2, -4, -1], [-3, -13, -4, 7], [4, -4, -20, 2], [46, -10, 29, 5], [18, 10, -30, 5], [8, 8, -14, 11], [40, 0, 6, 3], [-28, -4, 15, -3], [17, -7, 8, -4], [-12, -4, 16, 4], [12, 0, -6, 1], [24, 0, 14, 0], [43, 11, 0, 0], [0, 6, -42, 10], [2, -8, 10, 10], [-26, -3, -35, 10], [-9, 9, -24, -6], [22, 2, 34, -2], [4, 6, 12, 4], [-14, -2, -52, 8], [0, 16, -23, 0], [-6, -5, 31, -10], [-18, 1, -13, 12], [9, 7, 40, 3], [13, 3, -22, 2], [2, 8, -30, 11], [-12, 13, 14, 7], [-20, -13, -25, 14], [-32, -2, -44, -4], [24, -16, 25, -5], [-57, 5, -36, -4], [36, -8, 35, 0], [18, -8, 55, -5], [-61, 5, 4, 2], [-11, -4, -13, 16], [10, 0, 2, -6], [-8, -12, 20, -4], [8, 5, 32, -9], [30, 11, 33, -16], [-14, 8, 20, 4], [-18, -12, 24, -6], [-17, -11, -30, 4], [-10, 2, 40, -4], [-8, 8, 34, 11], [-36, 7, 31, 12], [16, 2, -9, -3], [-20, -12, 16, 3], [40, 8, 17, 1], [16, 0, -17, -10], [20, -5, 25, 8], [1, 15, -40, 5], [34, 10, 8, 3], [59, 0, 61, 0], [-32, 1, 26, -7], [-45, 3, -36, -6], [-12, -11, 4, 5], [48, -15, 25, 10], [34, -2, 64, 0], [-26, 0, 2, -6], [30, -14, 12, 15], [11, -3, -10, -12], [40, 4, -12, -2], [-16, 15, -21, -8], [8, -6, -24, 10], [46, 2, -18, 7], [-4, 6, 39, 5], [-36, 2, -28, 8], [16, -12, 26, -5], [-26, -3, -3, 0], [-38, 6, -1, -5], [12, 2, -1, 13], [-10, -6, 20, 5], [8, 7, 18, -15], [-44, 2, -2, -2], [19, 5, -6, -8], [-14, -16, 4, 0], [-15, -3, -10, -2], [-32, 14, -28, 0], [-6, 6, -8, 7], [8, -6, -10, 2], [8, 2, -30, 6], [14, -3, 3, 0], [-40, -8, 1, 4], [22, 4, 22, -10], [0, 9, 7, 4], [8, 8, 29, 4], [-25, -6, -1, 8], [-36, 12, -12, 4], [28, -14, -1, -3], [-22, 2, 26, -12], [-12, -4, -1, 1], [11, 5, -48, 0], [50, 2, 4, 8]];
    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_bs();".
// 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_bs(: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_bs();".
// 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_bs( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3630_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![116, 34, -29, -1, 1]>,<13,R![121, 42, -41, -2, 1]>],Snew);
    return Vf;
end function;