
// Make newform 483.2.q.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_483_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_483_q_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_483_2_q_b();".
// 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_483_2_q_b();".
// 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 := [1, -1, 1, -1, 1, -1, 1, -1, 1, -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_483_q();"
function MakeCharacter_483_q()
    N := 483;
    order := 11;
    char_gens := [323, 346, 442];
    v := [11, 11, 6];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_483_q_Hecke();"
function MakeCharacter_483_q_Hecke(Kf)
    N := 483;
    order := 11;
    char_gens := [323, 346, 442];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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, 1, -1, 0, -1, 0, -1, 1, 0, 0], [-1, 1, -1, 1, -1, 1, -1, 1, -1, 1], [-1, 0, 0, 0, -1, 0, -1, 1, -1, 1], [0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [1, 0, 0, -1, 0, -1, -1, 0, -1, -1], [-2, 2, 0, 0, 0, 1, -1, 1, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [-3, 1, -1, 1, -1, 3, 0, 2, 0, 2], [-2, 0, -2, 0, 0, 0, 1, 2, 2, -2], [1, -4, 0, 1, 0, -4, 1, 0, 1, -1], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 0, -1, 1, 0, 0, 3, 0, 2], [-1, 4, -7, 4, -1, 0, 0, 2, -2, 0], [3, 0, 3, -2, 1, -1, 2, -3, 0, -3], [-3, 0, -3, 0, -2, 1, -1, 2, 0, 3], [0, 3, -4, 1, 1, 5, 1, 1, -4, 3], [1, -1, 0, -3, 0, -4, 8, -4, 0, -3], [-4, 6, -4, 0, 0, 6, 0, 0, -6, 0], [-3, 2, 0, 0, -5, 0, 0, 2, -3, 0], [2, 1, -4, 2, -6, 2, -4, 1, 2, 0], [1, 2, 3, -3, -2, -1, 0, 1, -5, 1], [0, 0, 0, 5, -5, 0, -5, 0, -5, 5], [-4, 3, -3, 2, -2, 3, -3, 4, 0, 1], [3, -3, 11, -3, 5, -5, 3, -11, 3, -3], [-6, -2, 1, -2, -6, 0, -4, 1, -1, 4], [3, -5, 2, -9, 9, -2, 5, -3, 0, -7], [-2, 8, -8, 2, 0, 3, -8, 3, -8, 3], [-7, -1, -7, 0, 0, -1, -3, 3, 1, 0], [-2, -6, 0, 1, 0, -6, -2, 0, -7, 7], [-7, 7, -5, 6, -9, 6, -5, 7, -7, 0], [4, 2, -2, -4, 0, -5, 2, 2, 2, -5], [0, 0, -3, -2, 3, -4, 3, -2, -3, 0], [4, 0, -1, 9, 1, 6, -6, -1, -9, 1], [-9, 0, -6, 0, -6, 3, -3, 6, 0, 6], [-5, 1, -1, 5, 0, 3, -7, 4, -7, 3], [2, -2, 0, 3, 1, -7, -4, -7, 1, 3], [-4, 7, 3, -3, -7, 4, 0, 9, 1, 9], [4, 2, -8, 1, -1, 1, -8, 2, 4, 0], [-1, -5, -1, 0, -1, -5, -1, 0, 8, -8], [-10, 7, -7, 10, 0, 7, -1, 1, -1, 7], [-12, 9, -13, 9, -12, 0, -13, 12, -12, 13], [8, -5, 12, -14, 5, -5, 14, -12, 5, -8], [0, -2, -6, 2, 7, 1, 7, 2, -6, -2], [4, -6, 9, -1, 1, -9, 6, -4, 0, 8], [-6, 6, 0, 9, -7, 6, 2, 6, -7, 9], [7, -7, 7, 0, 3, -6, 7, -7, 6, -3], [6, -7, 10, -10, 7, -6, 0, -12, 11, -12], [0, 2, -6, 13, -10, 3, -10, 13, -6, 2], [8, -9, 7, -12, 7, -7, 12, -7, 9, -8], [-2, 0, -5, 5, 6, 7, -7, -6, -5, 5], [15, -13, 12, -6, 6, -6, 6, -12, 13, -15], [-5, -4, 1, 0, 0, -1, 4, 5, 0, -7], [-5, 0, 0, 5, 0, 5, 6, 1, 6, 5], [-1, 9, -2, 11, -6, 11, -2, 9, -1, 0], [-5, 2, 2, -2, -2, 5, 0, -4, -16, -4], [-3, 3, 0, 14, -6, 9, 1, 9, -6, 14], [0, 0, 11, -15, 8, -9, 8, -15, 11, 0], [9, -18, 0, -18, 9, 0, 2, -3, 3, -2], [1, 0, 5, 6, 4, -2, 2, -4, -6, -5], [-1, 0, 7, 0, -1, 0, 2, -12, 12, -2], [0, -5, 7, -5, -3, -1, -3, -5, 7, -5], [-3, 9, -17, 0, -17, 9, -3, 0, -8, 8], [-8, -4, -8, 0, -8, 6, -5, 5, -6, 8], [6, -1, 1, -6, 0, -10, 3, -2, 3, -10], [-2, 2, 2, 12, -12, -2, -2, 2, 0, 0], [-2, -9, -8, -9, -2, 0, -5, 7, -7, 5], [1, 10, -6, 6, -6, 6, -10, -1, 0, 9], [1, -7, 1, 0, -5, 0, 1, -1, 0, 5], [-2, 14, 4, 2, 0, 2, 4, 14, -2, 0], [6, -4, 0, 0, 4, -6, 0, -5, -15, -5], [-2, 18, -2, 0, -1, 2, -7, 7, -2, 1], [-9, -6, 4, 1, -1, -4, 6, 9, 0, 15], [2, 0, 2, -2, -2, 6, -6, 2, 2, -2], [14, -13, 6, -13, 14, 0, 4, -2, 2, -4], [6, -3, 3, -6, 0, -14, 3, -9, 3, -14], [-7, 5, -7, 0, -6, 9, -13, 13, -9, 6], [7, -13, 14, -10, 19, -10, 14, -13, 7, 0], [21, -16, 9, -9, 9, -16, 21, 0, 9, -9], [-3, 3, 0, 10, -11, 6, -1, 6, -11, 10], [2, -5, -9, -5, 2, 0, 4, 1, -1, -4], [-3, -8, -3, -8, -3, 0, -10, -7, 7, 10], [0, 8, 12, -8, 15, -1, 15, -8, 12, 8], [5, 2, 3, -21, 3, 2, 5, 0, 6, -6], [-10, 0, 0, 0, 0, 10, 0, 11, -3, 11], [4, -3, -13, 4, -3, 4, -13, -3, 4, 0], [3, -5, -2, -4, -2, -5, 3, 0, 11, -11], [13, -2, 2, -13, 0, -14, 7, -7, 7, -14], [19, -6, 23, -26, 13, -13, 26, -23, 6, -19], [10, 0, 6, -4, 14, 9, -9, -14, 4, -6], [8, -8, 21, 0, 21, -21, 0, -21, 8, -8], [0, -14, 0, -16, 12, -16, 12, -16, 0, -14], [0, 15, -11, 11, -15, 0, 0, 14, 1, 14], [-5, -2, 16, -16, 2, 5, 0, 1, -9, 1], [-4, 14, -4, 0, -2, 3, 3, -3, -3, 2], [0, 7, -7, 6, 8, 14, 8, 6, -7, 7], [24, -4, -1, -4, 5, -5, 4, 1, 4, -24], [-8, -3, 0, 4, 1, -1, -4, 0, 3, 8], [18, -21, 18, 0, 20, -9, 3, -3, 9, -20], [16, -7, 0, -4, 0, -7, 16, 0, 3, -3], [0, 1, -1, 0, 0, 10, -1, -6, -1, 10], [2, -1, -15, -1, 2, 0, -3, -4, 4, 3], [11, -1, -8, -1, 11, 0, 1, 9, -9, -1], [-3, -10, 10, 3, 0, -10, 9, -3, 9, -10], [7, 12, 15, -12, 15, 12, 7, 0, -3, 3], [-1, -3, -4, 4, 3, 1, 0, 3, 19, 3], [2, -3, 2, -5, -33, -5, 2, -3, 2, 0], [4, -15, 15, -4, 0, -3, -11, -7, -11, -3], [-16, 9, 10, 9, -16, 0, -5, 7, -7, 5], [-5, 0, 1, 11, -6, -3, 3, 6, -11, -1], [-15, 15, -12, 2, 9, -9, -2, 12, -15, 15], [-12, 12, 12, -3, 3, -12, -12, 12, 0, 7], [-2, -15, -2, 0, 6, -9, 0, 0, 9, -6], [-6, 28, -6, 6, -28, 6, 0, 5, 0, 5], [-28, 16, -19, 10, -24, 10, -19, 16, -28, 0], [10, -10, 0, 5, -1, 10, 8, 10, -1, 5], [-2, -14, -1, 4, 3, -3, -4, 1, 14, 2], [0, 0, 7, -18, 3, 2, -2, -3, 18, -7], [1, -11, -8, 15, -7, 7, -15, 8, 11, -1], [-13, 18, -9, 16, -16, 9, -18, 13, 0, -4], [28, -22, 28, 0, 8, -8, 11, -11, 8, -8], [2, -13, 12, -22, 12, -13, 2, 0, 9, -9], [26, -20, 15, -2, 15, -20, 26, 0, 8, -8], [-11, 11, 0, 13, -7, 19, -31, 19, -7, 13], [0, -1, -11, -1, -1, 7, -1, -1, -11, -1], [2, 0, 2, 12, -11, 11, -11, 11, -12, -2], [-6, -13, 13, 6, 0, -3, 9, -13, 9, -3], [-14, -1, -5, 5, 1, 14, 0, -10, 1, -10], [-8, -4, 5, 10, 5, -4, -8, 0, 9, -9], [-8, -10, 9, 10, -10, -9, 10, 8, 0, 3], [-30, 25, -20, 20, -15, 15, -20, 20, -25, 30], [-25, 11, -20, 13, -4, 4, -13, 20, -11, 25], [0, -2, 2, -16, -14, 7, -14, -16, 2, -2], [-15, 38, -15, 0, -7, 0, -4, 4, 0, 7], [-20, 18, -18, 23, -18, 23, -18, 18, -20, 0], [20, -15, -2, 2, -2, 2, -2, -15, 20, 0], [-14, 14, 0, 3, 2, 7, 7, 7, 2, 3], [-5, -2, 12, 15, -15, -12, 2, 5, 0, 3], [7, -8, 12, -8, 7, 0, 15, -12, 12, -15], [-15, 34, -15, 0, -13, 21, -22, 22, -21, 13], [13, 0, -5, 5, 0, -13, 0, -13, 19, -13], [-8, 19, -11, 24, -11, 19, -8, 0, -25, 25], [0, 22, -22, 15, -18, 15, -18, 15, -22, 22], [-9, 36, -14, 36, -9, 0, -10, 13, -13, 10], [33, 0, 20, -3, -4, -18, 18, 4, 3, -20], [5, 0, 7, 13, -5, -8, 8, 5, -13, -7], [3, -13, 13, -3, 0, 5, 2, -22, 2, 5], [3, -23, 12, -22, 0, -22, 12, -23, 3, 0], [20, -13, -2, -16, -2, -13, 20, 0, 10, -10], [4, 13, 4, 0, 21, 3, -2, 2, -3, -21], [0, 18, -18, 0, 0, 9, -20, 15, -20, 9], [-25, 9, 1, 8, 6, -6, -8, -1, -9, 25], [0, 12, -3, 12, 8, -12, 8, 12, -3, 12], [18, -17, 13, 1, -1, -13, 17, -18, 0, -22], [3, -3, 0, 10, 4, -3, 18, -3, 4, 10], [-25, 25, 0, 4, 0, 7, 2, 7, 0, 4], [29, -28, 31, -26, 26, -31, 28, -29, 0, -24], [-6, 0, -13, 7, 12, -8, 8, -12, -7, 13], [-11, 20, -10, 17, -17, 10, -20, 11, 0, 22], [25, -7, 12, 1, 12, -7, 25, 0, 3, -3], [-7, 13, -17, 12, -8, 12, -17, 13, -7, 0], [-9, 2, -5, 5, -2, 9, 0, 23, -11, 23], [0, 0, 0, -25, 9, -17, 31, -17, 9, -25], [1, 0, -18, 13, 5, -6, 6, -5, -13, 18], [-26, 25, -27, 25, -26, 0, -22, 21, -21, 22], [2, 7, -7, -2, 0, 2, -3, -22, -3, 2], [-23, 26, -30, 29, -30, 26, -23, 0, -31, 31], [22, -8, 14, -4, -17, -4, 14, -8, 22, 0], [-10, 17, -10, 0, 5, 27, 2, -2, -27, -5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_483_q_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_483_2_q_b();".
// 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_483_2_q_b(:prec:=10)
    chi := MakeCharacter_483_q();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_483_2_q_b();".
// 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_483_2_q_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_483_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -20, 147, -135, 126, -100, 75, -37, 14, -5, 1]>],Snew);
    return Vf;
end function;