
// Make newform 2475.2.a.bf in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2475_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_2475_2_a_bf();".
// 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_2475_2_a_bf();".
// 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, 4, -6, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, -2, 1, 0], [5, -3, -3, 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_2475_a();"
function MakeCharacter_2475_a()
    N := 2475;
    order := 1;
    char_gens := [551, 2377, 2026];
    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_2475_a_Hecke(Kf)
    return MakeCharacter_2475_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, 1, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-1, -1, 0, -1], [-1, 0, 0, 0], [-1, -1, 0, 1], [-1, -1, -2, -1], [2, -2, -2, 0], [2, 0, 2, 0], [-2, 0, -2, -2], [0, 2, 0, 2], [-2, 2, 2, 2], [-2, 4, 2, 2], [-3, 1, 0, 1], [2, -2, 0, 2], [-4, 0, 2, 0], [-8, 4, 2, 0], [0, 2, 0, -2], [-2, 4, -2, 0], [-4, 0, 2, 0], [-1, -1, 0, 1], [2, 2, 2, 0], [-1, -1, -2, 1], [-2, -4, 0, 0], [-4, 4, 2, 0], [-4, 2, -2, 0], [8, -6, -2, -2], [-5, 7, 2, 1], [6, -4, 0, 0], [-2, -2, 2, -2], [-5, -1, -4, -1], [2, -2, 0, 2], [6, 0, 0, 0], [-8, -4, -2, -2], [-4, -6, 2, 0], [2, -2, -2, 0], [0, -6, -4, -2], [0, 2, -2, -2], [-11, 1, 2, -1], [-13, -1, -2, -1], [4, -4, 0, 0], [4, 6, 2, 2], [-4, -4, 4, 0], [-11, 1, 4, -1], [-9, -1, 2, -1], [0, -4, 0, 0], [-6, -2, 2, -4], [6, 2, 4, 2], [-9, -5, -2, 1], [-2, 4, 0, -4], [-1, 3, 2, -1], [-6, 2, -4, -2], [0, 2, 0, 2], [0, 0, -6, 0], [4, -2, -4, -2], [-3, 9, 6, 3], [-8, -4, 0, 4], [-6, 6, -2, 0], [1, 5, 8, -1], [-14, -4, -6, -2], [11, 7, -4, 3], [-9, -1, 2, -5], [3, 7, 0, -1], [8, 0, 2, -4], [-6, 2, -2, 2], [-14, 6, -2, -2], [-8, 2, 4, 6], [-7, -3, -8, -1], [-7, -7, -2, -5], [-4, -10, -4, -6], [8, -4, -2, -4], [-2, 6, 4, 6], [18, -2, 0, -2], [-9, -5, 0, -3], [0, 4, 8, 4], [16, -6, -2, -2], [-16, 0, -4, -4], [4, 8, 2, 4], [-6, -4, -4, 4], [10, 0, 8, 4], [8, -4, 4, 0], [-8, -6, 2, -2], [-4, 4, 0, -4], [-4, 0, -2, 4], [-12, 4, 2, 2], [12, -2, -2, 2], [-6, -4, -4, -8], [-3, -3, -4, -5], [0, -6, -6, 4], [6, 6, 8, 6], [10, -4, 6, 4], [6, 6, 0, 6], [6, -4, -2, -8], [20, 0, -4, 0], [-16, 0, 0, -4], [15, 7, -2, 5], [0, 4, 4, 4], [-14, 12, 4, 4], [1, 5, 8, 5], [-10, -8, 4, 0], [-7, 1, -4, -7], [-3, -7, 2, 1], [17, -11, -6, -1], [24, 6, 6, 4], [-8, 0, 2, -2], [4, -6, 0, 6], [-10, 16, 6, 8], [-11, 1, -10, -7], [-20, -4, 0, 4], [-4, 6, 0, -6], [-5, -1, -4, -1], [9, 1, 0, -5], [10, 6, 10, 6], [-8, 2, 4, 6], [-12, 8, 0, -4], [2, 8, 4, 4], [4, -10, -10, -6], [-12, 6, 6, 2], [-12, 10, 4, 2], [0, 12, 0, 0], [6, 4, 8, 4], [-21, 11, 4, 1], [-9, 7, -2, 7], [18, -2, 4, 2], [12, -4, -12, 4], [18, -12, -6, 2], [-12, 2, -6, -2], [-8, -4, 0, -8], [12, -4, -8, 4], [-1, 7, 8, 5], [12, -8, -10, 2], [-3, -11, -2, -5], [24, -8, -4, -4], [0, 6, 8, -6], [-28, 2, -2, 4], [20, -10, -4, -6], [-8, -2, 8, 6], [9, -7, 4, 1], [2, -14, -6, -6], [18, 0, 6, 6], [16, -4, -2, -2], [2, -8, 6, 2], [20, -10, -6, 2], [-7, 13, -6, -1], [-12, 14, 6, 2], [12, -16, -10, -4], [19, -13, -4, -3], [29, 1, -6, 5], [8, 2, -4, -2], [36, 2, 2, -2], [1, -7, 8, 3], [-4, -8, 0, -4], [-24, -4, -8, -4], [-5, -5, 2, -3], [-14, 2, -4, -6], [-20, 16, 8, -4], [16, 4, 6, -2], [0, -8, -8, -4], [-13, -9, -8, -7], [-12, -2, -2, -4], [-14, 10, -4, -2], [1, -11, -10, 1], [19, 3, 0, 3], [4, 4, -4, 4], [-6, 4, -8, 4], [0, -2, 10, 2], [20, -14, -8, -6], [25, -15, 0, 3], [-12, 10, 8, -2], [-36, 0, -6, 0], [12, 8, 8, -4], [12, 2, -6, -2], [-6, -6, 0, -10], [-25, 11, 12, 1], [-16, -10, -4, -6], [4, 12, 4, 0], [-32, -8, -6, -2], [-8, -8, -4, -8], [-17, -1, -16, -9], [-46, 16, 4, 0], [-8, -4, -4, 4], [4, 8, 12, 12], [12, 6, 8, 10], [-17, 3, -2, -5], [-14, 2, -12, -2], [10, -8, -4, -4], [15, 7, 0, -3], [-10, 18, 4, 2], [22, -12, -4, 0], [12, -20, 4, 8], [8, 6, -8, 2], [1, -7, 6, -5], [36, -8, -4, -4], [-28, 0, -4, -4], [-7, 5, 10, 15], [-16, 6, 8, 6], [-4, 10, 4, -6], [-14, 8, -4, 0], [29, -7, -2, -3], [19, -5, -10, -7], [14, 0, 2, 6], [8, -4, 14, 6], [-14, -12, -12, -12], [0, 14, 0, 2], [-20, 8, 12, 4], [6, -10, 2, -2], [24, 8, 0, -8], [-17, -5, 2, 5], [2, 16, 6, 14], [4, 6, 8, 6], [17, -15, -8, -13], [-6, -12, 0, -8], [12, 0, -4, 0], [26, 8, -2, -8], [-2, 14, 12, -2], [-6, -4, 0, 12], [-33, -1, 4, -5], [0, 18, -6, -4], [-24, -12, 0, -12], [-28, -10, -8, -10], [-4, -6, -12, -10], [-14, 6, -10, 0], [20, -16, -8, -12], [-24, 2, 10, -2], [1, 9, 10, 15], [-22, 8, -4, 0], [-30, -4, -4, 4], [-4, 8, -2, -4], [25, -7, 8, 9], [22, 2, 0, -14], [4, -20, 10, 0], [8, 4, 10, -6], [-8, 16, -6, -10], [26, -20, 6, 2], [-6, 8, -2, -4], [17, -15, -10, -1], [0, 10, 2, 14], [-13, -5, -6, -1], [-40, 16, 6, 0], [16, 12, 4, 4], [28, -2, 14, 2], [18, -6, -2, -4], [16, -8, 4, 0], [8, -18, 6, 2], [-17, -9, -2, -5], [34, -14, -4, -2], [-22, -4, -6, -8], [-36, 12, 6, 12], [-4, -4, 14, 6], [-1, -9, 14, 5], [-7, -7, -12, -5], [0, 14, 2, -4], [-32, 8, 0, -8], [42, 8, 12, 8], [-19, 17, 10, 17], [16, -12, -8, -8], [-36, 6, -2, 2], [45, -15, -4, -3], [-34, 10, 6, -6], [-1, 15, 4, 1], [11, 3, 4, 7], [-3, -15, -6, 3], [-12, -14, 8, -10], [17, 13, 20, 3], [32, -4, -2, -12], [-4, 4, -8, -8], [-10, -4, -8, 8], [0, -16, 8, -4], [41, 1, 8, -3], [-1, -5, -6, -13], [-34, 0, 12, 0], [-30, -2, 0, -2], [4, 2, -4, 6], [48, -20, 2, 2], [1, 5, 8, 3], [2, -12, -14, 0], [22, 2, -12, 6], [-36, 2, 0, 2], [4, 14, 4, 10], [38, -30, -16, -6], [23, -9, 2, -7], [-24, 10, -4, 2], [-53, -5, -10, -7], [-42, 4, -4, -4], [19, 3, 0, -9], [4, 4, -16, 4], [-4, 4, -10, 8], [17, -19, 10, 1], [-28, -14, -8, -6], [32, -6, 2, -4], [28, 16, -4, 4], [0, -22, -22, -10], [5, 9, 2, 17], [6, -6, 12, -6], [33, -3, -4, -5], [4, -18, 10, 0], [16, -12, -16, -8], [-4, -6, -16, -6], [32, -12, -4, -12], [-47, 21, 12, 9], [1, -11, 4, -1], [3, -9, -6, 3], [16, 16, -6, 14], [-2, 28, 14, 20], [-20, -2, -12, -2], [-26, 6, 6, -2], [10, 14, 0, -6], [-6, -4, 0, -4], [24, -8, -8, 8], [13, 1, -20, 3], [-17, -5, -22, -15], [-44, 12, -8, -8], [12, 14, 2, 16], [18, -4, -2, 8], [35, 7, -6, 5], [4, 14, 4, -2], [-28, 12, 20, -4], [46, -10, 0, 6], [-20, -6, 12, 6], [4, 10, 14, -4], [2, -10, -10, -8], [20, 14, 0, 10], [-6, -12, -6, 6], [-34, 0, -2, -4], [-49, -5, -6, -1], [0, 18, 10, 14], [20, 18, 0, -6], [-12, 8, -8, -8], [51, -5, 10, 5], [25, -3, -2, 1], [24, 6, -8, 2], [-2, 16, 8, 8], [14, 10, -2, -12], [15, -9, -6, -3], [-2, -10, -2, 0], [-16, 10, 18, 6], [10, 8, 4, 16], [-49, -1, -2, 3], [-24, -22, -6, -18], [-11, 13, 4, 9], [-8, 8, 2, -12], [-4, 24, 14, 0], [-12, -22, 2, -8], [4, 6, 8, -14], [8, -14, -12, -22], [30, -6, -12, -10], [22, -24, -16, -8], [-40, 22, 2, 2], [-20, -12, -20, 4], [-12, 22, 16, 22], [-4, -20, -14, -14], [42, -12, 8, 4], [18, 0, 12, 12], [3, 3, 8, -5], [20, -22, -16, -6], [15, 15, 6, -9], [16, -4, 18, 4], [42, 14, -4, 2], [33, 5, -10, 1], [12, 0, 0, -8], [-17, 23, -4, -3], [-16, 6, 2, -4], [26, -16, 10, 0], [14, 10, 0, -6], [24, 12, -4, 4], [-44, 18, 12, 6], [-35, 5, 6, 13], [-23, 21, 24, 1], [12, 14, 12, 18], [8, 12, 26, 0], [8, -6, 0, -18], [-39, 9, 6, 7], [-12, -2, -2, -16], [-36, 20, 6, 6], [8, -18, 4, 2], [-32, 32, 0, 4], [-6, 2, -16, -10], [29, -15, 4, -5], [-16, -10, -26, -4], [-17, -5, 4, -15], [-32, -8, -4, 12], [-20, 2, 0, 10], [-27, -3, -4, 1], [31, -13, -6, -5], [-34, 10, -2, 12], [-8, 28, -4, -4], [-20, 20, 4, 0], [-16, 0, 10, 20], [-15, 1, -12, 1], [0, -8, 4, 8], [44, -2, 10, 2], [-40, 4, 18, -4], [0, -24, 6, 0], [-22, -12, -2, 8], [16, -4, -6, -8], [-35, 21, 12, 3], [-8, 8, 10, -2], [-24, 4, 16, -4], [8, 16, 4, 8], [-38, -12, -26, -14], [18, 12, 4, 8], [-10, -2, 6, -10], [-33, 15, 8, 7], [48, -18, 0, 6], [-48, 6, -6, 0], [-34, 6, 6, 0], [-4, 16, -10, -12], [8, 10, -6, 8], [-6, -12, -10, -8], [-12, 24, 6, 16], [13, 17, 20, 3], [-17, -13, 6, 7], [12, -28, -4, -4], [20, 28, 22, 22], [-43, -7, 0, -5], [10, -24, -8, 0], [-44, 20, 12, 16], [28, 6, 6, -6], [24, 12, 6, 12], [-15, -15, 6, 7], [-34, -8, -24, -12], [-5, 23, 8, 21], [12, -20, -8, 4], [70, 2, 0, -2], [58, -24, 0, 0], [-44, -2, -4, 14], [-20, -24, -8, -8], [4, -10, -30, 4], [-8, -12, -16, 4], [-70, 26, 4, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2475_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2475_2_a_bf();".
// 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_2475_2_a_bf(:prec:=4)
    chi := MakeCharacter_2475_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_2475_2_a_bf();".
// 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_2475_2_a_bf( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2475_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![3, -10, -6, 2, 1]>,<7,R![-36, -64, -16, 4, 1]>,<29,R![-144, -240, -88, 4, 1]>],Snew);
    return Vf;
end function;