
// Make newform 1225.2.a.s in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1225_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_1225_2_a_s();".
// 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_1225_2_a_s();".
// 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 := [-4, -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_1225_a();"
function MakeCharacter_1225_a()
    N := 1225;
    order := 1;
    char_gens := [1177, 101];
    v := [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_1225_a_Hecke(Kf)
    return MakeCharacter_1225_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, 1], [-1, 1], [0, 0], [0, 0], [1, -1], [3, -1], [-3, 1], [2, 2], [2, -2], [-1, 3], [0, 0], [-6, 0], [0, -2], [-6, 2], [-1, -3], [0, 2], [4, 0], [0, -6], [0, -4], [8, 0], [-2, -4], [-5, 1], [4, 0], [-4, 2], [-7, 5], [6, 4], [3, 1], [2, -6], [13, -3], [14, 0], [-4, 4], [6, -2], [12, 2], [10, 2], [2, 4], [11, -7], [10, 4], [2, 2], [11, -7], [-7, 1], [20, 0], [-8, 10], [-11, -1], [-4, -6], [4, -2], [12, -4], [-9, 9], [-5, 1], [19, -3], [-16, 2], [0, -2], [7, 5], [-6, 4], [14, -2], [10, -8], [18, -2], [-10, 12], [16, 0], [-10, 4], [15, -1], [-19, 3], [5, -3], [-27, 3], [-6, -10], [11, -13], [10, -8], [12, 0], [22, -8], [-2, -2], [10, -8], [-7, 5], [8, 0], [-1, -3], [-20, 6], [4, -8], [-4, 4], [-7, -7], [25, 1], [29, 1], [14, -8], [-16, -4], [5, -3], [-7, -5], [-2, 4], [6, -6], [18, -6], [11, -5], [12, 2], [8, 2], [8, -8], [25, -1], [-10, 2], [-2, 2], [-13, -11], [13, 11], [21, 7], [22, -4], [-10, 0], [20, 8], [-25, -5], [-8, 4], [8, 2], [-28, 0], [10, -16], [20, 8], [-1, -9], [-16, -4], [-25, -1], [1, -5], [14, -8], [-19, -9], [-10, -12], [-22, -4], [14, -18], [1, -5], [2, 0], [-1, 1], [16, 8], [-18, 20], [-19, 11], [0, 2], [4, -14], [-13, 7], [-24, 12], [4, 8], [11, 7], [-1, 11], [-34, 10], [32, 8], [-1, 3], [17, 7], [-8, 16], [35, -7], [-24, -6], [-38, -4], [22, 8], [19, 7], [-21, -11], [1, 9], [-27, 17], [6, -10], [17, -15], [16, 8], [-10, 6], [-40, 2], [34, -2], [-38, 12], [-26, 4], [-4, 8], [36, -4], [30, 4], [-24, 14], [-12, 8], [0, 8], [-22, 18], [-8, -8], [7, 13], [24, -18], [-7, -19], [-8, -14], [-4, 0], [28, 2], [30, 2], [12, -24], [18, -20], [13, 15], [-8, 8], [5, -3], [-7, -11], [-10, 0], [8, 4], [18, -16], [-3, -1], [-6, 12], [-38, -2], [-20, 18], [13, -13], [1, -15], [-4, 12], [-24, 2], [12, -12], [12, 16], [-14, 8], [-6, -16], [12, 4], [-18, 12], [-4, -18], [23, -23], [-51, 9], [-24, 24], [-34, -12], [-14, 10], [11, 5], [-52, -2], [-28, 8], [39, -9], [-45, 3], [0, 26], [-3, -15], [19, 17], [-26, 8], [-30, -10], [-18, 8], [2, -20], [-2, -2], [-35, -3], [-2, -6], [-4, 16], [6, -12], [-40, 12], [16, -26], [-26, -12], [-24, -16], [27, -19], [42, -10], [4, 10], [-14, -10], [24, -2], [-26, 10], [22, -4], [10, -20], [10, 6], [57, 5], [-34, -6], [-37, 5], [38, -16], [17, -27], [-7, 11], [-13, -15], [4, 8], [4, -10], [-10, -2], [16, -16], [-3, -7], [27, -3], [51, -7], [20, -14], [-32, 2], [-37, -3], [38, 10], [-16, -4], [32, -12], [-49, 5], [-2, 16], [39, -1], [38, -6], [56, -8], [58, -6], [15, -23], [-6, -18], [14, 4], [18, -4], [-8, 16], [-8, 6], [2, 28], [-43, 3], [-2, -8], [-12, 24], [3, -9], [39, 7], [26, -2], [-22, -6], [0, 18], [15, -13], [-18, 4], [30, 6], [-47, 9], [26, -4], [-64, 4], [-10, 20], [-36, -2], [48, -4], [9, 13], [60, -12], [42, 8], [-1, 5], [-20, -24], [18, -4], [-27, -15], [-22, 26], [-5, 1], [-29, 17], [-16, 24], [-46, 8], [-21, 13], [-7, 27], [2, -4], [54, 4], [40, 0], [0, -26], [33, -7], [33, -9], [20, -22], [-8, -20], [30, -28], [36, 6], [-18, 10], [59, -17], [-28, 32], [-33, 1], [37, 17], [22, 12], [21, 15], [24, 20], [-21, 13], [-30, 12], [42, -14], [16, -6], [-16, 8], [12, 22], [-32, 0], [-53, -1], [-2, 20], [26, 10], [2, 14], [-4, -14], [-70, 10], [-7, 11], [5, 9], [1, -19], [6, -18], [20, 10], [38, 12], [0, 16], [34, -28], [-24, 14], [-5, -27], [4, -24], [-12, 12], [-8, -6], [-34, 16], [16, 10], [2, 6], [13, -21], [12, 0], [-73, 1], [-3, -3], [-2, -12], [-42, 0], [63, -3], [42, 0], [-20, -14], [-2, 12], [-35, 31], [38, -20], [68, 0], [42, -16], [-28, 32], [46, -6], [-31, 33], [92, 0], [-32, -2], [1, -7], [-17, 13], [-47, 17], [2, 24], [-26, 18], [-6, 18], [-34, -16], [-70, 6], [24, -6], [20, 26], [-44, -4], [-35, 3], [4, -16], [-2, -8], [9, -15], [14, -14], [-31, -11], [-83, -5], [40, -4], [44, 4], [3, 7], [6, 26], [-42, 28], [0, -4], [-3, -9], [15, 7], [-38, 36], [-21, 19], [-82, 4], [30, -8], [40, 16], [-50, 16], [-4, -32], [-45, -15], [25, -37], [-13, 23], [-44, 0], [25, -37], [21, 25], [53, -19], [-1, 25], [-68, 8], [-7, 3], [-48, -2], [36, 12], [36, -22], [33, -1], [18, 20], [36, -26], [8, 30], [-14, 22], [1, -11], [4, -2], [34, -2], [10, 8], [-29, -13], [-20, 24], [-12, -24], [33, -19], [-38, -16], [54, 6], [-68, 8], [-34, 24], [32, -22], [65, 3], [4, -28], [-2, 36], [43, -23], [37, -11], [-6, -4], [30, -6], [-20, -8], [11, 3], [25, 1], [36, -8], [18, -24], [-52, 24], [-24, 32]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1225_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1225_2_a_s();".
// 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_1225_2_a_s(:prec:=2)
    chi := MakeCharacter_1225_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_1225_2_a_s();".
// 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_1225_2_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1225_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-4, -1, 1]>,<3,R![-4, 1, 1]>],Snew);
    return Vf;
end function;