
// Make newform 360.2.br.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_360_br();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_360_br_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_360_2_br_a();".
// 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_360_2_br_a();".
// 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, 0, -1, 0, 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_360_br();"
function MakeCharacter_360_br()
    N := 360;
    order := 12;
    char_gens := [271, 181, 281, 217];
    v := [12, 6, 10, 3];
    // 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_360_br_Hecke();"
function MakeCharacter_360_br_Hecke(Kf)
    N := 360;
    order := 12;
    char_gens := [271, 181, 281, 217];
    char_values := [[1, 0, 0, 0], [-1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    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 := [[-1, 0, 0, -1], [0, -1, 0, -1], [0, 2, -1, -2], [-2, -2, -1, 3], [-2, 0, 2, 0], [-2, 4, -2, -2], [-3, 0, 0, 3], [-3, -2, 0, 1], [-1, 2, -1, -1], [3, -2, 3, 0], [0, 3, 3, 3], [1, -4, 4, -1], [-4, -3, 2, 3], [-5, -5, 1, 4], [-3, 3, 0, -3], [3, 6, -6, -3], [-6, 7, 3, -7], [-5, -6, -5, 0], [-1, -5, -4, 4], [0, 0, 0, -12], [-8, -6, 6, 8], [-3, 3, -3, 0], [-4, 4, -5, -9], [6, -10, 0, 5], [-5, -5, 5, 0], [2, 0, -2, 0], [-12, -7, 5, -5], [2, 3, 3, 2], [0, 10, 0, -5], [-2, 4, -2, -2], [10, -3, -3, 10], [0, 3, -13, 3], [-4, 4, 8, 4], [0, -4, 0, -4], [0, -7, 0, 7], [16, 0, -16, 0], [0, -9, -9, 9], [1, 4, 4, 1], [3, 12, 9, -9], [-2, 2, 2, 0], [6, 0, -12, 4], [0, 0, 0, 3], [-9, -3, -9, 0], [0, 10, 10, -10], [-12, 0, 0, -12], [-3, 0, 6, -1], [0, 0, 0, 0], [-8, 8, 11, 3], [-3, -3, -9, 12], [0, -2, 15, -2], [1, -14, -14, 1], [0, 4, 12, 4], [24, -3, -24, 6], [11, 18, 0, -9], [4, -13, -17, 17], [7, 7, 1, -8], [0, 0, 0, 1], [-11, 18, 0, -9], [1, -1, -5, -4], [4, -9, 4, 0], [9, -9, 0, 0], [6, -3, -9, 9], [-14, -1, 1, 14], [6, 9, -3, -9], [-14, 14, -4, -18], [10, 10, -10, 0], [-9, 15, -9, 0], [6, -8, 2, 2], [-6, 3, 3, 3], [-33, -2, 33, 4], [8, 8, -4, -4], [-3, 6, 0, -3], [-7, -7, 7, 0], [10, -11, 1, 1], [21, 14, 0, -7], [20, -7, -13, -13], [-6, -7, -6, 0], [3, -6, 6, -3], [-28, -12, 14, 12], [0, -6, 0, 6], [12, 2, -6, -2], [-10, 12, -10, 0], [-11, 0, 22, 3], [-11, -12, 12, 11], [-12, 2, -12, 0], [-15, 15, 24, 9], [12, 24, 0, -12], [5, 5, 1, -6], [2, -9, -2, 18], [-6, 7, 13, -13], [25, -12, -12, 25], [-9, -7, 9, 14], [-13, 12, 12, -13], [0, -12, -4, -12], [0, -12, -6, -12], [19, -11, -11, 19], [-18, 8, 9, -8], [5, 0, -10, -6], [-8, 7, 7, -8], [2, 0, -4, -9], [-29, 29, 22, -7], [-9, 6, 6, -9], [-15, -18, -3, 3], [6, -2, -6, 4], [-4, 0, 2, 0], [-11, 12, 12, -11], [-25, -25, 28, -3], [1, 10, 10, 1], [0, -9, -15, -9], [-32, -6, 32, 12], [21, -8, -13, -13], [9, -24, -24, 9], [-4, 22, 26, -26], [9, -3, -9, 6], [-18, 0, 0, 0], [-12, -21, -12, 0], [-27, 18, 9, 9], [-12, -21, 21, 12], [12, 17, 5, -5], [-6, -2, -6, 0], [8, 0, -4, 0], [28, -28, -10, 18], [20, 20, 4, -24], [7, 0, 0, -7], [-1, -45, -1, 0], [-19, -36, 0, 18], [33, -8, -33, 16], [-21, 26, 0, -13], [1, 1, 2, -3], [-2, -5, 7, 7], [-21, -2, 0, 1], [1, -20, 19, 19], [0, 6, -24, 6], [-26, -22, 22, 26], [-6, 30, 3, -30], [-48, 7, 24, -7], [14, 6, -6, -14], [26, 19, -7, 7], [6, 8, -14, -14], [18, 32, 0, -16], [1, 0, -2, -15], [13, -24, -13, 48], [-3, -7, -4, 4], [-3, 9, 9, -3], [-6, -22, 0, 11], [-9, -11, 9, 22], [24, 24, 0, -24], [-26, 26, -8, -34], [0, -10, 0, -10], [-19, 23, 23, -19], [-10, -14, -4, 4], [-4, 0, 8, -9], [-4, 23, 23, -4], [16, -7, -23, 23], [-34, 34, 11, -23], [7, 15, 7, 0], [3, 0, -6, -15], [6, -6, -6, 12], [20, -18, -18, 20], [0, 3, -14, 3], [-18, -18, 21, -3], [14, 8, 8, 14], [-21, 17, 4, 4], [-35, 6, 0, -3], [-20, -22, -2, 2], [5, 5, 8, -13], [5, 6, 0, -3], [17, -17, -1, 16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_360_br_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_360_2_br_a();".
// 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_360_2_br_a(:prec:=4)
    chi := MakeCharacter_360_br();
    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_360_2_br_a();".
// 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_360_2_br_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_360_br();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![169, 104, 41, 10, 1]>,<11,R![4, 2, 1]>],Snew);
    return Vf;
end function;