
// Make newform 3696.2.a.bo in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3696_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_3696_2_a_bo();".
// 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_3696_2_a_bo();".
// 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, -4, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [-3, 1, 1], [-3, -1, 1]];
    Rf_basisdens := [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_3696_a();"
function MakeCharacter_3696_a()
    N := 3696;
    order := 1;
    char_gens := [463, 2773, 2465, 1585, 673];
    v := [1, 1, 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_3696_a_Hecke(Kf)
    return MakeCharacter_3696_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, 0, 0], [-1, 0, 0], [1, -1, 0], [1, 0, 0], [1, 0, 0], [-1, -1, 2], [2, 1, -3], [3, 0, 1], [-4, -1, -1], [-1, -1, 2], [2, 1, 3], [1, 1, 2], [4, 0, -2], [4, -3, 1], [-1, -2, -1], [0, -1, 1], [-1, 0, -3], [-2, 0, 0], [1, 2, -3], [4, -2, 2], [-7, 1, -2], [-4, 2, -2], [-2, 3, -3], [10, 2, 2], [0, 3, 1], [-2, -3, 1], [6, 0, 2], [-9, 0, -3], [10, 3, 3], [2, -2, 6], [6, 1, -1], [-4, 0, 0], [-2, -3, -3], [8, -2, 2], [-1, 3, -2], [6, 3, 1], [0, 4, 2], [1, 2, -3], [16, -1, 3], [4, 4, 6], [0, -4, 0], [8, 1, 3], [2, 3, 5], [-12, -3, -1], [-6, -6, -2], [-4, 1, 1], [10, 0, 2], [14, 2, -4], [6, 2, 0], [4, 1, -9], [2, 0, -8], [5, 4, -1], [-1, -3, -4], [-3, 4, 3], [1, -1, -4], [-13, -4, 1], [14, 4, -4], [23, 0, 1], [4, 4, -10], [7, -3, -4], [7, 4, 9], [0, -2, 4], [8, -4, 0], [-8, 0, 0], [14, 1, 1], [4, 5, 3], [-16, 2, -2], [0, -3, 3], [4, 0, 8], [7, -5, 6], [-17, 3, 2], [4, 2, -10], [-2, -2, 8], [6, -5, 7], [-13, -2, -1], [12, -4, 0], [8, -2, 4], [-10, 3, -9], [4, 1, 3], [14, -4, 8], [-3, -2, -3], [11, 7, 6], [7, 4, -3], [-8, -4, -10], [-13, -4, -7], [-8, -2, 10], [-8, 6, -8], [12, 7, 1], [0, 2, 0], [-23, 6, -3], [7, 2, 7], [6, -9, -3], [-12, 8, -4], [-29, 0, 1], [11, -10, 7], [-24, 3, -1], [-6, -4, -8], [9, 5, 10], [9, 6, 1], [8, 6, 4], [-4, -4, 12], [7, 1, 4], [-8, -3, 5], [10, 0, 0], [-20, -6, -6], [-10, -1, -9], [-3, 12, -5], [-20, -2, 4], [-20, 2, -2], [25, -5, 0], [7, 2, -5], [6, 7, -13], [-18, 2, -10], [-14, -3, -13], [-36, -2, 2], [-6, 3, -9], [12, 3, -1], [-7, 0, 7], [-16, 10, 0], [-17, 0, -3], [-20, 3, -7], [20, -3, -1], [-18, -2, 6], [-6, 11, -7], [-6, 6, 0], [-30, 0, -4], [-33, -5, -10], [-7, -2, 5], [8, 1, -11], [2, 4, 0], [24, 3, 3], [11, -8, 5], [-7, 12, -5], [-27, 5, 2], [12, 6, 4], [-13, -3, 8], [-7, -3, 14], [-11, -4, -5], [-9, 3, -10], [31, -3, 4], [17, 6, -7], [-3, 3, -16], [5, 0, -5], [-29, 6, -1], [-16, -2, 4], [11, 6, 3], [-22, -2, 2], [-10, -8, 4], [12, 4, -4], [-12, -5, 11], [8, 0, 22], [29, -1, 0], [11, 4, 13], [-10, -9, -11], [20, 0, 16], [22, 4, -10], [0, -3, -7], [19, 1, 4], [-2, -4, -8], [8, 4, -14], [22, 0, -2], [3, 7, 2], [-18, -4, 6], [1, 0, 11], [-28, 5, -9], [40, -6, 2], [-31, 0, 7], [-6, 8, -12], [2, -8, 8], [1, -13, 4], [7, 6, -5], [26, 2, 14], [-8, 8, 0], [-14, 1, 13], [-10, 14, 0], [-34, -4, 0], [-4, 11, -17], [-6, -10, -6], [5, 2, -11], [25, 3, -4], [-7, -4, 11], [8, -1, -1], [-10, -6, -14], [16, -12, 6], [-4, 12, -8], [13, -5, 0], [-28, 4, -2], [-6, -9, 21], [-10, 1, 9], [14, 1, -9], [10, 0, 8], [-31, -4, -17], [21, -2, -11], [11, 13, -12], [-29, 2, 3], [19, -9, 10], [2, -5, 7], [-2, -10, 14], [4, 7, 9], [0, 3, 7], [-10, -4, 20], [21, 4, -9], [-36, -1, 5], [5, 7, 8], [-40, -4, 8], [22, 0, 0], [4, -14, 6], [-23, 4, -9], [6, 14, 10], [-12, -7, -3], [-6, 8, 0], [19, 1, -8], [-1, 10, -13], [-16, 2, -2], [-14, 0, 6], [-26, -6, 6], [0, 4, 20], [-30, 7, -5], [-2, 5, 11], [-26, 3, -9], [-14, 5, -3], [1, -4, 11], [-28, 5, -9], [11, 4, -15], [-20, 6, -2], [-6, -4, 8], [3, 1, -20], [6, -8, 10], [-3, -12, -9], [21, -2, -11], [2, 12, 8], [9, 0, 11], [34, 2, 8], [-9, 17, -16], [-48, -1, -1], [34, 8, 6], [34, 3, -1], [19, 1, 0], [14, -3, -17], [-9, 8, -15], [24, -9, -1], [12, 16, 8], [14, 1, -1], [6, 10, -6], [12, -6, 0], [8, -14, 10], [-4, 2, -18], [-13, -8, -7], [12, 14, 14], [7, 14, -1], [14, 4, -28], [48, 3, -7], [-22, 2, 0], [34, -2, 14], [2, -2, -6], [52, 4, 4], [-3, 7, 20], [-23, 10, -11], [3, -7, 0], [17, 15, 12], [14, -4, -18], [-44, -4, -4], [22, 12, -4], [-35, -7, -10], [3, 15, -22], [10, 13, -13], [-22, -2, -22], [9, -9, 20], [0, 10, -10], [10, 12, -8], [-8, -6, 8], [5, 14, -11], [-44, -7, -13], [2, -3, -1], [1, 7, -4], [-6, -5, 21], [-12, -4, 20], [26, -14, -2], [-2, 9, -23], [8, 3, -13], [12, 17, -7], [-40, 0, 0], [2, 7, -23], [-3, 11, -8], [63, 2, 7], [-24, -3, 1], [-49, -1, 6], [-2, -6, 10], [26, -1, 9], [-14, -13, 7], [14, -7, -3], [-28, -4, 20], [-21, -1, 2], [-52, -7, 5], [-20, 4, 6], [4, -7, 15], [-2, -16, -14], [38, 8, 0], [4, 11, -9], [7, 18, -9], [-25, 15, -10], [9, 1, 2], [-8, 13, 9], [16, -9, -9], [-19, -10, -3], [-1, -7, 8], [-28, -11, -15], [12, -9, 5], [44, 5, -3], [16, 9, 3], [18, -4, -6], [36, 8, 10], [63, 7, -2], [44, 0, 0], [-40, -8, 0], [-10, -8, -4], [-4, 2, -6], [-56, 0, 0], [-40, -13, -7], [38, -2, 18], [23, 16, -3], [-25, 17, -16], [14, 2, 18], [24, -22, 10], [-11, 13, 14], [-28, 0, 14], [4, 12, -12], [-32, -15, -7], [49, -2, -11], [31, -9, 2], [16, 6, -20], [-48, -8, 22], [-1, -22, -5], [10, 5, 27], [24, 7, 15], [23, -4, -15], [23, 5, -24], [0, -7, 7], [6, -1, 31], [20, -4, 8], [6, -14, 10], [-6, -13, -1], [-10, 15, -13], [-25, -12, 13], [-34, -9, 3], [3, 12, -11], [2, 6, -14], [-40, 6, 10], [4, -14, 6], [-30, 4, 0], [28, -14, 10], [13, 3, -28], [-40, 4, 10], [-33, 2, -13], [6, -1, -5], [62, -7, 1], [-5, 16, -3], [32, -12, -8], [19, 1, -20], [-8, 3, 7], [-20, -14, 4], [40, 4, 14], [36, 14, 10], [31, -4, 17], [-4, 4, -12], [-21, -7, 36], [-10, 3, -33], [-14, 4, 10], [22, -16, 20], [30, 1, 11], [-42, -5, -11], [9, -8, 15], [53, -9, -4], [-20, 8, 4], [55, 13, 16], [-19, -4, 3], [25, -20, 3], [-7, 3, 16], [-14, -8, -16], [-10, -8, -20], [-13, -9, 30], [-52, -3, -17], [2, -4, -14], [-9, -5, 26], [4, -16, 8], [27, 14, -21], [-59, -14, -19], [6, 14, 6], [-22, 3, 9], [9, -6, 1], [45, 7, -16], [58, 6, 18], [4, -23, 5], [44, 19, 31], [22, 0, -6], [38, -5, 11], [-1, -14, 19], [15, -11, 4], [-8, -26, -6], [3, 5, -12], [-1, 11, -2], [30, -5, -5], [-18, -3, 19], [-42, -12, -16], [20, 1, -1], [-59, 8, 3], [35, -1, 14], [6, -20, 8], [-46, -16, -6], [32, 2, -10], [-14, -22, -14], [14, -14, 2], [-56, 10, -10], [24, 11, -5], [-6, -10, -2], [-31, 21, -10], [3, 2, 23], [22, -1, -3], [-46, 14, -2], [-22, -17, -7], [-14, -7, 25], [-77, 13, -4], [-46, -9, -23], [-30, 1, -25], [-6, -16, 0], [17, 25, 10], [-6, 15, -27], [7, 19, -22], [6, 14, -12], [4, -12, 24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3696_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3696_2_a_bo();".
// 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_3696_2_a_bo(:prec:=3)
    chi := MakeCharacter_3696_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_3696_2_a_bo();".
// 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_3696_2_a_bo( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3696_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![26, -7, -4, 1]>,<13,R![-94, -27, 4, 1]>,<17,R![328, -40, -8, 1]>],Snew);
    return Vf;
end function;