
// Make newform 735.2.y.b in Magma, downloaded from the LMFDB on 01 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_735_y();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_735_y_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_735_2_y_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_735_2_y_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, 0, 0, 0, -1, 0, 0, 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_735_y();"
function MakeCharacter_735_y()
    N := 735;
    order := 12;
    char_gens := [491, 442, 346];
    v := [6, 9, 4];
    // 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_735_y_Hecke();"
function MakeCharacter_735_y_Hecke(Kf)
    N := 735;
    order := 12;
    char_gens := [491, 442, 346];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, -1, 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, 0, 0, 0, 0, 0, 0, 2], [0, 0, -1, 0, -1, 0, 1, -1], [0, -1, -1, 0, -1, 2, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 2, 1, 2, 0, -2], [1, 0, -2, 1, -2, 0, 1, 0], [-1, -6, -1, 0, 1, 3, 0, 0], [-4, 3, 0, 0, 2, 0, 0, -3], [2, -2, 1, 0, -1, 0, 1, 0], [0, 0, -10, 0, 0, 0, 5, 0], [0, 0, 2, 0, 0, 0, -4, 0], [0, 0, -1, 4, -1, 0, 1, -2], [0, -2, 0, 2, 0, -2, 2, -4], [-3, -4, 0, 0, 0, -4, -3, 0], [0, 3, 1, 0, -1, -6, -1, 0], [-2, 0, 2, 0, -2, -2, -4, 0], [10, 0, 0, 0, -10, 0, 0, 0], [0, 6, 0, 0, 0, 0, 0, 6], [4, 0, -4, -2, -4, 0, 0, -2], [-4, 2, 0, -2, 8, -2, 0, 0], [0, 0, 0, -10, 0, 0, 0, 10], [0, 2, -7, 4, 0, -4, 7, -2], [-2, 0, 0, -6, 0, 0, 2, 12], [0, -2, 0, 4, -12, 4, 0, -2], [-1, 1, -2, 0, 2, -1, 1, 0], [0, 2, -6, 1, 0, -1, 0, 1], [-2, 0, 1, 0, 1, 0, 1, 9], [2, 0, -1, 0, -1, 0, -1, 8], [0, 8, 5, 4, 0, -4, 0, 4], [5, -6, 10, 0, -10, 6, -5, 0], [-4, 0, 0, -2, 0, 0, 4, 4], [0, -5, 0, -10, 0, 10, 0, 5], [2, 0, 2, 0, 2, 0, -4, 0], [-4, -1, 0, 1, 8, 1, 0, 0], [0, -14, -2, 0, 0, 0, -2, -14], [-7, 4, 0, -2, 7, -2, 0, -2], [2, 0, -2, 0, 2, -2, 4, 0], [0, -6, 3, 0, -3, 12, -3, 0], [-7, -5, 0, 0, 0, -5, -7, 0], [0, 0, 1, 6, 1, 0, -1, -3], [0, 0, 4, -2, 0, 2, -8, 2], [0, 0, 4, 0, 0, 0, -2, 0], [-2, -12, 0, 0, 1, 0, 0, 12], [0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 4, 6, 4, 0, -2, 0], [-10, 0, 0, -2, -10, -2, 0, 2], [-1, 4, 0, 4, 0, 4, 0, -8], [1, 0, -2, 11, -2, 0, 1, 0], [1, -6, 1, 0, -1, 3, 0, 0], [12, 0, 0, 0, -6, 0, 0, 0], [-12, 0, -6, 0, 6, 0, -6, 0], [0, -6, -6, -6, 0, 6, 3, 0], [0, 0, -8, -7, 0, 7, 16, 7], [0, -3, 0, 3, 0, -3, -12, -6], [0, 0, -8, 0, 8, 0, 8, 0], [1, 0, -1, 0, 1, 8, 2, 0], [4, 2, 0, -1, -4, -1, 0, -1], [0, 0, 8, 0, 0, 0, 8, 0], [3, 0, -3, -14, -3, 0, 0, -14], [11, -8, 0, 8, -22, 8, 0, 0], [5, 0, 5, -5, 5, 0, -10, 5], [-9, 0, 0, 5, 0, 0, 9, -10], [-1, 1, -2, 0, 2, -1, 1, 0], [0, 10, 10, 5, 0, -5, 0, 5], [6, 0, -3, 0, -3, 0, -3, 3], [10, 0, -5, 0, -5, 0, -5, 8], [0, 2, 0, -4, 14, -4, 0, 2], [7, 0, 0, 6, 0, 0, -7, -12], [9, 0, 9, -6, 9, 0, -18, 6], [-2, 9, 0, -9, 4, -9, 0, 0], [-3, 0, 3, 11, 3, 0, 0, 11], [0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 1, 0, -1, -19, -2, 0], [0, -4, -18, 0, 18, 8, 18, 0], [0, -4, 0, 4, 0, -4, 14, -8], [0, 0, 14, -16, 14, 0, -14, 8], [0, 0, -7, -14, 0, 14, 14, 14], [14, 3, 7, 0, -7, 0, 7, 0], [-6, -6, 0, 0, 3, 0, 0, 6], [-6, 0, 0, 1, -6, 1, 0, -1], [6, -6, 0, -6, 0, -6, 0, 12], [-11, -4, 0, -4, 0, -4, 0, 8], [-9, 0, 0, 2, -9, 2, 0, -2], [12, 0, -24, 2, -24, 0, 12, 0], [-16, -3, 0, 0, 8, 0, 0, 3], [-18, 0, -9, 0, 9, 0, -9, 0], [0, -14, 6, -14, 0, 14, -3, 0], [0, 0, -24, -4, -24, 0, 24, 2], [0, 4, 0, -4, 0, 4, -4, 8], [-6, 12, 0, 0, 0, 12, -6, 0], [0, 1, -5, 0, 5, -2, 5, 0], [0, -10, 0, 5, 0, 5, 0, 5], [9, 0, -9, 18, -9, 0, 0, 18], [-5, 20, 0, -20, 10, -20, 0, 0], [0, 0, 25, 0, 0, 0, -25, 0], [-15, 0, 0, 5, 0, 0, 15, -10], [0, -5, 0, 10, 0, 10, 0, -5], [0, -6, -22, -3, 0, 3, 0, -3], [-12, 0, 6, 0, 6, 0, 6, 4], [0, 4, 0, -8, -11, -8, 0, 4], [22, 0, 0, -4, 0, 0, -22, 8], [12, 0, 12, 0, 12, 0, -24, 0], [16, 0, -16, 2, -16, 0, 0, 2], [0, 10, -10, 0, 0, 0, -10, 10], [-10, 4, 0, -2, 10, -2, 0, -2], [1, 0, -1, 0, 1, -1, 2, 0], [-18, -4, 0, 0, 0, -4, -18, 0], [0, 0, -9, 10, -9, 0, 9, -5], [0, 0, -13, 14, 0, -14, 26, -14], [0, -21, -20, -21, 0, 21, 10, 0], [-18, 9, -9, 0, 9, 0, -9, 0], [15, 0, 15, 0, -15, 0, 0, 0], [-1, 0, 2, 6, 2, 0, -1, 0], [-4, 0, 0, -5, -4, -5, 0, 5], [-1, -14, 0, -14, 0, -14, 0, 28], [4, 0, 0, -12, 4, -12, 0, 12], [-9, 0, 18, 1, 18, 0, -9, 0], [0, 28, 0, 0, 0, -14, 0, 0], [10, 36, 5, 0, -5, 0, 5, 0], [0, -8, -38, -8, 0, 8, 19, 0], [0, 0, 0, 6, 0, -6, 0, -6], [-16, -8, 0, 0, 0, -8, -16, 0], [0, 5, -13, 0, 13, -10, 13, 0], [6, 0, -6, 0, 6, 0, 12, 0], [0, -4, -24, 0, 0, 0, -24, -4], [3, -14, 0, 14, -6, 14, 0, 0], [0, 0, -25, 0, 0, 0, 25, 0], [0, 13, 0, -26, -12, -26, 0, 13], [-6, 16, -12, 0, 12, -16, 6, 0], [6, 0, -3, 0, -3, 0, -3, -7], [0, 8, 27, 4, 0, -4, 0, 4], [-4, -30, -8, 0, 8, 30, 4, 0], [0, 6, 0, -12, 19, -12, 0, 6], [-2, 0, 0, -16, 0, 0, 2, 32], [0, 13, 12, 26, 0, -26, -12, -13], [-10, 17, 0, -17, 20, -17, 0, 0], [-11, 0, 11, -17, 11, 0, 0, -17], [15, 0, -15, 0, 15, 5, 30, 0], [7, 21, 0, 0, 0, 21, 7, 0], [0, 0, -11, 8, 0, -8, 22, -8], [0, 5, 16, 5, 0, -5, -8, 0], [2, 2, 0, 0, -1, 0, 0, -2], [22, -28, 22, 0, -22, 14, 0, 0], [-5, 0, 10, 18, 10, 0, -5, 0], [10, 0, 0, 22, 10, 22, 0, -22], [-24, 9, 0, 9, 0, 9, 0, -18], [-10, 0, 20, 0, 20, 0, -10, 0], [32, -8, 32, 0, -32, 4, 0, 0], [32, 0, 0, 0, -16, 0, 0, 0], [40, 6, 20, 0, -20, 0, 20, 0], [0, 0, 15, -40, 15, 0, -15, 20], [0, -6, 0, 6, 0, -6, -24, -12], [12, -4, 0, 0, 0, -4, 12, 0], [0, -14, 12, 0, -12, 28, -12, 0], [-11, 0, 11, 18, 11, 0, 0, 18], [-12, -4, 0, 4, 24, 4, 0, 0], [0, -6, -3, -12, 0, 12, 3, 6], [0, 12, 0, -24, 2, -24, 0, 12], [19, 1, 38, 0, -38, -1, -19, 0], [0, -20, 20, -10, 0, 10, 0, -10], [4, 0, -2, 0, -2, 0, -2, -6], [10, -26, 20, 0, -20, 26, -10, 0], [-9, 0, 0, -12, 0, 0, 9, 24], [0, 2, 18, 4, 0, -4, -18, -2], [-15, 0, -15, -12, -15, 0, 30, 12], [7, 0, -7, -9, -7, 0, 0, -9], [2, 32, 0, -16, -2, -16, 0, -16], [3, 0, -3, 0, 3, -13, 6, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_735_y_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_735_2_y_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_735_2_y_b(:prec:=8)
    chi := MakeCharacter_735_y();
    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_735_2_y_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_735_2_y_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_735_y();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![256, 0, 0, 0, -16, 0, 0, 0, 1]>,<11,R![25, 30, 7, -6, 1]>,<13,R![625, 0, 0, 0, 146, 0, 0, 0, 1]>,<17,R![390625, -62500, 5000, -5800, -161, 232, 8, 4, 1]>],Snew);
    return Vf;
end function;