
// Make newform 1470.2.i.v in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1470_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1470_i_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_1470_2_i_v();".
// 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_1470_2_i_v();".
// 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, 0, 2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 2, 2];
    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_1470_i();"
function MakeCharacter_1470_i()
    N := 1470;
    order := 3;
    char_gens := [491, 1177, 1081];
    v := [3, 3, 2];
    // 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_1470_i_Hecke();"
function MakeCharacter_1470_i_Hecke(Kf)
    N := 1470;
    order := 3;
    char_gens := [491, 1177, 1081];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [-1, 0, -1, 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 := [[-1, 0, -1, 0], [0, 0, -1, 0], [1, 0, 1, 0], [0, 0, 0, 0], [0, -1, 2, -1], [0, 0, 0, 0], [0, -1, 0, -1], [0, -2, 0, 0], [-2, -2, -2, 0], [4, 0, 0, -3], [0, -5, 2, -5], [0, 1, 0, 0], [6, 0, 0, -2], [6, 0, 0, 1], [-2, -5, -2, 0], [0, -8, 2, -8], [0, -6, 6, -6], [-6, -4, -6, 0], [0, 7, -2, 7], [8, 0, 0, -2], [0, -4, -2, -4], [0, 8, 0, 0], [8, 0, 0, 2], [2, -6, 2, 0], [-6, 0, 0, 2], [0, -4, 12, -4], [4, -10, 4, 0], [4, 0, 4, 0], [0, 8, -6, 8], [-2, 0, 0, -8], [0, 0, 0, 10], [-14, -2, -14, 0], [0, 0, -16, 0], [-12, 0, 0, 4], [4, 7, 4, 0], [0, 14, -2, 14], [0, 4, -6, 4], [-2, -9, -2, 0], [6, 0, 0, 5], [10, 6, 10, 0], [0, -1, 2, -1], [6, 0, 0, -2], [0, 8, 0, 0], [0, -14, -2, -14], [-18, 0, 0, 2], [0, 5, -10, 5], [4, 0, 0, 4], [-12, 0, 0, 4], [0, 4, 8, 4], [6, 14, 6, 0], [10, -12, 10, 0], [12, 0, 0, -8], [0, 3, 12, 3], [12, 0, 0, -4], [-8, 7, -8, 0], [0, -14, -2, -14], [0, -4, -2, -4], [-10, -3, -10, 0], [0, 7, 12, 7], [-22, 0, 0, -6], [0, -6, -22, -6], [6, 0, 0, -16], [-4, 0, 0, -8], [0, -2, 32, -2], [26, 2, 26, 0], [18, 0, 18, 0], [-4, 16, -4, 0], [-6, 0, 0, -2], [0, 4, -4, 4], [-14, 0, 0, -14], [0, 9, -12, 9], [-16, -2, -16, 0], [0, -6, -8, -6], [-4, -5, -4, 0], [-16, 0, 0, 6], [6, 7, 6, 0], [0, 5, 20, 5], [-2, 16, -2, 0], [14, 8, 14, 0], [0, 13, 0, 13], [12, 0, 0, -4], [-6, 0, 0, -16], [0, 10, 0, 10], [-6, 0, 0, 2], [18, 11, 18, 0], [-8, -4, -8, 0], [6, 0, 0, 22], [-10, 14, -10, 0], [14, 0, 0, 0], [28, 0, 0, -4], [-12, -6, -12, 0], [0, -6, -8, -6], [0, 6, 16, 6], [2, 0, 0, -7], [4, -14, 4, 0], [-14, 0, 0, 7], [-24, 4, -24, 0], [0, 10, -2, 10], [10, -22, 10, 0], [-6, 6, -6, 0], [26, 0, 0, -1], [0, 16, -10, 16], [0, 10, 12, 10], [-18, -12, -18, 0], [0, -8, 16, -8], [0, 10, -2, 10], [0, 0, 0, 14], [16, 15, 16, 0], [0, -8, 16, -8], [-4, 0, 0, 27], [12, -12, 12, 0], [0, -15, 16, -15], [-16, 0, 0, 20], [0, 10, -16, 10], [22, 0, 0, -2], [0, 0, -30, 0], [-6, 0, 0, 30], [0, -5, -26, -5], [-30, -2, -30, 0], [30, 0, 0, 7], [0, 4, 22, 4], [2, 0, 0, -28], [14, -2, 14, 0], [0, 14, 12, 14], [16, -20, 16, 0], [4, 0, 0, 11], [14, 22, 14, 0], [16, -20, 16, 0], [-40, 0, 0, 4], [18, -24, 18, 0], [0, -22, -12, -22], [0, 0, 0, -4], [-34, 6, -34, 0], [28, 0, 0, 3], [-2, -12, -2, 0], [32, 0, 0, 1], [0, 4, -34, 4], [0, -12, -12, -12], [-18, 0, 0, 6], [0, 22, -6, 22], [20, 0, 0, 12], [20, -11, 20, 0], [0, 20, -12, 20], [20, 0, 0, 22], [0, 8, 34, 8], [-8, 0, 0, -2], [-18, 0, 0, -8], [0, 9, -40, 9], [0, 12, 0, 0], [-16, 12, -16, 0], [-20, -1, -20, 0], [-26, 0, 0, 4], [14, 0, 0, 17], [6, -21, 6, 0], [0, 15, 22, 15], [0, 0, 0, -4], [10, -26, 10, 0], [-6, -4, -6, 0], [34, 0, 0, 12], [0, -20, -8, -20], [-28, -6, -28, 0], [16, 0, 0, 28], [-16, 0, 0, 20], [28, 12, 28, 0], [0, 8, -6, 8], [0, 19, -10, 19], [0, 2, 18, 2], [0, -8, 14, -8], [-52, 0, 0, 4], [-2, 30, -2, 0], [0, -17, 10, -17], [10, 0, 0, -22], [24, 16, 24, 0], [0, 14, -30, 14], [0, 15, -22, 15], [6, 0, 0, 12], [44, 0, 0, -14], [0, -11, 28, -11], [32, 0, 0, 8], [-22, 28, -22, 0], [32, 14, 32, 0], [-6, 0, 0, -26], [44, 0, 0, -7], [16, 1, 16, 0], [0, 30, 18, 30], [0, -20, 6, -20], [0, -11, 0, -11], [0, -22, 14, -22], [42, 8, 42, 0], [0, -16, -24, -16], [-38, -10, -38, 0], [-40, 0, 0, 0], [-16, -30, -16, 0], [46, 4, 46, 0], [0, 2, -24, 2], [0, 25, 8, 25], [0, -28, 12, -28], [24, 23, 24, 0], [-16, 0, 0, 3], [-26, 15, -26, 0], [0, 19, -8, 19], [-34, 0, 0, -5], [-8, 0, -8, 0], [0, 3, 40, 3], [-28, 0, 0, 0], [0, -8, -42, -8], [10, 27, 10, 0], [28, 22, 28, 0], [2, 0, 0, 0], [0, 8, -8, 8], [14, -20, 14, 0], [2, 0, 0, -24], [-28, 0, 0, 24], [8, 10, 8, 0], [0, 22, -8, 22], [56, -9, 56, 0], [0, 6, -40, 6], [0, -2, -38, -2], [30, -10, 30, 0], [-14, 0, 0, 10], [18, -14, 18, 0], [10, 0, 0, -15], [10, 30, 10, 0], [28, -6, 28, 0], [24, 0, 0, 6], [-30, 0, 0, -22], [0, 5, 0, 0], [0, -24, -32, -24], [8, -4, 8, 0], [38, -19, 38, 0], [0, 7, 40, 7], [0, -6, -36, -6], [-16, 0, 0, 20], [0, 0, -58, 0], [-20, 0, 0, -12], [0, -33, 30, -33], [-16, 9, -16, 0], [-14, -6, -14, 0], [-50, 0, 0, -13], [20, 0, 0, -2], [0, 16, -52, 16], [12, 2, 12, 0], [0, 0, 52, 0], [18, -14, 18, 0], [20, 0, 0, 33], [56, 12, 56, 0], [8, 0, 0, -12], [0, -38, 22, -38], [0, -16, -36, -16], [-16, 0, 0, 20], [4, 0, 0, -31], [22, -18, 22, 0], [0, -4, 0, -4], [-52, 0, 0, 1], [0, 8, 6, 8], [54, 5, 54, 0], [0, -48, -6, -48], [0, 26, -38, 26], [6, 0, 0, -44], [-30, -26, -30, 0], [0, 36, 8, 36], [-4, 0, 0, -12], [0, -4, 26, -4], [56, 0, 0, 0], [0, 21, 52, 21], [60, 4, 60, 0], [40, 0, 0, -1], [-30, 0, 0, 10], [18, 0, 0, 39], [0, 22, -6, 22], [30, 22, 30, 0], [0, -31, 10, -31], [0, 10, -30, 10], [0, 8, 0, 0], [34, 0, 0, 26], [-56, -2, -56, 0], [-16, 40, -16, 0], [0, 10, 42, 10], [-20, -36, -20, 0], [-34, -18, -34, 0], [0, 9, 58, 9], [0, 24, 0, 24], [34, 0, 0, -9], [34, 0, 0, 12], [14, 26, 14, 0], [0, -6, 0, 0], [0, -30, -22, -30], [38, 0, 0, -12], [0, -5, -54, -5], [-2, 0, 0, 52], [0, -35, -16, -35], [0, -36, 28, -36], [-22, -4, -22, 0], [2, 0, 0, -10], [72, 0, 0, 7], [0, -12, 16, -12], [-2, -19, -2, 0], [-14, 0, 0, 14], [20, 0, 20, 0], [-12, 0, 0, 4], [-30, -40, -30, 0], [46, 14, 46, 0], [0, -12, 32, -12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1470_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1470_2_i_v();".
// 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_1470_2_i_v(:prec:=4)
    chi := MakeCharacter_1470_i();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1470_2_i_v();".
// 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_1470_2_i_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1470_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![4, 8, 14, 4, 1]>,<13,R![0, 1]>,<17,R![4, 0, 2, 0, 1]>,<19,R![64, 0, 8, 0, 1]>,<31,R![2116, -184, 62, 4, 1]>],Snew);
    return Vf;
end function;