
// Make newform 630.2.t.a in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_630_t();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_630_t_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_630_2_t_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_630_2_t_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_630_t();"
function MakeCharacter_630_t()
    N := 630;
    order := 6;
    char_gens := [281, 127, 451];
    v := [5, 6, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_630_t_Hecke();"
function MakeCharacter_630_t_Hecke(Kf)
    N := 630;
    order := 6;
    char_gens := [281, 127, 451];
    char_values := [[1, 0, -1, 0], [1, 0, 0, 0], [0, 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 := [[0, -1, 0, 0], [0, -1, 0, 2], [-1, 0, 0, 0], [2, 0, -3, 0], [-1, 0, 2, 3], [2, 0, 2, 0], [0, 0, 0, 0], [2, -3, -1, 3], [2, 0, -4, 6], [-4, -3, 2, 3], [-6, -3, 3, 3], [0, -3, -1, -3], [-9, 0, 9, 0], [0, -3, 2, -3], [-9, 0, 9, 0], [1, -9, 1, 0], [0, 3, -3, 3], [4, 6, 4, 0], [0, 0, 4, 0], [1, 0, -2, -3], [4, 0, 4, 0], [1, -9, -1, 18], [0, 3, 6, 3], [0, -6, 6, -6], [-4, 12, 2, -12], [-6, -6, 0, 3], [-7, 0, 14, 0], [8, 9, -4, -9], [2, 3, -2, -6], [-9, 3, -9, 0], [-1, -12, 0, 6], [-6, 12, 0, -6], [-1, 0, 2, -9], [11, 3, 11, 0], [0, 0, 0, 12], [1, -6, 0, 3], [-10, -9, 5, 9], [0, 6, -2, 6], [0, 6, 0, -12], [3, 3, -3, -6], [9, 9, 9, 0], [6, 0, -12, -3], [2, 0, 2, 0], [0, 3, -19, 3], [5, 0, -10, -15], [4, 0, 4, 0], [5, 9, -5, -18], [2, 12, -1, -12], [-6, -12, 0, 6], [4, 0, -8, 21], [-2, 3, 1, -3], [7, -9, 7, 0], [5, 0, -10, 12], [-18, 0, 0, 0], [18, 12, 0, -6], [-1, 0, 2, 12], [18, 0, -18, 0], [18, -9, -9, 9], [-19, -6, 0, 3], [10, 6, -5, -6], [-16, -3, 8, 3], [-6, -6, 6, 12], [-12, 0, 24, 3], [0, 3, -21, 3], [-3, -15, -3, 0], [-2, 24, -2, 0], [8, 0, -8, 0], [-10, 0, 10, 0], [28, -3, -14, 3], [12, 6, -6, -6], [15, -18, 0, 9], [6, -21, -3, 21], [3, 0, -6, -24], [5, 6, 0, -3], [-20, 12, 0, -6], [9, 12, 0, -6], [16, 0, -32, 3], [-18, -15, 9, 15], [-9, 0, 18, -12], [-26, 12, 13, -12], [0, 6, 0, -12], [0, 15, 4, 15], [-12, -18, -12, 0], [0, 0, 0, -18], [5, 21, 5, 0], [0, 9, 0, 0], [-9, 0, 18, -6], [8, -6, -8, 12], [0, -3, 0, -3], [13, -6, -13, 12], [0, -3, 12, -3], [0, -12, 0, 6], [-34, 0, 34, 0], [-7, -15, -7, 0], [-11, -6, 0, 3], [3, -12, 0, 6], [-6, 18, 0, -9], [21, 0, -21, 0], [0, 33, 0, -33], [0, 0, -8, 0], [0, -9, 10, -9], [12, 12, 12, 0], [0, 6, -24, 6], [0, -6, 0, 0], [0, 6, -32, 6], [0, 24, 0, 0], [0, -3, 36, -3], [0, -12, -6, -12], [12, -6, -6, 6], [16, 18, -8, -18], [-7, 0, 14, -6], [-14, 12, 14, -24], [14, -6, 14, 0], [-6, 0, 12, 6], [2, 12, 0, -6], [-12, 0, 24, -24], [16, -9, 16, 0], [15, -6, -15, 12], [8, 0, -16, -36], [-34, -9, 17, 9], [-16, 27, 8, -27], [0, 15, -1, 15], [-3, 15, 3, -30], [-2, 39, -2, 0], [4, -12, 4, 0], [-10, 0, 20, 27], [26, 12, -26, -24], [0, 24, 0, 24], [-28, -24, 14, 24], [-11, 0, 22, 15], [-2, -18, 2, 36], [5, -36, 5, 0], [-32, 12, 0, -6], [7, -18, 0, 9], [-33, 0, 0, 0], [9, 0, 9, 0], [-18, 0, 18, 0], [-28, 15, 14, -15], [12, -18, -12, 36], [13, 30, 13, 0], [2, 0, -4, 48], [16, 9, 16, 0], [0, 6, -41, 6], [20, 0, -40, 3], [8, 21, 8, 0], [0, -3, 9, -3], [24, -24, -12, 24], [-33, 18, 0, -9], [3, 0, -6, 21], [20, 12, -10, -12], [5, 54, 0, -27], [-6, -12, 0, 6], [-8, 42, 0, -21], [-39, -12, 0, 6], [28, -6, 0, 3], [10, 3, -5, -3], [0, -9, 11, -9], [-9, -24, 9, 48], [9, 0, -18, -27], [0, -15, 0, -15], [-24, 18, -24, 0], [-1, 0, 2, -3], [16, 0, -16, 0], [0, 12, -6, 12], [-8, 12, 4, -12], [-21, -12, 0, 6], [29, -9, -29, 18], [13, 0, -26, -9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_630_t_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_630_2_t_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_630_2_t_a(:prec:=4)
    chi := MakeCharacter_630_t();
    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_630_2_t_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_630_2_t_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_630_t();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![36, 0, 24, 0, 1]>],Snew);
    return Vf;
end function;