
// Make newform 8030.2.a.q in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8030_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_8030_2_a_q();".
// 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_8030_2_a_q();".
// 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, -1, 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_8030_a();"
function MakeCharacter_8030_a()
    N := 8030;
    order := 1;
    char_gens := [1607, 2191, 881];
    v := [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_8030_a_Hecke(Kf)
    return MakeCharacter_8030_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 := [[1, 0], [0, 1], [-1, 0], [-2, -1], [1, 0], [-2, 0], [-1, 2], [1, 0], [1, -3], [-7, 0], [3, -2], [1, -2], [3, -3], [3, -1], [8, -2], [-4, 5], [-6, 2], [-8, 1], [-12, 0], [4, -7], [-1, 0], [-2, 0], [4, 0], [1, 0], [-8, -2], [1, 1], [8, -6], [-7, 3], [2, 0], [-10, 0], [-3, 7], [-16, 3], [-18, 0], [-4, 0], [6, 3], [0, 0], [-10, 3], [16, -1], [-8, 9], [7, -5], [-2, 0], [-4, 4], [0, -8], [13, 0], [20, -2], [-1, 0], [-13, 2], [-13, 3], [12, 0], [5, 3], [2, -9], [3, -1], [0, 6], [-21, 1], [4, 2], [0, -5], [-24, 0], [8, -4], [-10, -2], [-2, 12], [-6, -2], [15, -7], [-1, 1], [-14, 5], [-7, 13], [17, 4], [-10, 8], [13, 6], [26, 2], [2, 8], [2, 6], [10, 0], [-8, -6], [2, -10], [-8, 10], [-32, 0], [-30, -2], [-6, 12], [-1, 0], [16, -2], [-7, 9], [-11, -1], [-1, 1], [6, 2], [0, -12], [1, -9], [20, -10], [-10, 3], [-38, -1], [12, 8], [-1, 12], [-32, 6], [-3, 11], [10, -7], [-31, 3], [19, 1], [-24, 6], [-2, 4], [10, 4], [-7, -4], [18, -4], [-19, 7], [-16, -6], [24, -4], [10, -21], [3, 15], [-6, 13], [-22, 4], [-13, -6], [4, 6], [12, -7], [12, 8], [21, -11], [17, 5], [19, -16], [17, 10], [-1, 4], [4, 2], [30, -5], [15, 0], [10, 4], [-36, -5], [32, 2], [5, -20], [26, -16], [-1, -16], [-18, 0], [-15, 0], [-21, 7], [-29, 11], [32, 4], [-24, 13], [-4, 1], [22, 4], [22, 10], [0, 0], [8, -7], [-38, -6], [35, 3], [29, -13], [16, 11], [12, -2], [40, -4], [0, -8], [-6, 12], [10, -6], [-17, 15], [-28, -9], [-4, 4], [-32, 2], [6, 0], [4, 14], [-8, 13], [20, 4], [-28, 11], [-6, 15], [-29, 3], [-22, -1], [-40, -6], [-12, 19], [-12, -13], [16, -7], [29, 12], [4, -2], [-1, -1], [4, -20], [16, 0], [7, -11], [-2, 18], [34, 5], [16, 1], [0, -16], [-14, 14], [23, -22], [-23, -5], [-11, -2], [18, -6], [-17, -3], [-15, -8], [13, -6], [24, -26], [0, 1], [-11, 0], [-40, 7], [5, 7], [-31, 5], [14, -4], [-1, 9], [24, -16], [29, -17], [-7, -9], [-1, -13], [-12, -8], [-21, 1], [-14, 6], [-6, 2], [-60, 4], [30, -20], [-6, -8], [-17, -2], [-4, -13], [17, 13], [15, -6], [-61, -1], [-8, 4], [39, 0], [-20, 12], [20, -10], [-15, 11], [13, 9], [-30, 17], [-23, -7], [28, -16], [17, -24], [-8, 16], [6, -5], [56, -5], [36, -10], [26, -4], [-10, 5], [-28, -10], [14, 8], [0, 11], [-48, 14], [9, -15], [-26, 17], [-16, -20], [-36, -5], [40, -20], [5, -13], [16, 10], [-8, 7], [15, -1], [14, 18], [12, 8], [-7, 24], [24, -15], [-16, -6], [42, 0], [9, -12], [-27, -12], [-15, -1], [-32, -10], [-10, -4], [-21, 11], [36, -2], [8, -12], [-20, 12], [-19, 16], [31, -2], [-5, -1], [28, 16], [-71, 1], [-11, -22], [34, 0], [-4, 27], [-57, -5], [-13, -19], [34, 4], [-6, 25], [-33, -11], [28, -26], [-19, -18], [-48, 12], [40, 2], [-38, -14], [2, 16], [-24, -9], [-23, 25], [39, -9], [-35, 7], [25, -32], [12, -16], [-27, 5], [4, -9], [-52, 18], [-8, 12], [-44, 10], [39, -21], [-44, 17], [24, 3], [-26, -10], [28, -12], [2, 27], [0, -32], [-2, -7], [-4, -16], [56, 0], [46, -1], [49, 1], [34, 10], [12, -8], [35, -14], [22, -37], [22, 2], [-19, 1], [-8, -8], [37, -24], [21, 11], [-20, 22], [25, -22], [-36, 18], [63, -11], [0, 12], [4, 6], [16, 0], [-1, -7], [-2, 12], [-23, -4], [-47, 15], [21, 23], [-70, 10], [-1, -21], [9, 2], [14, 11], [-28, -8], [36, -4], [8, -22], [-30, -6], [-33, 3], [-36, -19], [-37, 19], [14, -24], [24, -4], [24, 21], [60, -7], [6, 18], [-17, 9], [28, 13], [-7, -21], [31, -18], [32, 14], [-10, -4], [-21, 25], [28, -3], [34, -9], [60, 6], [-9, -22], [-21, 6], [-56, -5], [-38, 28], [49, -17], [-62, -11], [18, -1], [27, 24], [-51, 10], [-46, 26], [8, 13], [-40, -18], [-50, 5], [-63, 12], [-32, -14], [-26, -20], [-24, -5], [2, -7], [34, 7], [-19, -8], [16, -25], [42, 16], [66, 0], [-56, 14], [-25, -1], [-29, 27], [27, -36], [57, -18], [52, -14], [40, -12], [34, -5], [-12, 24], [-43, -21], [35, -15], [-56, 24], [-4, 1], [-60, 10], [-51, 29], [-4, -2], [-46, 6], [-20, -5], [-7, 19], [-18, 1], [-3, 3], [-70, 16], [-39, 3], [41, -26], [-22, -6], [23, -9], [-23, 39], [26, 7], [-28, 8], [52, -16], [-39, 4], [9, 28], [-57, 19], [-26, -2], [7, -7], [-2, -20], [-24, -12], [-34, 31], [-66, 16], [46, 9], [58, -7], [9, -15], [4, 28], [-54, 2], [50, 4], [-60, -10], [13, -28], [-1, 33], [-59, -12], [-14, 26], [-55, 1], [26, 30], [-28, 36], [-85, -3], [9, 9], [-17, -13], [24, 8], [-92, 2], [-50, 6], [-36, 5], [65, 5], [-63, 15], [6, -24], [-44, -19], [-31, -1], [-22, 15], [4, -3], [34, -18], [-24, -8], [38, 12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8030_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8030_2_a_q();".
// 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_8030_2_a_q(:prec:=2)
    chi := MakeCharacter_8030_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_8030_2_a_q();".
// 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_8030_2_a_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8030_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-4, -1, 1]>,<7,R![2, 5, 1]>],Snew);
    return Vf;
end function;