
// Make newform 5070.2.a.ba in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5070_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_5070_2_a_ba();".
// 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_5070_2_a_ba();".
// 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 := [-3, 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_5070_a();"
function MakeCharacter_5070_a()
    N := 5070;
    order := 1;
    char_gens := [1691, 4057, 1861];
    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_5070_a_Hecke(Kf)
    return MakeCharacter_5070_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], [-1, 0], [1, 0], [-3, 0], [2, -1], [0, 0], [-4, 0], [4, 1], [0, -2], [-2, 2], [-2, 4], [1, -4], [4, 0], [-6, 0], [3, 2], [-2, 1], [8, 2], [-4, 2], [2, -2], [-6, -4], [0, 4], [10, -4], [6, 2], [-2, -7], [-2, 6], [-2, -6], [-4, 9], [-6, -4], [0, -6], [-12, 0], [4, 5], [-3, 10], [-10, -2], [1, -12], [-2, 12], [-2, -12], [16, 3], [-4, 4], [3, -2], [-10, 7], [-16, 4], [-10, -4], [0, -10], [4, -8], [1, 6], [18, 4], [-15, 4], [5, 8], [10, -2], [-8, 2], [-18, 0], [0, -2], [20, 3], [-23, 2], [-18, 2], [-2, -13], [-24, 4], [14, 6], [4, -9], [0, 4], [-4, 6], [-5, -14], [-4, -14], [-12, -4], [-16, -10], [-27, -2], [4, 6], [16, 6], [-6, 6], [-14, 2], [-18, 6], [-6, 4], [4, 10], [16, -4], [8, 1], [-16, 12], [-4, -8], [-1, 4], [-10, -7], [-20, -11], [-4, 8], [-8, -8], [0, 8], [-12, 12], [-4, 10], [-6, -8], [-6, 7], [4, 2], [-22, 6], [-20, -12], [-14, -6], [2, -14], [21, 0], [5, -6], [-16, -10], [2, 1], [-12, -10], [-17, 6], [-14, -16], [-6, 12], [12, -10], [-33, 6], [-2, 10], [-1, 10], [3, -8], [-8, 10], [2, 4], [-24, -12], [22, 4], [3, 16], [8, -17], [-45, 0], [-32, -2], [-20, 13], [8, 0], [-11, -2], [-8, 12], [34, 1], [14, -9], [-12, -16], [-32, 6], [-4, 12], [4, 4], [-24, -10], [4, -11], [-6, 6], [10, 10], [-16, 0], [-4, 5], [-7, -8], [-4, 23], [4, 4], [-10, 18], [20, 1], [18, -15], [12, -16], [-43, 6], [2, -2], [24, -8], [14, -12], [-8, -9], [32, -10], [24, -9], [-2, 28], [22, -10], [12, -18], [10, -20], [22, 4], [-13, 4], [4, -4], [-6, 16], [-1, -22], [-4, 10], [54, 1], [-14, -4], [-8, 12], [4, -30], [44, 8], [6, -16], [12, 28], [24, 6], [36, 10], [15, -8], [-31, -6], [2, 2], [15, 18], [8, 18], [36, 15], [20, 19], [-12, -24], [28, 10], [26, 12], [-6, -16], [48, 6], [6, 16], [-3, 22], [40, -6], [48, -4], [0, -2], [-8, 18], [28, -8], [7, -2], [16, -11], [-4, -32], [-15, -10], [-18, -12], [20, 1], [26, -14], [-32, -5], [-18, -4], [2, 32], [14, -18], [-39, -8], [42, -4], [12, 2], [-6, -4], [-52, 0], [44, 13], [12, 4], [8, 18], [-12, -2], [16, -14], [26, 0], [21, -8], [22, 15], [24, -8], [-56, 0], [-34, 2], [-10, 25], [-49, -4], [38, -14], [-12, 18], [40, -1], [34, 0], [-18, 4], [24, 25], [-12, -2], [13, 18], [37, 2], [-33, 14], [-22, 14], [-26, -24], [-22, -15], [-20, -16], [-36, 10], [30, -2], [-60, 2], [-10, 0], [28, -14], [22, -9], [12, -13], [41, 0], [42, 18], [5, 2], [-4, -16], [35, 10], [-32, 0], [-30, 20], [17, -10], [-34, 12], [-2, -16], [16, -12], [-20, 13], [34, -12], [-42, -8], [-18, 26], [16, -16], [-38, 5], [-28, 21], [-18, 3], [-18, -12], [18, -11], [9, -34], [-26, -12], [-46, -3], [54, 7], [20, 6], [-16, 18], [30, -3], [-2, -24], [-4, -25], [44, 16], [28, 12], [48, -4], [-6, 12], [-8, -8], [28, 2], [2, -11], [-2, 14], [-48, 8], [-42, -16], [-34, -22], [-14, 10], [-22, 14], [-56, -12], [-20, -20], [42, -4], [-8, -38], [-20, -16], [-4, 8], [-62, -5], [26, -6], [16, 6], [-28, 12], [28, 6], [-36, 6], [18, -32], [-13, 34], [-24, -2], [67, -2], [-28, -12], [12, 4], [-32, -12], [38, 17], [8, 32], [-10, -36], [-2, 8], [46, -11], [-8, 36], [26, -10], [-44, -4], [14, 20], [0, 16], [32, -6], [4, 24], [-22, 0], [16, -8], [56, 10], [10, 14], [52, -17], [-66, 1], [-44, 26], [61, 2], [-22, 30], [15, -10], [35, 0], [24, 22], [-56, 2], [14, -12], [6, 8], [12, -16], [30, 2], [2, 10], [-8, -40], [-18, -4], [-2, 44], [-16, 18], [62, 12], [-38, 5], [60, 8], [40, -16], [6, -3], [16, -16], [14, 26], [-40, 6], [28, -24], [68, -8], [-70, -6], [36, -9], [12, -7], [37, 24], [62, -6], [-20, -30], [12, -28], [-23, -22], [9, -34], [-28, -32], [-10, 24], [-18, -20], [2, -7], [48, 9], [-22, -2], [12, -10], [-76, 3], [-32, 0], [6, 0], [38, 22], [-14, -21], [20, 4], [-41, -2], [-11, 24], [9, -6], [-56, -14], [14, 9], [32, 14], [-12, 0], [-18, 12], [1, 12], [-18, -16], [4, 24], [-43, -24], [-24, 8], [8, 20], [-28, 24], [-32, 24], [6, -11], [-54, -12], [38, 0], [-39, -10], [10, 14], [14, -8], [6, -30], [-47, -12], [10, -30], [-20, -23], [-1, -10], [-2, 0], [-40, -24], [56, -6], [48, 18], [-40, -33], [27, -10], [22, -17], [-44, 16], [-20, 36], [-4, 22], [28, -18], [-50, -28], [24, 4], [0, -22], [-22, 36], [-58, -14], [20, -28], [-56, -8], [-16, 22], [-82, 12], [-38, 24], [-30, 33], [-10, 4], [66, 3], [-28, -18], [-18, -9], [20, -44], [13, 12], [-54, 4], [12, 50], [-8, -34], [28, 9], [22, 8], [68, 18], [-42, -16], [-5, 0], [9, 18], [7, 34], [46, 6], [-27, -30], [-68, -4], [-10, 3], [28, -9], [24, 26]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5070_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5070_2_a_ba();".
// 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_5070_2_a_ba(:prec:=2)
    chi := MakeCharacter_5070_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_5070_2_a_ba();".
// 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_5070_2_a_ba( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5070_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![3, 1]>,<11,R![1, -4, 1]>,<17,R![4, 1]>,<31,R![-44, 4, 1]>],Snew);
    return Vf;
end function;