
// Make newform 5070.2.a.x in Magma, downloaded from the LMFDB on 23 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_x();".
// 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_x();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
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], [1], [1], [5], [3], [0], [-8], [5], [-4], [-4], [2], [7], [-6], [6], [3], [1], [-12], [2], [-8], [-2], [0], [-2], [-8], [11], [0], [-8], [-7], [-6], [-14], [8], [-21], [-19], [12], [7], [2], [-22], [15], [20], [23], [5], [4], [-2], [2], [-24], [-3], [22], [-15], [3], [0], [-14], [-14], [18], [25], [15], [28], [-15], [4], [-4], [-15], [-18], [-10], [9], [6], [-12], [-6], [23], [-4], [14], [-16], [-8], [-16], [18], [-8], [38], [25], [28], [-32], [-25], [19], [-25], [-12], [-12], [-12], [16], [10], [6], [27], [-30], [-8], [-8], [-24], [28], [-37], [21], [-4], [-11], [-10], [5], [-10], [-34], [6], [-15], [36], [-15], [-33], [-2], [-18], [-20], [34], [37], [-29], [-25], [14], [-17], [-12], [27], [44], [3], [27], [36], [10], [-32], [-22], [-30], [-17], [0], [-32], [-20], [-11], [43], [-19], [16], [-8], [17], [-9], [34], [-1], [28], [-10], [-26], [-25], [22], [-35], [-6], [-16], [14], [26], [38], [3], [24], [54], [-15], [30], [41], [10], [-20], [-2], [-2], [30], [0], [-58], [-54], [31], [15], [-20], [-23], [10], [-1], [-7], [42], [-12], [-38], [-50], [46], [-2], [21], [-32], [20], [0], [54], [-28], [-17], [9], [24], [23], [30], [-49], [44], [3], [-58], [34], [36], [53], [-50], [38], [38], [-2], [11], [-12], [-8], [22], [-14], [62], [-27], [33], [-62], [48], [16], [11], [41], [32], [-22], [-13], [-6], [-30], [-55], [-64], [-39], [-9], [-23], [-52], [30], [41], [-4], [54], [-8], [74], [-34], [56], [1], [19], [21], [40], [-15], [-72], [-61], [-46], [18], [45], [-6], [6], [-12], [43], [-70], [-18], [-16], [8], [-13], [-1], [-65], [2], [-67], [15], [-26], [-15], [-39], [-14], [-50], [-11], [26], [-31], [-36], [12], [-14], [-46], [0], [54], [-51], [-4], [50], [34], [28], [20], [-12], [32], [16], [10], [-6], [22], [4], [-11], [-8], [68], [28], [-22], [66], [66], [-39], [-30], [-1], [48], [-32], [-24], [75], [30], [18], [14], [27], [-28], [4], [-76], [78], [20], [18], [-68], [42], [-72], [2], [-24], [7], [-33], [6], [65], [-32], [-29], [61], [56], [-10], [-70], [18], [-4], [-4], [-24], [16], [54], [-82], [-20], [38], [-71], [90], [-36], [3], [-40], [-76], [68], [84], [8], [-12], [-85], [7], [41], [56], [-10], [8], [-41], [-77], [20], [-38], [-54], [-45], [23], [-12], [-26], [-3], [48], [62], [84], [25], [68], [93], [25], [-77], [24], [-27], [70], [52], [2], [59], [58], [-12], [71], [-8], [84], [84], [46], [57], [6], [-70], [-63], [28], [86], [-48], [-53], [-44], [67], [-3], [38], [86], [62], [68], [-63], [-51], [-71], [40], [60], [14], [-62], [78], [-52], [2], [-14], [52], [88], [-12], [30], [42], [34], [-105], [-38], [15], [-34], [-27], [84], [-3], [-62], [34], [-58], [7], [-46], [56], [-30], [97], [13], [-11], [-96], [21], [64], [5], [-3], [42]];
    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_x();".
// 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_x(:prec:=1)
    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_x();".
// 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_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5070_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-5, 1]>,<11,R![-3, 1]>,<17,R![8, 1]>,<31,R![-2, 1]>],Snew);
    return Vf;
end function;