
// Make newform 2070.2.a.b in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2070_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_2070_2_a_b();".
// 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_2070_2_a_b();".
// 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_2070_a();"
function MakeCharacter_2070_a()
    N := 2070;
    order := 1;
    char_gens := [461, 1657, 1891];
    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_2070_a_Hecke(Kf)
    return MakeCharacter_2070_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], [-4], [-2], [6], [4], [1], [2], [0], [-2], [-10], [-4], [0], [-6], [4], [-10], [-12], [8], [10], [-8], [4], [-18], [2], [-6], [0], [12], [-10], [6], [-16], [-20], [-2], [-20], [10], [8], [-10], [20], [-8], [-14], [-4], [-2], [0], [2], [-6], [-16], [4], [16], [20], [14], [6], [-16], [18], [-20], [14], [24], [-14], [0], [-10], [-18], [-4], [-22], [4], [-8], [10], [-30], [28], [2], [-12], [14], [-34], [-8], [-8], [-18], [12], [32], [-38], [-2], [-10], [26], [-12], [-2], [32], [34], [-8], [4], [30], [10], [18], [16], [-12], [-32], [40], [4], [20], [40], [34], [14], [-36], [14], [4], [-14], [36], [-18], [-4], [2], [-28], [30], [40], [-38], [-16], [-18], [-18], [-4], [16], [-26], [20], [-8], [-14], [20], [-50], [2], [-38], [36], [-28], [50], [-34], [32], [16], [6], [-12], [24], [-40], [-18], [22], [-14], [-54], [4], [34], [-26], [28], [42], [-40], [12], [46], [-24], [6], [-42], [44], [32], [-50], [-10], [20], [24], [44], [-32], [-32], [30], [-22], [-14], [-4], [-2], [-24], [-52], [-42], [-24], [0], [38], [-46], [10], [-20], [46], [8], [26], [-32], [30], [44], [-38], [-32], [54], [0], [36], [-26], [22], [-16], [-22], [14], [52], [42], [0], [34], [-28], [20], [-30], [52], [-34], [-14], [-26], [-26], [8], [-46], [64], [-42], [18], [-4], [-30], [-40], [-36], [54], [-52], [50], [42], [-64], [12], [72], [-38], [0], [30], [24], [66], [70], [32], [-18], [-56], [12], [-42], [62], [0], [-48], [-12], [-50], [-60], [40], [-26], [44], [-48], [-14], [-22], [-12], [56], [36], [-4], [40], [-18], [30], [8], [16], [-44], [-4], [-32], [-42], [22], [-40], [58], [-78], [12], [-34], [12], [-38], [-6], [-8], [-12], [22], [6], [14], [20], [-46], [30], [-68], [-70], [62], [4], [10], [8], [50], [-72], [68], [-2], [42], [52], [32], [32], [-8], [-66], [28], [0], [50], [42], [-40], [-50], [-46], [-60], [70], [12], [14], [34], [-8], [42], [36], [-76], [26], [18], [-56], [44], [12], [-30], [-44], [86], [-8], [22], [16], [-70], [54], [-12], [-72], [-70], [12], [0], [-14], [46], [84], [-22], [-30], [48], [-82], [-78], [-60], [12], [-80], [58], [-2], [2], [32], [12], [28], [52], [38], [-74], [42], [-56], [-26], [30], [-6], [64], [-46], [-4], [-90], [-68], [32], [-6], [-76], [74], [-78], [56], [14], [-90], [48], [60], [78], [56], [14], [70], [-80], [-68], [84], [-86], [18], [-16], [-22], [-20], [-20], [48], [26], [-80], [46], [-100], [48], [34], [6], [-6], [-30], [30], [40], [46], [20], [-24], [0], [-74], [-84], [0], [82], [-70], [20], [52], [24], [-38], [-88], [78], [12], [-54], [-34], [62], [24], [62], [-38], [-8], [-58], [62], [20], [-20], [82], [10], [-68], [4], [58], [50], [64], [72], [-42], [56], [-30], [-2], [-96], [-28], [10], [-46], [4], [-26], [-20], [40]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2070_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2070_2_a_b();".
// 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_2070_2_a_b(:prec:=1)
    chi := MakeCharacter_2070_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_2070_2_a_b();".
// 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_2070_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2070_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![4, 1]>,<13,R![2, 1]>,<17,R![-6, 1]>,<29,R![-2, 1]>],Snew);
    return Vf;
end function;