
// Make newform 1710.2.a.f in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1710_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_1710_2_a_f();".
// 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_1710_2_a_f();".
// 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_1710_a();"
function MakeCharacter_1710_a()
    N := 1710;
    order := 1;
    char_gens := [191, 1027, 1351];
    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_1710_a_Hecke(Kf)
    return MakeCharacter_1710_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], [4], [4], [-2], [2], [-1], [8], [-6], [4], [-10], [2], [12], [0], [-6], [0], [-10], [-4], [8], [2], [-12], [8], [-6], [18], [10], [16], [12], [10], [6], [8], [12], [-14], [-12], [-22], [4], [-6], [-4], [16], [-14], [0], [2], [0], [18], [18], [16], [-12], [16], [12], [22], [-6], [8], [2], [28], [-18], [-24], [-6], [-24], [18], [-30], [-4], [10], [28], [-24], [-30], [-30], [-12], [34], [-24], [-34], [-6], [-32], [-12], [30], [-20], [0], [-6], [2], [18], [-6], [-12], [-14], [0], [2], [-4], [24], [-6], [-6], [-14], [-36], [-8], [16], [-32], [-28], [-4], [-32], [10], [10], [4], [-18], [-20], [-30], [20], [18], [-4], [2], [40], [10], [-32], [-6], [24], [-6], [-30], [-4], [8], [10], [-28], [32], [-14], [16], [18], [-14], [-6], [36], [-44], [34], [22], [40], [12], [-14], [20], [-48], [52], [-38], [30], [2], [26], [52], [2], [-18], [12], [-22], [52], [-52], [10], [40], [-46], [-42], [-4], [-32], [6], [30], [4], [-48], [-20], [-8], [-24], [-18], [-38], [50], [40], [-10], [52], [24], [-50], [-48], [-20], [-14], [2], [-22], [-24], [-38], [32], [-30], [-4], [30], [52], [-6], [-40], [-2], [60], [-8], [54], [22], [16], [-54], [-26], [4], [-6], [8], [-46], [32], [-60], [2], [40], [14], [2], [10], [42], [-8], [-30], [-52], [-42], [26], [-28], [42], [-24], [52], [-10], [-20], [-22], [10], [-68], [-12], [32], [-62], [28], [-46], [-48], [-46], [-22], [12], [-6], [12], [12], [-10], [-26], [0], [-48], [-28], [10], [-36], [68], [58], [20], [0], [34], [-6], [-4], [72], [-4], [76], [-4], [50], [-26], [48], [-12], [48], [44], [16], [-46], [-50], [16], [-14], [-70], [36], [-66], [28], [-54], [-78], [-16], [4], [-50], [-58], [18], [-36], [-46], [54], [20], [-54], [-70], [-68], [50], [-8], [2], [-36], [72], [2], [-38], [28], [-40], [-8], [16], [50], [52], [-24], [18], [-22], [56], [-14], [18], [-56], [14], [8], [-58], [34], [-60], [2], [24], [52], [34], [18], [0], [84], [-12], [50], [28], [82], [-24], [26], [-8], [26], [-46], [-76], [24], [50], [-28], [-8], [10], [-26], [4], [-78], [-86], [56], [26], [18], [-76], [-44], [0], [-54], [6], [2], [-48], [72], [92], [64], [-30], [62], [-6], [-20], [-66], [42], [66], [-44], [18], [-24], [-34], [-4], [-48], [-78], [76], [10], [66], [40], [34], [86], [0], [-60], [90], [-16], [66], [-2], [72], [-24], [-92], [58], [-86], [32], [-46], [-52], [20], [-72], [-30], [0], [74], [40], [-48], [50], [78], [10], [-94], [-6], [44], [18], [4], [24], [40], [82], [20], [8], [10], [26], [-68], [-20], [0], [-38], [-20], [-70], [-52], [10], [-86], [98], [96], [-42], [34], [-80], [-70], [-54], [28], [-52], [34], [42], [4], [60], [34], [2], [40], [8], [-54], [-56], [-30], [-18], [80], [56], [42], [-30], [84], [6], [60], [-40]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1710_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1710_2_a_f();".
// 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_1710_2_a_f(:prec:=1)
    chi := MakeCharacter_1710_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_1710_2_a_f();".
// 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_1710_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1710_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-4, 1]>,<11,R![-4, 1]>,<13,R![2, 1]>,<53,R![6, 1]>],Snew);
    return Vf;
end function;