
// Make newform 1710.2.a.p 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_p();".
// 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_p();".
// 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], [-6], [0], [4], [-1], [-4], [-10], [-2], [-4], [10], [-12], [0], [-6], [4], [-10], [-8], [0], [6], [-10], [14], [10], [-6], [10], [18], [0], [-8], [-2], [18], [-10], [12], [-12], [-10], [-18], [-18], [12], [-8], [18], [0], [-16], [16], [-2], [10], [20], [-4], [14], [-8], [-18], [4], [-12], [-22], [2], [-6], [20], [-14], [-4], [10], [-10], [28], [26], [-20], [0], [-14], [-30], [28], [-26], [6], [10], [24], [0], [8], [12], [20], [-24], [-18], [22], [-6], [2], [26], [28], [-24], [6], [10], [6], [-26], [-38], [-2], [-32], [-10], [-24], [-2], [-26], [-36], [-28], [-18], [42], [-16], [-22], [4], [22], [-4], [-34], [-20], [-2], [42], [-44], [0], [26], [22], [14], [8], [-20], [-36], [42], [28], [-12], [-42], [-36], [-40], [-10], [2], [36], [28], [-30], [-34], [16], [-28], [-18], [-4], [8], [46], [-26], [24], [-14], [30], [-52], [-2], [36], [20], [50], [-16], [-36], [0], [-24], [2], [30], [-28], [-48], [4], [-20], [20], [-16], [32], [32], [-52], [-36], [14], [-50], [-38], [34], [12], [-60], [-18], [-24], [-34], [-10], [-6], [-10], [48], [12], [-24], [10], [58], [24], [-52], [6], [-6], [-10], [-56], [56], [4], [-46], [-24], [14], [52], [36], [18], [-36], [14], [-6], [44], [14], [-18], [22], [-50], [-26], [52], [48], [50], [10], [32], [-2], [-54], [18], [0], [-4], [20], [24], [-18], [6], [-32], [16], [40], [6], [-20], [34], [-8], [-42], [-40], [10], [42], [-8], [-52], [-34], [-14], [-16], [-2], [-14], [70], [-8], [34], [-6], [52], [-12], [22], [22], [6], [24], [0], [28], [-20], [-76], [-6], [60], [-72], [32], [-40], [28], [-26], [-24], [64], [14], [-14], [34], [70], [-68], [6], [10], [-50], [36], [-30], [16], [-8], [44], [-46], [-64], [56], [-42], [4], [-20], [66], [16], [78], [16], [78], [12], [18], [-18], [-16], [24], [-12], [76], [-28], [60], [38], [-58], [8], [-34], [-34], [14], [-54], [56], [64], [-50], [30], [74], [-16], [12], [-26], [46], [48], [-8], [84], [-14], [-32], [88], [-24], [-6], [-40], [66], [18], [-16], [-32], [-26], [-18], [48], [-14], [-36], [-80], [-22], [-42], [38], [-20], [-30], [20], [-84], [-32], [30], [-6], [-6], [44], [-66], [68], [14], [-40], [-42], [-30], [52], [84], [12], [-38], [30], [-2], [-60], [-86], [-88], [-48], [-62], [16], [34], [-46], [82], [-32], [-26], [60], [-46], [28], [-32], [54], [60], [8], [0], [44], [-6], [-62], [-54], [-42], [-38], [80], [-92], [74], [-4], [-14], [12], [-100], [46], [-12], [-22], [66], [56], [8], [-12], [-84], [72], [72], [26], [4], [48], [70], [70], [30], [28], [-16], [-54], [-18], [-2], [40], [-42], [-44], [-24], [-34], [-6], [-6], [80], [2], [-2], [56], [6], [-70], [-10], [52], [28], [18], [-46], [40], [-42], [76], [24], [50], [-60], [-16], [76], [86], [18], [72], [32], [-76], [-36]];
    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_p();".
// 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_p(: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_p();".
// 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_p( : 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![6, 1]>,<13,R![0, 1]>,<53,R![6, 1]>],Snew);
    return Vf;
end function;