
// Make newform 1710.2.a.c 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_c();".
// 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_c();".
// 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], [-2], [0], [6], [-8], [1], [4], [-2], [-2], [-2], [12], [4], [-12], [-10], [-6], [-14], [-12], [8], [-10], [14], [-2], [0], [2], [6], [-16], [16], [-12], [-6], [-8], [0], [-12], [-12], [-10], [18], [-4], [8], [-8], [-2], [-10], [-12], [0], [-2], [-18], [4], [-16], [-4], [-20], [6], [-24], [8], [-14], [12], [30], [12], [30], [-4], [4], [20], [20], [18], [-28], [-24], [6], [2], [20], [-2], [10], [14], [-20], [0], [34], [-14], [-16], [0], [-18], [8], [0], [-6], [28], [4], [-24], [-34], [-10], [34], [-36], [10], [18], [22], [22], [0], [40], [16], [28], [16], [-2], [0], [-20], [-18], [-36], [2], [-24], [-4], [12], [-30], [26], [-36], [36], [-14], [24], [-16], [-20], [4], [16], [20], [-28], [16], [42], [6], [-4], [2], [-6], [36], [28], [-14], [-18], [8], [42], [8], [20], [-40], [-2], [40], [-22], [-18], [-10], [-20], [18], [-38], [16], [38], [14], [-32], [28], [0], [-28], [-42], [-28], [-8], [18], [-42], [-52], [-32], [-36], [-20], [-24], [-34], [2], [-30], [-2], [6], [-34], [6], [30], [0], [50], [44], [-6], [-6], [-34], [20], [-48], [30], [-38], [2], [20], [-54], [36], [-58], [-42], [2], [-18], [54], [-36], [-66], [26], [52], [-46], [-32], [2], [-18], [24], [42], [-2], [18], [22], [52], [20], [-36], [-46], [-30], [22], [46], [0], [54], [-20], [56], [-34], [-16], [-58], [10], [66], [-60], [-28], [-34], [54], [-16], [0], [-14], [0], [30], [12], [70], [36], [70], [30], [40], [48], [56], [76], [-20], [30], [40], [-40], [20], [-34], [-6], [36], [-40], [4], [28], [26], [24], [10], [24], [-46], [-58], [16], [12], [56], [6], [16], [-58], [38], [-36], [2], [-52], [2], [14], [28], [-56], [10], [-22], [-24], [76], [18], [6], [36], [26], [-52], [-44], [14], [20], [78], [-22], [2], [-8], [-54], [-24], [24], [-40], [16], [-12], [-8], [8], [10], [-66], [-48], [-40], [50], [-58], [-78], [46], [82], [-30], [-26], [30], [-42], [-68], [78], [-2], [-8], [-56], [-12], [2], [32], [-40], [-16], [32], [16], [18], [8], [12], [-60], [-34], [60], [60], [-74], [66], [24], [62], [-50], [-64], [24], [34], [80], [-76], [24], [-54], [-70], [-66], [-80], [-18], [20], [-18], [40], [-90], [-14], [-2], [26], [-24], [-50], [-14], [26], [-14], [50], [-60], [-40], [30], [-4], [62], [78], [40], [4], [62], [80], [-8], [36], [48], [68], [42], [-80], [-26], [-12], [-70], [86], [-40], [78], [64], [4], [40], [54], [-92], [20], [-22], [80], [26], [6], [74], [42], [24], [82], [-56], [64], [72], [76], [100], [-48], [48], [62], [-86], [4], [40], [28], [-18], [-14], [-72], [-56], [70], [-52], [-96], [-20], [14], [-2], [-36], [-92], [44], [76], [16], [34], [-62], [44], [-20], [14], [14], [-68], [100], [-68], [0], [82], [-70], [-84], [-26], [2], [42], [-48], [-18], [-20], [-56]];
    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_c();".
// 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_c(: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_c();".
// 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_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1710_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<11,R![0, 1]>,<13,R![-6, 1]>,<53,R![10, 1]>],Snew);
    return Vf;
end function;