
// Make newform 1785.2.a.b in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1785_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_1785_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_1785_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_1785_a();"
function MakeCharacter_1785_a()
    N := 1785;
    order := 1;
    char_gens := [596, 1072, 766, 1261];
    v := [1, 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_1785_a_Hecke(Kf)
    return MakeCharacter_1785_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], [1], [4], [2], [1], [8], [-4], [2], [-8], [2], [-6], [4], [4], [6], [12], [-2], [-12], [-8], [-14], [4], [0], [14], [2], [10], [0], [16], [-2], [-6], [0], [4], [-6], [12], [6], [0], [-14], [-4], [8], [14], [20], [22], [24], [-10], [-10], [-8], [8], [-16], [-28], [14], [-22], [8], [2], [-12], [-6], [8], [6], [20], [10], [-6], [12], [22], [-20], [-16], [-22], [22], [-12], [-18], [16], [-18], [-30], [-24], [32], [-34], [-16], [28], [30], [30], [-10], [-30], [28], [-10], [-8], [-2], [32], [36], [-18], [-38], [-14], [40], [-8], [40], [-40], [-36], [0], [16], [26], [-14], [20], [-2], [4], [-34], [8], [-6], [-16], [38], [0], [-22], [0], [-38], [-8], [-2], [26], [12], [0], [6], [12], [28], [6], [-12], [6], [-18], [-26], [-16], [28], [14], [-18], [0], [24], [34], [4], [-12], [-36], [-2], [38], [42], [-18], [12], [-18], [-18], [52], [-22], [-32], [-8], [14], [0], [-2], [42], [-56], [24], [10], [42], [44], [-8], [-12], [24], [48], [-22], [22], [-42], [-8], [-54], [-8], [-20], [18], [24], [12], [-18], [-46], [-10], [-12], [-42], [-16], [-6], [-36], [58], [16], [2], [-32], [-26], [-8], [12], [10], [-14], [-64], [18], [18], [60], [-14], [-32], [-22], [-24], [-20], [-14], [-24], [-6], [-22], [-54], [-62], [-60], [-38], [56], [-70], [50], [44], [14], [-4], [12], [30], [-48], [38], [-30], [56], [24], [-56], [10], [-64], [30], [-12], [30], [22], [-32], [-54], [-8], [-24], [-42], [-38], [-24], [-8], [-52], [-34], [-60], [64], [-54], [-44], [-4], [-62], [22], [52], [0], [24], [-32], [56], [-50], [-30], [0], [-8], [-12], [-44], [-64], [62], [-6], [48], [-54], [54], [-44], [-58], [12], [-42], [38], [-16], [60], [-10], [14], [-78], [-80], [70], [54], [44], [-10], [62], [-28], [46], [-32], [38], [-56], [-76], [-2], [-38], [60], [-60], [-76], [-32], [-66], [28], [40], [38], [78], [-20], [22], [-30], [-68], [10], [36], [-14], [-26], [36], [6], [-20], [4], [22], [62], [52], [48], [-80], [-74], [44], [30], [72], [-26], [-48], [-6], [-18], [44], [-64], [-70], [36], [-48], [74], [-6], [12], [30], [66], [-24], [42], [82], [-48], [-20], [36], [86], [-58], [-58], [72], [-48], [16], [-76], [30], [50], [-38], [88], [78], [-6], [6], [-8], [-66], [-20], [38], [-60], [-72], [-10], [64], [-82], [-2], [-8], [-10], [-70], [-40], [-36], [-54], [-64], [82], [58], [24], [36], [20], [10], [62], [16], [34], [-52], [92], [60], [-34], [44], [-14], [60], [40], [30], [-42], [-66], [-74], [58], [80], [-14], [-76], [-48], [8], [-14], [4], [-12], [-70], [54], [52], [-4], [48], [78], [92], [-10], [0], [86], [46], [-78], [32], [58], [42], [-72], [30], [-70], [-44], [76], [-62], [22], [76], [-64], [50], [30], [88], [32], [-46], [-4], [14], [-58], [-20], [44], [58], [-18], [32], [30], [40], [-56]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1785_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1785_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_1785_2_a_b(:prec:=1)
    chi := MakeCharacter_1785_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_1785_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_1785_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1785_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 1]>,<11,R![-4, 1]>,<13,R![-2, 1]>],Snew);
    return Vf;
end function;