
// Make newform 2850.2.a.y in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2850_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_2850_2_a_y();".
// 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_2850_2_a_y();".
// 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_2850_a();"
function MakeCharacter_2850_a()
    N := 2850;
    order := 1;
    char_gens := [1901, 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_2850_a_Hecke(Kf)
    return MakeCharacter_2850_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], [0], [-2], [-6], [0], [-2], [-1], [-4], [-8], [-8], [4], [-4], [6], [12], [-6], [-4], [2], [8], [0], [-6], [8], [-4], [-4], [-12], [-10], [0], [12], [-14], [-14], [12], [22], [-6], [0], [22], [0], [-6], [-6], [16], [-18], [12], [2], [-22], [8], [-14], [20], [-28], [16], [28], [18], [-14], [-18], [-22], [-14], [-6], [-16], [8], [20], [2], [16], [-22], [-10], [8], [-6], [14], [-6], [-20], [20], [-36], [-2], [18], [-6], [10], [24], [20], [-24], [-30], [-22], [0], [14], [-14], [-2], [36], [0], [-8], [24], [-4], [14], [-34], [26], [8], [18], [-16], [-34], [24], [-40], [36], [24], [-20], [-10], [-40], [22], [-4], [-20], [16], [38], [-24], [-14], [-36], [-10], [-40], [10], [38], [-8], [0], [-12], [26], [24], [18], [-48], [26], [28], [-22], [36], [-8], [6], [50], [14], [-14], [-18], [-28], [8], [40], [38], [-18], [-14], [-6], [-44], [-2], [-30], [-28], [-50], [-14], [36], [-30], [24], [-26], [-18], [20], [-24], [32], [14], [-14], [8], [28], [16], [32], [-6], [34], [-40], [-44], [34], [18], [-24], [-42], [-24], [8], [58], [-6], [-58], [24], [54], [-54], [-34], [16], [-54], [-40], [6], [-48], [14], [-34], [-32], [-16], [26], [36], [-2], [-28], [-36], [-42], [-18], [-32], [0], [20], [-56], [24], [-14], [-2], [-10], [-50], [36], [-32], [-8], [16], [10], [30], [66], [48], [20], [46], [-60], [-18], [42], [26], [-68], [-4], [-42], [26], [-40], [40], [-30], [14], [-56], [36], [14], [-28], [-10], [-14], [52], [-4], [-58], [-58], [4], [16], [-36], [-10], [60], [-50], [22], [54], [12], [-12], [-68], [-10], [26], [-18], [6], [-42], [52], [-28], [-56], [62], [-30], [-32], [-34], [34], [-46], [-38], [-76], [18], [-22], [-40], [12], [54], [-16], [-2], [-76], [40], [-38], [76], [42], [-26], [56], [42], [-44], [-12], [-46], [60], [-30], [-42], [-42], [80], [-72], [-12], [70], [22], [42], [-26], [50], [-64], [-20], [70], [68], [-66], [-8], [8], [-22], [72], [-46], [-20], [-42], [50], [-38], [0], [52], [72], [-28], [-44], [-74], [-54], [-66], [-4], [18], [0], [-56], [40], [-38], [54], [48], [14], [78], [-44], [82], [-48], [28], [22], [90], [-52], [-60], [88], [-6], [-18], [66], [92], [-60], [-64], [-76], [38], [6], [30], [-34], [-24], [-54], [-28], [48], [10], [-72], [10], [-80], [12], [22], [52], [56], [58], [32], [82], [22], [-66], [-14], [-26], [64], [42], [-18], [80], [24], [-38], [-24], [22], [-48], [78], [-58], [-100], [64], [-20], [32], [-22], [60], [22], [38], [-30], [28], [-84], [86], [-14], [90], [-12], [48], [36], [-38], [-46], [48], [22], [-14], [90], [26], [4], [-72], [0], [8], [52], [-30], [70], [78], [100], [-78], [-96], [56], [-86], [68], [64], [42], [100], [2], [76], [64], [42], [-2], [-4], [60], [82], [72], [52], [0], [44], [92], [64], [-42], [36], [-14], [8], [54]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2850_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2850_2_a_y();".
// 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_2850_2_a_y(:prec:=1)
    chi := MakeCharacter_2850_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_2850_2_a_y();".
// 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_2850_2_a_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2850_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<11,R![6, 1]>,<13,R![0, 1]>,<23,R![4, 1]>],Snew);
    return Vf;
end function;