
// Make newform 2850.2.a.q in Magma, downloaded from the LMFDB on 17 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_q();".
// 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_q();".
// 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], [0], [-2], [0], [1], [0], [-6], [2], [-2], [0], [-8], [0], [-6], [-6], [2], [4], [0], [-14], [2], [-6], [-12], [10], [-6], [-8], [0], [-16], [6], [16], [0], [12], [-4], [-6], [-10], [4], [4], [0], [18], [6], [-16], [0], [-2], [-18], [-4], [8], [4], [-12], [-10], [-24], [24], [-22], [12], [-6], [24], [18], [20], [28], [24], [-32], [6], [-20], [0], [10], [30], [20], [-2], [-18], [-10], [-12], [24], [10], [-14], [8], [24], [-30], [-8], [12], [-22], [12], [8], [-24], [-2], [-22], [30], [24], [22], [30], [-26], [-6], [0], [16], [0], [20], [12], [-6], [12], [4], [-10], [4], [-6], [24], [24], [20], [-2], [-18], [-36], [12], [26], [-32], [16], [-12], [20], [32], [-24], [-32], [-12], [18], [6], [32], [-14], [30], [12], [-28], [30], [-10], [-48], [34], [-32], [44], [48], [-22], [-32], [6], [-34], [18], [-44], [-42], [54], [-16], [42], [22], [-48], [8], [-24], [-44], [18], [-4], [0], [-38], [-30], [16], [48], [-44], [60], [-40], [-6], [-26], [6], [18], [-30], [-26], [-42], [42], [-24], [-58], [-44], [2], [18], [6], [-16], [-48], [-14], [62], [6], [-4], [6], [28], [-10], [46], [-30], [-26], [-30], [24], [-30], [-38], [-20], [50], [-24], [34], [-54], [8], [6], [-54], [6], [-58], [4], [-12], [48], [30], [-10], [-26], [14], [0], [30], [20], [-24], [-54], [-64], [58], [-42], [10], [12], [-12], [-34], [22], [12], [0], [-30], [-4], [26], [0], [-26], [12], [-34], [-30], [24], [-8], [24], [28], [44], [-22], [60], [28], [0], [-34], [-54], [-12], [72], [-36], [-52], [10], [20], [-42], [-24], [-38], [-18], [56], [-48], [-32], [-54], [-12], [62], [6], [60], [26], [76], [-42], [10], [-44], [-72], [-46], [-14], [-24], [68], [30], [18], [-44], [-54], [8], [52], [10], [-4], [46], [10], [-66], [20], [50], [72], [48], [-40], [-12], [-64], [4], [-24], [22], [-6], [32], [36], [-18], [-30], [-18], [-42], [-14], [30], [26], [30], [-66], [40], [58], [18], [56], [-24], [-4], [-38], [48], [-28], [-24], [-8], [36], [-66], [36], [4], [-72], [-34], [-84], [-36], [58], [-18], [-40], [10], [-6], [-8], [-48], [74], [80], [76], [72], [66], [26], [-30], [-64], [18], [-52], [-54], [44], [66], [50], [70], [10], [48], [-30], [62], [78], [-78], [-70], [-44], [48], [-42], [20], [-26], [18], [-80], [-40], [66], [-72], [-72], [60], [24], [28], [-90], [-72], [30], [-56], [-62], [-18], [64], [62], [48], [-4], [-36], [-54], [44], [-20], [18], [48], [70], [18], [-14], [30], [-72], [-38], [72], [-16], [-24], [68], [76], [-20], [72], [86], [78], [84], [-92], [60], [22], [14], [60], [-40], [-54], [56], [48], [4], [-78], [-78], [68], [28], [72], [4], [-48], [-14], [66], [36], [44], [-86], [18], [84], [28], [-12], [24], [6], [-38], [24], [30], [10], [78], [96], [-30], [-4], [-24]];
    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_q();".
// 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_q(: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_q();".
// 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_q( : 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![0, 1]>,<13,R![2, 1]>,<23,R![0, 1]>],Snew);
    return Vf;
end function;