
// Make newform 2850.2.a.r in Magma, downloaded from the LMFDB on 20 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_r();".
// 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_r();".
// 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], [3], [-2], [-6], [1], [-3], [3], [-7], [-2], [-6], [10], [-12], [9], [6], [-1], [-5], [0], [1], [-13], [-15], [-15], [-8], [6], [7], [0], [2], [-9], [-11], [-15], [-12], [-4], [12], [20], [-14], [22], [18], [3], [0], [-10], [3], [-20], [-6], [2], [-13], [-5], [18], [-13], [-6], [0], [-10], [24], [9], [21], [30], [20], [-11], [3], [-8], [15], [19], [12], [13], [3], [5], [-32], [-12], [-19], [6], [0], [-8], [22], [20], [-12], [-6], [-17], [9], [-34], [12], [-10], [18], [16], [35], [-27], [-33], [-2], [-12], [16], [39], [27], [40], [-12], [14], [-36], [33], [-15], [-20], [-43], [-11], [18], [12], [-6], [-4], [-23], [45], [36], [30], [8], [-23], [34], [18], [-46], [-40], [-18], [-14], [15], [0], [6], [-4], [34], [33], [-12], [-28], [-24], [35], [15], [34], [-5], [-40], [-24], [-4], [-29], [12], [11], [18], [49], [30], [-30], [41], [36], [-26], [18], [-4], [-6], [46], [-18], [-22], [12], [22], [-24], [-56], [-12], [-8], [-30], [-40], [0], [-2], [-51], [-36], [33], [-38], [-6], [54], [-36], [-25], [19], [-22], [36], [12], [-22], [24], [-53], [-31], [-30], [32], [-24], [-44], [-1], [4], [0], [-8], [-27], [-45], [54], [16], [-11], [-40], [9], [-56], [51], [65], [-9], [-36], [-42], [41], [-29], [-18], [-33], [-45], [-28], [-38], [38], [-63], [-60], [56], [42], [54], [44], [13], [-24], [-62], [-18], [-12], [14], [-14], [33], [42], [30], [50], [17], [-66], [40], [24], [-46], [21], [-72], [-32], [-27], [-62], [-49], [5], [45], [28], [-72], [-10], [24], [0], [0], [-18], [20], [58], [20], [-30], [33], [16], [66], [29], [-45], [-41], [-60], [-3], [-28], [0], [-39], [-10], [-47], [30], [58], [37], [-6], [-7], [4], [-78], [-43], [66], [18], [-47], [-66], [44], [79], [73], [-22], [-32], [-62], [33], [-82], [-52], [27], [-6], [14], [-24], [44], [-68], [12], [79], [-45], [-16], [-69], [-48], [69], [42], [-72], [-44], [-60], [80], [42], [42], [34], [-77], [33], [-16], [48], [-58], [16], [18], [68], [69], [49], [-21], [-60], [45], [13], [-36], [-70], [27], [66], [-14], [24], [20], [22], [-78], [64], [-18], [-52], [20], [28], [-78], [-84], [53], [-27], [68], [-3], [-10], [-9], [-46], [39], [-67], [-20], [82], [-36], [-45], [23], [69], [30], [17], [37], [-12], [66], [77], [58], [-72], [31], [32], [51], [-57], [45], [-24], [-6], [-53], [-18], [-12], [-90], [-20], [-14], [54], [76], [-28], [33], [23], [-72], [51], [-88], [-2], [-24], [-9], [-59], [0], [-86], [90], [-6], [-56], [66], [-1], [30], [44], [22], [52], [12], [-22], [57], [-45], [70], [48], [64], [-76], [15], [23], [30], [26], [-6], [-80], [-87], [21], [98], [37], [99], [31], [69], [-68], [-18], [-6], [8], [-11], [90], [72], [91], [48], [-48], [30], [-2], [-63], [90], [94], [78], [24], [-72], [-58], [9]];
    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_r();".
// 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_r(: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_r();".
// 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_r( : 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![-3, 1]>,<13,R![2, 1]>,<23,R![3, 1]>],Snew);
    return Vf;
end function;