
// Make newform 3850.2.a.k in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3850_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_3850_2_a_k();".
// 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_3850_2_a_k();".
// 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_3850_a();"
function MakeCharacter_3850_a()
    N := 3850;
    order := 1;
    char_gens := [2927, 2201, 1751];
    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_3850_a_Hecke(Kf)
    return MakeCharacter_3850_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], [2], [0], [-1], [-1], [4], [0], [-4], [0], [-6], [-10], [-2], [-12], [4], [-6], [6], [-6], [-4], [4], [12], [4], [8], [-12], [18], [10], [0], [-14], [-12], [2], [6], [-8], [0], [-6], [-4], [-18], [-16], [22], [4], [-24], [-12], [12], [-10], [0], [-14], [6], [2], [-4], [10], [24], [-10], [6], [-24], [-4], [-18], [-18], [0], [6], [20], [22], [-18], [16], [-24], [-32], [18], [-2], [-6], [20], [22], [-12], [20], [6], [-24], [34], [22], [-16], [6], [30], [-2], [-6], [32], [-6], [26], [24], [-14], [-28], [24], [6], [10], [-36], [40], [-42], [12], [-32], [-12], [-40], [12], [6], [-42], [16], [-10], [28], [6], [12], [6], [-4], [-38], [-18], [12], [36], [-4], [40], [10], [-6], [-10], [8], [30], [-2], [6], [18], [12], [26], [-2], [-12], [24], [2], [-30], [38], [30], [-2], [40], [-4], [-24], [-28], [-38], [24], [-16], [18], [-44], [-42], [-30], [-16], [30], [4], [12], [26], [30], [4], [12], [-10], [-12], [34], [6], [16], [-12], [28], [36], [-16], [-42], [52], [-36], [-48], [18], [-32], [30], [-18], [42], [20], [52], [2], [18], [-12], [50], [-24], [22], [-22], [-18], [44], [-54], [-44], [56], [28], [-60], [-38], [-24], [-36], [-66], [-2], [46], [-58], [0], [34], [60], [8], [42], [-12], [54], [-58], [22], [-12], [-36], [-18], [8], [10], [32], [-66], [-66], [-46], [-36], [-6], [-46], [-2], [54], [-44], [-6], [-36], [26], [40], [0], [24], [66], [50], [44], [-18], [-20], [48], [50], [42], [-72], [-20], [12], [-2], [-16], [8], [-54], [-26], [-12], [-34], [-66], [-48], [-42], [60], [-4], [-26], [2], [0], [-72], [-2], [42], [-4], [0], [-26], [0], [-24], [26], [-24], [-60], [74], [28], [30], [-56], [40], [36], [-4], [-20], [-66], [-34], [42], [-42], [-68], [-6], [2], [-68], [-38], [32], [4], [34], [-48], [14], [2], [60], [-48], [44], [-60], [-64], [-44], [-12], [34], [78], [-46], [12], [18], [6], [54], [-84], [-74], [60], [-82], [-30], [-60], [-44], [-44], [66], [56], [-60], [-40], [-38], [-24], [-46], [48], [-26], [24], [30], [6], [-8], [24], [56], [-66], [48], [-14], [-66], [56], [70], [36], [-20], [-30], [-70], [80], [22], [-24], [42], [62], [-18], [-40], [60], [-28], [-18], [2], [-24], [-22], [4], [-62], [-78], [0], [80], [-30], [84], [38], [28], [72], [-30], [44], [58], [78], [-44], [-58], [72], [-6], [-72], [90], [-84], [94], [36], [-12], [-36], [-2], [-38], [36], [-56], [2], [48], [2], [0], [-18], [-40], [-2], [30], [48], [-32], [12], [16], [-6], [90], [-8], [42], [-28], [66], [-16], [22], [-92], [42], [62], [-30], [72], [-26], [48], [46], [-52], [42], [-40], [42], [-4], [102], [-32], [-18], [-24], [56], [-86], [42], [-86], [42], [100], [-30], [-24], [-52], [-50], [-78], [-24], [70], [-66], [48], [30], [88], [-72], [-96], [70], [-6], [-60], [-18], [-70], [108]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3850_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3850_2_a_k();".
// 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_3850_2_a_k(:prec:=1)
    chi := MakeCharacter_3850_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_3850_2_a_k();".
// 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_3850_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3850_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 1]>,<13,R![-4, 1]>,<17,R![0, 1]>,<19,R![4, 1]>],Snew);
    return Vf;
end function;