
// Make newform 4200.2.a.k in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4200_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_4200_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_4200_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_4200_a();"
function MakeCharacter_4200_a()
    N := 4200;
    order := 1;
    char_gens := [3151, 2101, 2801, 1177, 3601];
    v := [1, 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_4200_a_Hecke(Kf)
    return MakeCharacter_4200_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 := [[0], [-1], [0], [1], [2], [-2], [0], [2], [-8], [2], [-6], [-8], [-10], [0], [12], [2], [0], [2], [-4], [14], [2], [4], [-16], [-6], [-2], [6], [-16], [-4], [-6], [-14], [12], [12], [-10], [-14], [6], [16], [-22], [8], [0], [-24], [2], [-2], [18], [-16], [6], [10], [-8], [0], [-8], [-30], [18], [-10], [-10], [-8], [12], [0], [-18], [14], [-24], [6], [-12], [-8], [-20], [-24], [14], [18], [-32], [32], [-20], [-22], [24], [34], [16], [-20], [-4], [-24], [30], [30], [-14], [-22], [12], [-38], [22], [-18], [-6], [12], [-30], [12], [-30], [4], [0], [8], [20], [26], [-4], [-16], [34], [10], [4], [46], [-4], [-6], [12], [18], [28], [-10], [-28], [8], [-14], [-42], [16], [16], [-38], [-14], [-4], [30], [12], [8], [30], [-42], [2], [20], [24], [4], [30], [-38], [-38], [52], [-48], [30], [-36], [24], [20], [-44], [22], [-26], [40], [52], [24], [6], [18], [18], [-20], [36], [-26], [36], [22], [4], [-26], [-24], [32], [10], [44], [-36], [-20], [42], [-36], [-34], [2], [26], [36], [42], [56], [-48], [26], [4], [4], [-42], [50], [48], [26], [-42], [-18], [-32], [-10], [22], [-48], [62], [56], [30], [-32], [-24], [-36], [-24], [24], [10], [8], [-20], [46], [-60], [62], [-60], [28], [-18], [60], [-16], [34], [-20], [20], [-56], [-30], [-62], [-46], [22], [-56], [32], [70], [-52], [-34], [-2], [52], [-54], [64], [-56], [-4], [-2], [-4], [38], [40], [-10], [26], [-14], [30], [24], [4], [26], [-12], [26], [-40], [18], [-20], [-30], [-24], [-34], [44], [12], [-70], [54], [-22], [-60], [-4], [-22], [32], [-10], [12], [52], [-8], [-12], [0], [64], [-48], [54], [0], [10], [-80], [-30], [50], [-28], [40], [6], [-68], [20], [34], [26], [-24], [-42], [-70], [-22], [36], [30], [-26], [56], [46], [76], [30], [16], [-28], [-30], [6], [8], [-4], [76], [-44], [70], [76], [-30], [-76], [-30], [10], [-50], [58], [-8], [-22], [12], [-36], [-30], [14], [36], [-40], [-20], [74], [-10], [56], [-12], [-28], [-48], [-36], [34], [6], [-44], [-36], [18], [82], [-28], [-8], [30], [36], [-46], [-42], [26], [-50], [4], [-34], [-48], [-26], [14], [-92], [-36], [-8], [-70], [-2], [46], [70], [76], [-24], [-20], [-50], [-76], [78], [0], [92], [-46], [-26], [-8], [78], [54], [-38], [-12], [-12], [-8], [58], [-52], [-54], [-72], [14], [16], [-44], [-52], [26], [-64], [76], [90], [8], [6], [20], [-20], [-12], [-16], [-10], [-30], [46], [48], [86], [94], [-32], [-4], [-54], [-26], [-14], [58], [-18], [-46], [92], [-58], [50], [-24], [-60], [-82], [-28], [60], [-42], [-60], [-2], [44], [62], [-64], [22], [-30], [-84], [-14], [6], [50], [24], [-68], [-30], [38], [-32], [-22], [52], [-56], [26], [90], [-36], [56], [-88], [-50], [-90], [-16], [8], [-4], [-34], [70], [48], [-24], [102], [-24], [52], [-46], [38], [20]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4200_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4200_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_4200_2_a_k(:prec:=1)
    chi := MakeCharacter_4200_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_4200_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_4200_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4200_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-2, 1]>,<13,R![2, 1]>,<17,R![0, 1]>,<19,R![-2, 1]>,<23,R![8, 1]>],Snew);
    return Vf;
end function;