
// Make newform 4200.2.a.z in Magma, downloaded from the LMFDB on 20 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_z();".
// 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_z();".
// 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], [4], [2], [6], [4], [-8], [-2], [0], [2], [10], [-4], [0], [-14], [12], [-2], [4], [0], [-2], [-8], [4], [-6], [6], [6], [8], [-20], [14], [14], [16], [4], [22], [-20], [6], [16], [18], [4], [8], [2], [12], [-10], [8], [-18], [18], [-8], [-4], [16], [-28], [-26], [6], [-24], [2], [12], [6], [8], [-18], [-32], [2], [26], [-20], [26], [-28], [-8], [-2], [10], [20], [-34], [12], [30], [6], [0], [-16], [-14], [-12], [-16], [22], [34], [18], [-6], [-12], [-10], [-24], [-26], [8], [-20], [-30], [6], [14], [16], [4], [0], [24], [-28], [-36], [-8], [30], [-38], [-36], [-2], [-28], [10], [36], [-6], [-12], [6], [-36], [6], [16], [10], [-32], [2], [38], [12], [-16], [2], [20], [-24], [-6], [-20], [-42], [-34], [42], [44], [-12], [-2], [-10], [16], [8], [34], [44], [24], [-40], [-46], [42], [-14], [26], [-28], [-14], [-22], [-4], [-42], [-56], [-52], [-34], [-8], [26], [46], [-4], [-16], [10], [-14], [4], [-24], [12], [-56], [16], [50], [14], [-18], [-12], [-42], [40], [60], [-18], [-40], [24], [-38], [18], [-6], [4], [-34], [0], [-26], [32], [-38], [-12], [54], [-56], [14], [32], [4], [34], [-50], [0], [-10], [42], [4], [-22], [0], [-10], [28], [44], [62], [-12], [14], [-46], [-38], [-58], [-40], [46], [0], [-22], [-14], [-20], [-62], [0], [36], [10], [-4], [-18], [22], [-8], [12], [-24], [-38], [48], [66], [-56], [-54], [22], [-24], [-62], [32], [-44], [-74], [-18], [72], [-40], [-12], [-54], [-44], [-8], [-54], [28], [48], [50], [50], [52], [-56], [-12], [-20], [40], [30], [-26], [24], [-32], [-44], [20], [0], [58], [50], [8], [74], [-14], [-36], [54], [-52], [26], [-34], [-16], [68], [-26], [2], [-58], [20], [-50], [10], [-20], [-30], [-18], [52], [62], [-56], [-58], [-56], [60], [-18], [74], [36], [0], [32], [72], [-58], [12], [24], [62], [-30], [-8], [-78], [-50], [52], [-10], [44], [42], [30], [32], [-38], [-4], [36], [62], [-6], [24], [68], [-28], [-34], [-4], [-50], [-80], [2], [-64], [-42], [-30], [36], [-56], [-38], [36], [-40], [22], [-78], [36], [6], [-34], [64], [-26], [-62], [44], [44], [32], [82], [94], [42], [64], [52], [68], [-36], [-2], [-42], [58], [64], [50], [6], [6], [-48], [-6], [28], [38], [60], [96], [10], [-28], [-58], [-18], [-16], [38], [78], [-80], [76], [30], [-40], [-30], [26], [-32], [68], [-92], [38], [98], [-72], [-70], [44], [44], [0], [6], [40], [-38], [-12], [24], [38], [-30], [94], [30], [-58], [-72], [-34], [-28], [8], [24], [-54], [-20], [0], [-62], [90], [-60], [-28], [-48], [-26], [-80], [42], [20], [54], [62], [46], [16], [62], [-90], [-24], [42], [82], [52], [-28], [-26], [-30], [28], [12], [70], [-50], [8], [40], [54], [56], [78], [58], [0], [-84], [14], [-78], [-60], [-6], [-20], [40]];
    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_z();".
// 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_z(: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_z();".
// 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_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4200_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-4, 1]>,<13,R![-2, 1]>,<17,R![-6, 1]>,<19,R![-4, 1]>,<23,R![8, 1]>],Snew);
    return Vf;
end function;