
// Make newform 3800.2.a.b in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3800_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_3800_2_a_b();".
// 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_3800_2_a_b();".
// 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_3800_a();"
function MakeCharacter_3800_a()
    N := 3800;
    order := 1;
    char_gens := [951, 1901, 1977, 401];
    v := [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_3800_a_Hecke(Kf)
    return MakeCharacter_3800_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], [-2], [0], [-4], [-4], [0], [-6], [-1], [-8], [-6], [-8], [8], [-2], [0], [-12], [-4], [8], [-14], [2], [-8], [2], [4], [-12], [6], [0], [-6], [6], [-18], [-2], [16], [18], [-20], [10], [4], [-6], [12], [6], [-24], [6], [20], [0], [-10], [-24], [-16], [-6], [8], [16], [-2], [6], [-6], [-10], [24], [-2], [12], [-12], [24], [6], [-16], [-26], [30], [4], [-4], [22], [-8], [14], [-32], [-16], [-8], [4], [-10], [-6], [8], [-36], [12], [4], [-22], [6], [14], [10], [-14], [20], [2], [4], [-16], [12], [-36], [-6], [22], [-18], [-16], [4], [24], [-26], [12], [-12], [-24], [26], [-30], [-22], [10], [-26], [-2], [18], [-22], [4], [-18], [-28], [6], [-44], [30], [-18], [-34], [-6], [-28], [-16], [-2], [-28], [-24], [-26], [-16], [-46], [-44], [-20], [34], [12], [2], [-50], [-40], [20], [-6], [28], [-30], [20], [34], [-42], [14], [-48], [-34], [-44], [-6], [44], [-2], [52], [-6], [-42], [-36], [-22], [28], [-4], [34], [-52], [14], [8], [18], [46], [-44], [16], [-30], [-22], [-10], [-52], [-36], [40], [-28], [28], [34], [-20], [22], [6], [6], [12], [6], [-32], [22], [16], [38], [4], [-6], [-6], [-22], [40], [12], [56], [-28], [-32], [-50], [-28], [42], [50], [24], [-28], [-20], [20], [-22], [-48], [-8], [58], [66], [6], [36], [18], [44], [16], [-10], [20], [-42], [16], [54], [50], [12], [54], [26], [40], [10], [32], [-66], [-28], [-46], [10], [-54], [-38], [40], [38], [40], [38], [-10], [52], [-8], [50], [-12], [2], [-4], [28], [30], [-4], [-28], [-66], [-46], [12], [-20], [42], [-28], [20], [34], [72], [-72], [-12], [56], [-76], [-4], [58], [2], [16], [-46], [-38], [-36], [-58], [-22], [-28], [-38], [-34], [42], [42], [-44], [-66], [-64], [-30], [30], [-34], [58], [-58], [34], [-6], [-80], [-24], [4], [64], [26], [78], [-44], [-70], [-24], [-48], [-50], [20], [-32], [-26], [-24], [56], [26], [-42], [-56], [72], [16], [-44], [58], [48], [-30], [84], [64], [-46], [52], [24], [54], [60], [16], [34], [10], [-40], [66], [36], [26], [18], [26], [40], [38], [-44], [84], [-82], [-30], [52], [2], [-78], [10], [18], [6], [48], [-78], [-6], [6], [18], [36], [-56], [32], [20], [40], [-82], [-84], [-54], [-32], [44], [66], [78], [-48], [-12], [8], [22], [-22], [-64], [-66], [-28], [60], [-46], [-22], [2], [28], [40], [28], [-46], [26], [-10], [6], [-74], [-68], [-28], [48], [-2], [66], [78], [12], [64], [68], [-2], [8], [-34], [-12], [48], [-34], [50], [64], [-42], [-18], [-92], [-18], [-72], [62], [-72], [-2], [16], [-18], [34], [12], [4], [-36], [44], [12], [-84], [30], [4], [-10], [-50], [-26], [38], [-36], [54], [56], [38], [-30], [-94], [-4], [76], [66], [-66], [-60], [-78], [14], [-100], [-30], [66], [34], [54], [36], [-60], [12], [56], [-28], [18], [14], [20], [32]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3800_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3800_2_a_b();".
// 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_3800_2_a_b(:prec:=1)
    chi := MakeCharacter_3800_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_3800_2_a_b();".
// 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_3800_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<7,R![4, 1]>],Snew);
    return Vf;
end function;