
// Make newform 3840.2.a.k in Magma, downloaded from the LMFDB on 27 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_3840_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_3840_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_3840_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_3840_a();"
function MakeCharacter_3840_a()
    N := 3840;
    order := 1;
    char_gens := [511, 2821, 2561, 1537];
    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_3840_a_Hecke( : Kf := Kf)
    return MakeCharacter_3840_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]];
    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], [1], [2], [-2], [-2], [-8], [8], [-8], [10], [8], [2], [-6], [4], [4], [-2], [6], [0], [12], [8], [6], [8], [8], [6], [10], [-14], [-14], [-8], [-8], [4], [-14], [18], [12], [-4], [6], [12], [-10], [4], [0], [18], [14], [16], [-8], [6], [-6], [4], [20], [26], [20], [-20], [8], [-24], [22], [-6], [-4], [8], [-6], [-12], [2], [6], [28], [-6], [-12], [8], [22], [2], [-8], [-10], [12], [16], [-32], [24], [-30], [14], [4], [20], [14], [-14], [-10], [10], [10], [-4], [24], [14], [-36], [-12], [-34], [-22], [-18], [-34], [0], [16], [22], [-6], [-40], [-36], [-6], [-18], [20], [-16], [-12], [10], [12], [14], [-8], [18], [-8], [12], [0], [-10], [14], [-30], [16], [-28], [8], [42], [-36], [-36], [46], [-30], [32], [-34], [38], [16], [8], [-46], [-28], [-24], [-26], [-22], [16], [-20], [32], [18], [14], [-14], [26], [36], [-46], [-38], [28], [-54], [-34], [-20], [40], [-24], [26], [-32], [20], [16], [46], [-50], [4], [-48], [4], [24], [0], [14], [2], [22], [-12], [-12], [-30], [-46], [0], [4], [52], [-10], [6], [42], [-2], [12], [0], [46], [8], [-2], [-52], [-2], [46], [-28], [-34], [26], [-54], [-36], [-12], [14], [30], [4], [-10], [48], [-34], [-4], [-4], [-42], [0], [12], [-46], [-50], [-12], [52], [-66], [-64], [-2], [42], [6], [18], [16], [28], [46], [-40], [-58], [30], [42], [12], [-24], [-42], [26], [54], [-12], [-22], [-52], [-36], [34], [-10], [40], [-20], [36], [-16], [-26], [62], [38], [-20], [20], [-18], [52], [-8], [50], [-26], [-26], [0], [52], [48], [-42], [16], [56], [0], [-38], [-66], [-16], [-56], [-6], [-14], [48], [-42], [6], [-14], [72], [44], [-66], [-34], [-62], [-36], [-60], [-70], [60], [64], [30], [-30], [-20], [38], [28], [-28], [-26], [48], [-82], [-6], [12], [-40], [-38], [-6], [0], [-4], [36], [60], [36], [32], [-22], [-42], [60], [-54], [-26], [8], [8], [78], [-6], [-58], [16], [6], [78], [44], [46], [-18], [-68], [72], [80], [-22], [32], [4], [0], [46], [0], [-86], [-18], [-20], [-52], [82], [6], [-56], [46], [2], [-24], [-10], [-30], [-26], [-16], [38], [12], [-84], [44], [90], [-84], [-54], [64], [40], [36], [48], [20], [24], [-58], [2], [22], [48], [-70], [-52], [54], [-42], [-12], [-68], [32], [-26], [-4], [-38], [10], [42], [48], [-24], [56], [-18], [-4], [24], [38], [-22], [-36], [-42], [28], [-86], [-42], [2], [10], [-66], [-92], [20], [-30], [-28], [-14], [38], [-72], [-62], [62], [-78], [-2], [76], [82], [-24], [68], [0], [48], [-22], [-44], [0], [34], [34], [-42], [68], [-96], [-30], [4], [-14], [56], [-50], [92], [-12], [34], [60], [74], [8], [-62], [70], [60], [106], [-38], [-42], [-56], [24], [14], [-62], [24], [-62], [-60], [64], [-10], [-98], [68], [58], [86], [66], [-56], [-6], [16], [88]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3840_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3840_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_3840_2_a_k(:prec:=1)
    chi := MakeCharacter_3840_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_3840_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_3840_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3840_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-2, 1]>,<11,R![2, 1]>,<13,R![2, 1]>,<17,R![8, 1]>,<19,R![-8, 1]>],Snew);
    return Vf;
end function;