
// Make newform 3840.2.a.d in Magma, downloaded from the LMFDB on 13 March 2026.
// 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_d();".
// 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_d();".
// 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], [4], [0], [-6], [-4], [4], [6], [10], [4], [-10], [-4], [-4], [-10], [8], [8], [12], [4], [-10], [-14], [0], [-14], [-10], [14], [2], [4], [-4], [6], [-6], [0], [2], [-4], [-6], [2], [-20], [-20], [-24], [10], [0], [-20], [-8], [-14], [-10], [6], [-12], [10], [0], [-4], [6], [16], [-30], [12], [-2], [0], [-10], [-2], [-16], [10], [-4], [-2], [-28], [-8], [-6], [-18], [4], [30], [16], [-8], [-18], [-12], [-18], [20], [-36], [24], [10], [-20], [30], [6], [4], [20], [-36], [14], [-2], [16], [-18], [-18], [34], [-22], [8], [4], [30], [-40], [20], [12], [-18], [6], [-20], [44], [4], [42], [0], [-2], [-20], [-2], [12], [22], [-12], [14], [2], [-20], [6], [20], [18], [14], [12], [-12], [-30], [0], [32], [-2], [-38], [48], [-12], [22], [44], [-36], [14], [24], [-12], [48], [26], [-8], [-10], [-6], [6], [-28], [-30], [-10], [-4], [-10], [-42], [-4], [44], [-4], [-44], [-6], [-44], [24], [-44], [-6], [28], [-16], [-52], [-48], [26], [46], [-22], [26], [36], [26], [-46], [12], [-22], [32], [2], [-28], [46], [42], [-32], [8], [48], [-14], [-14], [58], [4], [-6], [-30], [-40], [62], [-4], [8], [14], [-40], [50], [0], [-60], [-22], [-16], [26], [-36], [44], [18], [36], [-18], [-10], [-16], [6], [-8], [42], [-58], [8], [-34], [44], [-42], [-34], [0], [-54], [-36], [-30], [18], [26], [-4], [-36], [22], [22], [-42], [24], [-30], [-28], [-34], [-66], [-6], [0], [-24], [-42], [36], [18], [0], [-40], [60], [-26], [-38], [4], [60], [-34], [-30], [-60], [-52], [-16], [36], [62], [0], [-66], [60], [70], [0], [68], [56], [48], [26], [4], [-6], [78], [-8], [44], [12], [22], [38], [-62], [-44], [-28], [-44], [26], [44], [-38], [-30], [60], [-46], [-64], [-68], [18], [-6], [-38], [-38], [-32], [-40], [-82], [-24], [-36], [30], [12], [48], [-12], [-24], [-30], [34], [-14], [-54], [50], [28], [-66], [56], [-28], [-30], [38], [6], [28], [28], [-22], [42], [74], [52], [44], [34], [48], [28], [-36], [68], [24], [-42], [6], [-4], [-24], [10], [72], [-84], [-34], [18], [68], [-86], [-34], [-10], [-6], [-50], [52], [76], [-32], [-6], [-60], [82], [-10], [-68], [-68], [72], [-8], [70], [-58], [42], [76], [-14], [-42], [62], [-42], [12], [-52], [76], [-36], [38], [4], [62], [-42], [-14], [0], [46], [80], [-80], [-6], [-24], [28], [58], [-72], [-84], [4], [22], [14], [-10], [54], [-92], [-4], [96], [22], [58], [-8], [-100], [-8], [2], [6], [-70], [-86], [10], [-50], [6], [28], [56], [74], [-32], [-52], [-12], [82], [-54], [0], [84], [48], [14], [-42], [-82], [68], [82], [-32], [-54], [-38], [-30], [30], [-30], [-32], [-34], [68], [52], [46], [-62], [96], [-60], [-26], [78], [-88], [70], [-30], [84], [-86], [28], [24], [-84], [-54], [38], [40], [-6], [-20], [-36]];
    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_d();".
// 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_d(: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_d();".
// 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_d( : 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![-4, 1]>,<13,R![0, 1]>,<17,R![6, 1]>,<19,R![4, 1]>],Snew);
    return Vf;
end function;