
// Make newform 2730.2.a.j in Magma, downloaded from the LMFDB on 18 October 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2730_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_2730_2_a_j();".
// 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_2730_2_a_j();".
// 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_2730_a();"
function MakeCharacter_2730_a()
    N := 2730;
    order := 1;
    char_gens := [911, 547, 2341, 1471];
    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_2730_a_Hecke( : Kf := Kf)
    return MakeCharacter_2730_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 := [[-1], [-1], [1], [1], [2], [1], [6], [2], [4], [-4], [0], [-2], [-6], [0], [8], [10], [-8], [4], [2], [8], [-14], [-8], [14], [-6], [6], [2], [-16], [4], [-18], [-2], [-2], [-12], [-2], [16], [18], [0], [10], [14], [20], [8], [12], [8], [10], [-12], [-10], [2], [4], [16], [22], [-6], [-6], [0], [0], [4], [30], [-12], [-10], [0], [-8], [0], [20], [6], [4], [8], [6], [6], [-28], [18], [12], [30], [-16], [-12], [-8], [-32], [-20], [-20], [0], [-22], [36], [-12], [36], [6], [-8], [-2], [-10], [4], [8], [0], [-30], [-24], [4], [24], [24], [0], [-8], [-2], [-30], [-2], [28], [2], [-8], [-10], [-44], [-14], [12], [-2], [26], [-48], [-38], [-2], [20], [-38], [26], [-42], [-8], [-34], [24], [38], [-18], [20], [-14], [-18], [8], [-36], [-2], [0], [26], [8], [24], [-14], [0], [-8], [32], [24], [-34], [0], [6], [8], [28], [-30], [-6], [-18], [-14], [-44], [16], [-8], [54], [2], [56], [-48], [-38], [-6], [-4], [58], [-56], [22], [-48], [-42], [10], [-30], [48], [30], [-44], [-4], [-46], [-32], [44], [6], [-30], [-48], [2], [10], [-6], [56], [-26], [-26], [24], [2], [-24], [60], [28], [52], [-40], [-56], [24], [-18], [-16], [-16], [34], [-48], [-6], [-60], [20], [50], [60], [6], [-22], [4], [-12], [-50], [46], [46], [-22], [62], [-56], [-40], [-16], [60], [-8], [48], [66], [-6], [6], [-26], [32], [8], [-70], [62], [-24], [-30], [-22], [-16], [4], [-24], [16], [58], [50], [-42], [-52], [22], [20], [-64], [-8], [18], [68], [8], [68], [-6], [0], [-48], [-32], [4], [-32], [-46], [16], [56], [32], [36], [4], [12], [-18], [-78], [-24], [26], [48], [74], [-26], [-36], [-12], [38], [14], [12], [-30], [50], [28], [-32], [46], [-38], [34], [-26], [-32], [2], [-26], [-36], [-38], [24], [-48], [22], [-2], [44], [34], [24], [-58], [10], [-16], [30], [-34], [-54], [-36], [-54], [24], [68], [-22], [40], [28], [-54], [-6], [84], [-4], [-68], [-86], [14], [-36], [-44], [-52], [64], [52], [-32], [20], [28], [6], [-30], [-30], [32], [-80], [-34], [24], [12], [62], [-86], [64], [-68], [14], [4], [6], [46], [16], [88], [-48], [26], [-54], [62], [-70], [54], [20], [-30], [50], [-68], [64], [-20], [6], [-22], [6], [-28], [2], [-24], [72], [-46], [40], [24], [-14], [-12], [-58], [-48], [-78], [-26], [-24], [-52], [38], [-48], [-34], [-86], [-8], [42], [-20], [30], [66], [-4], [-10], [-48], [60], [-72], [-48], [-62], [-52], [-60], [42], [-58], [78], [86], [-94], [-6], [-64], [-66], [-62], [72], [-16], [30], [-2], [10], [10], [-78], [18], [-52], [72], [86], [-60], [54], [-28], [-66], [-26], [2], [-96], [0], [102], [34], [-70], [-48], [-64], [-32], [-18], [50], [-84], [-52], [-42], [-42], [24], [24], [-76], [-78], [36], [-102], [16], [-44], [-10], [102], [84], [0], [-42], [-24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2730_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2730_2_a_j();".
// 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_2730_2_a_j(:prec:=1)
    chi := MakeCharacter_2730_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_2730_2_a_j();".
// 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_2730_2_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2730_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-2, 1]>,<17,R![-6, 1]>,<19,R![-2, 1]>,<23,R![-4, 1]>,<29,R![4, 1]>],Snew);
    return Vf;
end function;