
// Make newform 3240.2.a.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3240_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_3240_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_3240_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_3240_a();"
function MakeCharacter_3240_a()
    N := 3240;
    order := 1;
    char_gens := [2431, 1621, 3161, 1297];
    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_3240_a_Hecke(Kf)
    return MakeCharacter_3240_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], [0], [-1], [0], [-5], [0], [3], [5], [6], [-10], [-2], [4], [-3], [3], [4], [-6], [-3], [2], [-11], [-14], [-15], [10], [-12], [14], [-13], [-12], [0], [-17], [-12], [-6], [10], [-4], [7], [7], [4], [22], [0], [16], [-22], [-2], [0], [-4], [12], [19], [-8], [-24], [12], [2], [13], [-16], [-11], [6], [-23], [11], [-31], [-12], [-30], [24], [-14], [10], [-4], [-10], [-7], [10], [7], [28], [20], [-31], [-3], [32], [25], [-12], [-8], [-26], [-35], [12], [-24], [8], [-7], [-7], [0], [38], [28], [27], [-14], [-29], [9], [-25], [4], [-4], [-11], [26], [34], [15], [-23], [34], [44], [-27], [-4], [24], [-9], [0], [5], [-15], [-3], [-31], [-17], [30], [16], [-1], [-32], [-2], [25], [-19], [2], [33], [-41], [30], [34], [44], [0], [-26], [28], [-13], [-8], [-6], [-46], [0], [16], [28], [5], [34], [20], [-26], [10], [-54], [-30], [28], [32], [-15], [35], [4], [12], [-12], [40], [14], [32], [-2], [-11], [-26], [22], [-2], [-47], [-6], [13], [6], [14], [30], [-42], [-16], [-21], [-13], [-6], [12], [23], [52], [28], [-42], [-2], [-30], [11], [28], [-18], [17], [-6], [-50], [1], [32], [-16], [-8], [-52], [37], [4], [57], [8], [18], [-38], [-40], [41], [-50], [1], [-51], [-39], [-16], [9], [5], [-25], [-58], [1], [-24], [-40], [44], [28], [15], [-9], [-32], [28], [-39], [-33], [-17], [7], [18], [26], [-52], [42], [-15], [-22], [25], [-54], [-54], [14], [-26], [17], [46], [-65], [-48], [42], [-6], [58], [12], [14], [56], [-58], [59], [-4], [-42], [26], [16], [61], [32], [-41], [-20], [26], [-10], [49], [-36], [-32], [3], [-20], [34], [42], [-45], [-30], [46], [-20], [-33], [50], [-52], [8], [43], [-2], [33], [22], [66], [54], [-17], [-6], [-33], [29], [24], [-42], [52], [61], [10], [-23], [-34], [12], [-40], [43], [-33], [-18], [14], [-48], [-10], [-53], [-58], [-66], [12], [-24], [-53], [-26], [-37], [-63], [21], [62], [-18], [-26], [42], [84], [-20], [7], [-6], [0], [-84], [52], [-2], [28], [74], [54], [14], [6], [-16], [26], [52], [-20], [3], [-45], [-68], [13], [-34], [-27], [3], [56], [50], [6], [-35], [-7], [-5], [52], [42], [2], [-36], [-22], [-3], [-83], [48], [38], [69], [63], [42], [14], [62], [0], [36], [-84], [-11], [-82], [-27], [0], [-22], [60], [-37], [50], [-88], [76], [34], [66], [-36], [-50], [20], [-92], [-57], [6], [-49], [29], [9], [38], [-92], [45], [80], [29], [100], [-8], [-56], [32], [45], [48], [-70], [57], [62], [50], [-13], [86], [-19], [-61], [80], [-64], [32], [-52], [-22], [-13], [20], [-29], [67], [-32], [-50], [52], [-87], [-69], [-36], [22], [-41], [10], [-42], [18], [-28], [-80], [33], [47], [28], [-47], [-28], [-17], [79], [-42], [76], [90], [-38], [23], [26], [-106], [-6], [-22], [-39], [10], [24], [53], [62], [-16], [-68]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3240_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3240_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_3240_2_a_b(:prec:=1)
    chi := MakeCharacter_3240_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_3240_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_3240_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3240_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![5, 1]>,<17,R![-3, 1]>],Snew);
    return Vf;
end function;