
// Make newform 2646.2.a.bb in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2646_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_2646_2_a_bb();".
// 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_2646_2_a_bb();".
// 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_2646_a();"
function MakeCharacter_2646_a()
    N := 2646;
    order := 1;
    char_gens := [785, 1081];
    v := [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_2646_a_Hecke(Kf)
    return MakeCharacter_2646_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 := [[1], [0], [2], [0], [5], [-6], [4], [4], [-4], [7], [-3], [8], [6], [8], [-6], [-6], [-7], [0], [10], [-4], [-13], [-3], [7], [-6], [5], [5], [-8], [12], [16], [4], [0], [-13], [-8], [8], [9], [-17], [-14], [2], [14], [-1], [-15], [0], [-18], [-19], [-25], [19], [26], [1], [27], [-4], [-22], [0], [1], [21], [0], [12], [-31], [0], [2], [2], [-16], [-27], [2], [-22], [-10], [-33], [32], [-27], [-27], [20], [30], [16], [4], [-20], [10], [4], [1], [18], [-18], [10], [-12], [-18], [-18], [7], [-3], [11], [-14], [-26], [23], [-29], [-7], [-26], [-13], [12], [-8], [-6], [-3], [0], [-26], [-30], [-18], [-11], [-20], [-12], [0], [-43], [-20], [-36], [30], [10], [9], [-6], [2], [-4], [5], [14], [-14], [18], [18], [41], [38], [2], [-9], [21], [-8], [-14], [8], [-6], [32], [36], [-6], [-6], [12], [-18], [-8], [-1], [38], [-18], [-33], [34], [50], [-49], [13], [51], [-22], [4], [46], [-30], [16], [-38], [34], [54], [2], [42], [-42], [36], [13], [-36], [-14], [-13], [25], [2], [8], [-12], [-18], [-36], [8], [58], [38], [3], [7], [-4], [24], [21], [-24], [-46], [54], [-19], [37], [34], [27], [-32], [34], [20], [-20], [27], [-54], [10], [39], [-24], [35], [-57], [-32], [55], [51], [46], [-1], [4], [-66], [54], [18], [27], [64], [3], [3], [57], [-56], [-12], [-6], [4], [17], [-13], [48], [53], [-12], [67], [-53], [-44], [2], [-34], [-50], [56], [-24], [-11], [60], [32], [-26], [-38], [27], [-24], [24], [14], [1], [66], [-38], [-12], [-55], [-27], [-3], [26], [-16], [58], [19], [60], [-32], [-24], [-39], [-23], [-16], [18], [58], [-18], [30], [59], [37], [53], [32], [6], [5], [-26], [43], [16], [-4], [-16], [36], [-10], [57], [-58], [64], [-51], [30], [2], [6], [-7], [11], [63], [68], [0], [31], [-12], [-30], [15], [-36], [16], [-72], [-22], [-78], [-58], [73], [0], [-18], [0], [-42], [-73], [8], [-21], [19], [81], [45], [-72], [-10], [-63], [61], [-15], [56], [-25], [-17], [-80], [8], [56], [-12], [-41], [64], [44], [22], [46], [-45], [-74], [7], [30], [-68], [49], [-18], [19], [4], [13], [-58], [-58], [-64], [-55], [-40], [-19], [51], [-44], [28], [41], [-84], [-58], [10], [-1], [80], [54], [30], [15], [15], [21], [-68], [-4], [48], [33], [-50], [-58], [42], [-96], [-38], [40], [-34], [-24], [38], [50], [-80], [34], [-32], [-31], [92], [-13], [-90], [-39], [-62], [20], [14], [-6], [7], [47], [-38], [12], [-22], [14], [-8], [-62], [-57], [16], [-56], [54], [68], [92], [44], [-40], [22], [-68], [101], [54], [47], [-26], [-8], [27], [53], [100], [90], [-82], [4], [-12], [-68], [62], [-87], [-104], [-2], [-96], [4], [-32], [53], [53], [32], [-102], [-55], [-74], [74], [8], [96], [-4], [29], [34], [-52], [51], [-17], [-43], [-56], [78], [-18], [-14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2646_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2646_2_a_bb();".
// 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_2646_2_a_bb(:prec:=1)
    chi := MakeCharacter_2646_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_2646_2_a_bb();".
// 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_2646_2_a_bb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2646_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<11,R![-5, 1]>,<13,R![6, 1]>,<17,R![-4, 1]>],Snew);
    return Vf;
end function;