
// Make newform 2646.2.a.d 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_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_2646_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_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], [2], [1], [3], [4], [-3], [-7], [11], [-6], [-6], [1], [-8], [-1], [7], [14], [-11], [11], [8], [4], [0], [-1], [-16], [16], [13], [16], [16], [10], [14], [-15], [8], [-6], [5], [4], [7], [9], [14], [8], [-6], [-7], [-24], [-19], [18], [5], [5], [8], [-13], [10], [8], [0], [8], [28], [14], [23], [10], [7], [16], [-30], [-16], [6], [2], [8], [-24], [-30], [11], [-13], [6], [27], [33], [19], [11], [-20], [24], [-18], [6], [-38], [18], [-18], [19], [24], [4], [28], [-3], [-18], [42], [23], [12], [20], [-28], [-30], [-20], [30], [-36], [-22], [38], [35], [-26], [-2], [-32], [-3], [-1], [-30], [-21], [-22], [-29], [-6], [-9], [10], [-33], [8], [-2], [-46], [-30], [28], [42], [-18], [-25], [-20], [10], [23], [30], [-14], [-22], [-14], [22], [-36], [-17], [-13], [-20], [-15], [-30], [10], [1], [34], [18], [38], [-30], [36], [22], [35], [34], [12], [-22], [-18], [-17], [9], [30], [3], [-36], [-47], [-5], [0], [28], [-8], [48], [-6], [-42], [6], [-4], [-30], [22], [-23], [-52], [-62], [-34], [37], [-53], [60], [42], [3], [-45], [-21], [32], [-31], [-37], [-9], [16], [41], [-36], [11], [6], [57], [-1], [-34], [2], [24], [25], [38], [28], [64], [-25], [36], [40], [-11], [13], [4], [-53], [30], [17], [13], [22], [10], [39], [-36], [35], [54], [34], [4], [3], [48], [-50], [3], [-44], [46], [-32], [30], [-51], [-1], [-8], [0], [-60], [38], [-60], [74], [19], [-32], [-15], [66], [-18], [-28], [22], [32], [18], [-44], [50], [-71], [-74], [16], [16], [44], [-23], [-52], [-45], [-60], [-67], [-12], [-23], [-53], [-12], [-59], [33], [38], [12], [-4], [32], [62], [30], [-68], [-76], [40], [10], [54], [27], [46], [6], [-54], [-13], [-26], [51], [16], [-8], [28], [-38], [-35], [2], [-14], [-74], [33], [44], [8], [8], [-47], [-58], [-48], [62], [-75], [17], [-21], [-66], [7], [42], [31], [36], [-28], [-16], [-32], [-45], [-58], [-10], [63], [-44], [36], [63], [17], [-46], [67], [48], [-14], [-72], [-15], [-36], [-61], [13], [-38], [-53], [32], [56], [-44], [-26], [14], [60], [-2], [-4], [-78], [-9], [54], [57], [-22], [44], [-30], [72], [44], [-7], [-13], [-14], [23], [52], [-36], [-46], [72], [68], [-6], [41], [-42], [-75], [-11], [78], [-12], [-78], [47], [35], [-89], [32], [-26], [6], [52], [46], [-64], [-10], [-69], [-3], [-74], [-34], [-6], [90], [-4], [-55], [-48], [-56], [-85], [-49], [5], [-38], [-89], [-83], [-98], [-62], [-76], [36], [12], [42], [86], [82], [48], [72], [-33], [71], [12], [59], [86], [2], [-82], [43], [97], [-45], [-9], [-43], [75], [-59], [40], [-82], [-55], [-18], [-20], [-44], [-72], [46], [81], [46], [-81], [38], [-53], [15], [60], [45], [8], [-61], [102], [83], [-43], [24], [33], [74], [-34], [42], [6], [38], [-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_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_2646_2_a_d(: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_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_2646_2_a_d( : 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![-2, 1]>,<13,R![-1, 1]>,<17,R![-3, 1]>],Snew);
    return Vf;
end function;