
// Make newform 2646.2.a.h in Magma, downloaded from the LMFDB on 24 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_h();".
// 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_h();".
// 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], [0], [0], [6], [5], [6], [-4], [6], [6], [-1], [-1], [-6], [-1], [-6], [-6], [-6], [-1], [-1], [12], [2], [-1], [6], [0], [17], [12], [-19], [18], [5], [12], [-13], [-12], [-6], [-7], [-12], [17], [-22], [11], [-18], [0], [0], [-10], [0], [-1], [-12], [11], [-13], [8], [-6], [5], [24], [24], [17], [18], [-6], [6], [0], [-7], [17], [12], [-13], [18], [17], [-12], [-22], [18], [20], [2], [-24], [-1], [-36], [6], [-28], [26], [-1], [6], [-12], [-13], [-6], [29], [-24], [14], [-18], [-25], [8], [-24], [-24], [-19], [18], [8], [12], [12], [-40], [-30], [-25], [6], [30], [42], [23], [-10], [-37], [0], [-42], [-36], [-16], [23], [0], [36], [36], [47], [-16], [11], [-30], [-25], [-7], [18], [-25], [-12], [-18], [0], [50], [-34], [12], [12], [17], [48], [17], [12], [5], [17], [-49], [18], [-16], [29], [0], [-10], [-24], [-31], [-30], [-30], [-16], [6], [-31], [-12], [14], [18], [-46], [-12], [23], [36], [-13], [30], [-40], [-18], [-37], [-6], [-55], [48], [-37], [18], [18], [-6], [35], [-36], [6], [-24], [5], [23], [-31], [-54], [12], [50], [0], [-22], [-31], [-6], [-7], [12], [-7], [35], [-4], [-48], [29], [-42], [36], [60], [-7], [-4], [-31], [-36], [14], [36], [47], [-6], [-18], [36], [-10], [53], [36], [18], [-12], [35], [41], [-19], [-60], [-42], [29], [24], [12], [-31], [-37], [-6], [-19], [-48], [18], [-61], [-4], [12], [-60], [42], [41], [-61], [48], [-16], [6], [-73], [6], [42], [32], [54], [-34], [20], [-31], [-30], [5], [54], [-46], [-36], [54], [-54], [24], [20], [-40], [-25], [-30], [-6], [11], [24], [-4], [-12], [29], [30], [-78], [29], [18], [18], [5], [65], [78], [-34], [56], [42], [17], [35], [66], [11], [18], [-60], [-4], [-24], [-70], [-25], [-7], [-1], [14], [-25], [12], [-25], [11], [-42], [-18], [17], [-54], [-25], [-13], [-48], [-7], [-42], [-7], [-48], [42], [60], [24], [6], [59], [18], [-55], [12], [30], [-37], [-49], [-84], [-52], [-42], [29], [5], [-30], [50], [-6], [-22], [60], [72], [6], [23], [-6], [29], [30], [-24], [53], [6], [-28], [-82], [42], [29], [-84], [17], [-19], [-49], [-18], [-6], [-7], [36], [-40], [42], [-55], [-30], [47], [60], [77], [-1], [35], [-84], [-30], [32], [54], [72], [-55], [44], [72], [42], [-67], [65], [-12], [8], [35], [-48], [-60], [-84], [-36], [48], [-73], [-12], [12], [-24], [65], [35], [-54], [29], [11], [-30], [-55], [-42], [6], [-64], [-79], [-66], [-42], [-61], [-30], [26], [-66], [-66], [-43], [-90], [5], [48], [5], [77], [-40], [-18], [-55], [-42], [60], [59], [24], [-67], [-79], [12], [89], [-6], [50], [-48], [32], [-18], [18], [83], [53], [-48], [17], [18], [-10], [-102], [-90], [-1], [-79], [36], [36], [-88], [-36], [-72], [54], [11], [72], [-84], [38], [-102], [18], [30], [92], [0]];
    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_h();".
// 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_h(: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_h();".
// 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_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2646_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<11,R![-6, 1]>,<13,R![-5, 1]>,<17,R![-6, 1]>],Snew);
    return Vf;
end function;