
// Make newform 2646.2.a.r 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_r();".
// 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_r();".
// 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], [-3], [0], [0], [4], [-6], [4], [6], [-3], [-8], [8], [6], [8], [-6], [9], [3], [10], [-10], [6], [7], [17], [12], [-6], [10], [15], [-8], [-3], [-4], [-6], [5], [12], [12], [-2], [-21], [8], [-14], [2], [-6], [9], [-15], [10], [-18], [26], [-15], [-11], [-4], [1], [-3], [-14], [18], [0], [1], [-9], [0], [-18], [-21], [25], [-28], [-18], [4], [3], [-8], [-12], [-5], [-18], [2], [-7], [33], [10], [30], [36], [19], [-10], [-10], [-6], [-9], [-32], [-18], [25], [-12], [32], [12], [7], [-8], [-39], [6], [-1], [33], [-19], [33], [24], [-13], [27], [2], [24], [-18], [30], [4], [-10], [-28], [-6], [-15], [18], [20], [-23], [15], [-6], [0], [25], [-11], [-16], [-18], [16], [20], [24], [16], [-42], [3], [21], [-32], [-43], [21], [-9], [-8], [-39], [8], [24], [-23], [-44], [14], [24], [-43], [2], [-48], [-26], [18], [22], [-18], [24], [-20], [21], [23], [21], [-32], [24], [-44], [0], [-14], [42], [14], [-6], [32], [42], [8], [-54], [-37], [-6], [1], [-18], [45], [-48], [-37], [3], [12], [24], [23], [-32], [23], [33], [12], [-14], [-6], [41], [31], [24], [-16], [-54], [7], [-26], [32], [3], [-16], [-30], [0], [-18], [-34], [-50], [14], [6], [-5], [48], [38], [-15], [21], [36], [-1], [-46], [24], [-6], [3], [37], [4], [-17], [3], [27], [-56], [33], [-36], [4], [17], [-3], [-37], [-12], [18], [7], [-13], [6], [12], [-54], [50], [16], [36], [29], [-45], [62], [24], [42], [67], [36], [-16], [34], [41], [-24], [22], [-42], [-35], [-57], [-3], [-54], [-36], [-2], [4], [20], [-72], [-24], [16], [-3], [14], [48], [-52], [42], [60], [79], [-78], [-12], [32], [-74], [-30], [79], [-7], [51], [46], [-26], [66], [-20], [27], [72], [14], [-21], [-20], [2], [-29], [8], [-14], [-47], [-27], [20], [11], [48], [0], [80], [-66], [16], [-32], [-72], [47], [-33], [-47], [-30], [-18], [75], [-12], [27], [68], [9], [-71], [-9], [-75], [-62], [-50], [-18], [56], [-60], [-34], [-85], [-27], [40], [-72], [-4], [-12], [-51], [-6], [-76], [42], [-74], [0], [-84], [-53], [90], [82], [-91], [87], [29], [54], [-62], [62], [-68], [-54], [-30], [20], [-9], [-59], [36], [8], [-39], [56], [12], [-5], [64], [80], [24], [-45], [-85], [-90], [-9], [-8], [26], [78], [63], [-50], [47], [-18], [19], [-28], [60], [36], [21], [48], [-90], [-10], [-66], [-72], [69], [52], [-73], [-30], [11], [-7], [45], [-26], [-36], [42], [37], [2], [-48], [-12], [-11], [-18], [-62], [-27], [-54], [-16], [-66], [58], [-18], [-76], [-50], [32], [12], [26], [-6], [-3], [94], [-78], [-13], [-47], [0], [-10], [-57], [-86], [-72], [32], [42], [-102], [-29], [-82], [24], [64], [-27], [-62], [-42], [-63], [8], [65], [-69], [-66], [-92], [-54], [-84], [-21], [34], [-42], [36], [43], [87], [9], [-12], [52], [66]];
    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_r();".
// 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_r(: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_r();".
// 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_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2646_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![3, 1]>,<11,R![0, 1]>,<13,R![-4, 1]>,<17,R![6, 1]>],Snew);
    return Vf;
end function;