
// Make newform 6422.2.a.j in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6422_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_6422_2_a_j();".
// 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_6422_2_a_j();".
// 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_6422_a();"
function MakeCharacter_6422_a()
    N := 6422;
    order := 1;
    char_gens := [4903, 4733];
    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_6422_a_Hecke(Kf)
    return MakeCharacter_6422_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], [3], [3], [-3], [0], [0], [5], [-1], [6], [-8], [-8], [5], [10], [7], [1], [-10], [-6], [-6], [4], [5], [-8], [12], [8], [0], [2], [-18], [-18], [16], [5], [6], [-10], [-5], [-12], [-5], [22], [-3], [-22], [-10], [0], [26], [5], [-24], [6], [24], [-3], [14], [5], [-7], [6], [-23], [-11], [5], [-2], [20], [-21], [14], [18], [-7], [10], [6], [-16], [-19], [-4], [18], [7], [-22], [8], [-11], [7], [-33], [-16], [16], [0], [-38], [-38], [-23], [36], [-18], [-6], [-16], [33], [-11], [35], [-39], [14], [-15], [-2], [22], [-17], [-24], [-8], [25], [16], [15], [-10], [-20], [-18], [17], [4], [33], [35], [-31], [-31], [21], [-31], [-18], [12], [32], [-8], [-13], [-8], [26], [42], [4], [43], [-10], [12], [-18], [-12], [-16], [26], [-13], [-36], [18], [-28], [6], [-6], [26], [0], [-11], [-40], [7], [16], [14], [-2], [-34], [-27], [-18], [-2], [51], [-2], [-13], [24], [26], [16], [-16], [17], [-6], [20], [-45], [11], [25], [9], [48], [-15], [50], [-32], [-4], [14], [51], [42], [9], [-37], [3], [38], [-17], [22], [-22], [20], [-8], [2], [31], [20], [12], [-10], [-25], [2], [6], [-26], [-26], [-36], [44], [-8], [28], [48], [18], [10], [-64], [34], [24], [31], [66], [-29], [-30], [-23], [14], [-2], [26], [22], [38], [-3], [22], [47], [-33], [54], [54], [-29], [-36], [32], [-53], [38], [-4], [-14], [-54], [-27], [-20], [32], [47], [-52], [57], [24], [-47], [36], [3], [3], [-46], [21], [4], [34], [-22], [68], [52], [72], [-55], [1], [69], [-6], [-19], [-4], [-36], [16], [27], [28], [-25], [-46], [-18], [-65], [0], [-66], [16], [-13], [-54], [-33], [13], [52], [-62], [-8], [16], [-72], [28], [-50], [41], [-55], [-72], [24], [5], [-43], [50], [-64], [-28], [54], [68], [26], [54], [54], [-16], [44], [-70], [70], [-11], [-68], [51], [26], [-30], [12], [30], [-3], [63], [-27], [-66], [12], [35], [-26], [-36], [-6], [18], [12], [54], [71], [-34], [55], [-15], [-46], [-48], [45], [42], [45], [58], [-48], [-8], [-24], [61], [-87], [-68], [7], [35], [-54], [12], [-12], [-42], [-52], [6], [64], [-31], [-54], [-50], [38], [64], [64], [16], [50], [-24], [16], [8], [50], [68], [38], [-64], [-58], [14], [-3], [3], [9], [2], [-13], [-47], [68], [-60], [-83], [-86], [-50], [44], [23], [94], [-48], [54], [-5], [58], [83], [-51], [-54], [-6], [36], [-36], [92], [-29], [37], [76], [-65], [-67], [43], [77], [-36], [46], [-34], [10], [42], [64], [50], [-18], [18], [22], [-92], [-52], [96], [-25], [-11], [-8], [-16], [-82], [-68], [55], [-16], [67], [19], [59], [49], [-45], [16], [51], [41], [50], [41], [12], [11], [20], [37], [-16], [-56], [18], [51], [-24], [-9], [-61], [-19], [40], [5], [-56], [20], [-60], [-84], [10], [-48], [-81], [-40], [-18], [105], [76], [14], [-82]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6422_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6422_2_a_j();".
// 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_6422_2_a_j(:prec:=1)
    chi := MakeCharacter_6422_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_6422_2_a_j();".
// 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_6422_2_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6422_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-3, 1]>,<5,R![-3, 1]>,<7,R![3, 1]>],Snew);
    return Vf;
end function;