
// Make newform 6400.2.a.cj in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6400_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_6400_2_a_cj();".
// 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_6400_2_a_cj();".
// 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
        poly := [-3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_6400_a();"
function MakeCharacter_6400_a()
    N := 6400;
    order := 1;
    char_gens := [4351, 4101, 5377];
    v := [1, 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_6400_a_Hecke(Kf)
    return MakeCharacter_6400_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 := [[0, 0], [1, 1], [0, 0], [1, -1], [-2, 0], [0, 2], [0, -2], [-4, 2], [1, 3], [0, 4], [2, 2], [-2, 0], [2, -2], [7, -1], [5, -1], [8, 2], [-4, -2], [-2, -4], [9, 1], [2, 2], [4, 2], [8, -4], [3, -1], [-2, -4], [4, 6], [4, -4], [-7, -5], [-1, -1], [10, 4], [6, 4], [15, 1], [6, 8], [-2, 4], [-4, 2], [-6, 8], [-2, -6], [10, -4], [5, -3], [15, -3], [-2, 0], [12, -2], [-16, 0], [2, -10], [-4, 2], [-16, -2], [12, -8], [-6, -12], [-11, 3], [-9, 11], [4, 0], [12, -10], [20, 0], [6, 6], [18, 4], [2, 0], [3, 5], [-2, -4], [2, 10], [2, 0], [-14, 2], [1, 5], [-2, -8], [1, -15], [14, 10], [-18, 8], [-12, 2], [-14, 0], [6, 8], [-7, 5], [-28, 0], [-6, -4], [12, 4], [15, -7], [2, -16], [-12, -14], [9, 7], [14, -12], [-8, -14], [6, -8], [-6, 10], [-8, -6], [-14, 8], [18, 2], [16, 2], [20, 12], [5, -15], [-6, -10], [-6, -12], [-4, -4], [-13, 1], [-17, -5], [8, -8], [9, -9], [-10, -4], [-28, -2], [9, -5], [12, -8], [30, 8], [17, -3], [-20, -4], [35, -1], [6, -12], [-3, -11], [10, 2], [-14, 12], [2, -24], [9, -3], [2, -20], [0, -20], [-22, 2], [5, 15], [16, 6], [-24, 2], [-16, 10], [-6, -10], [-10, 6], [27, 7], [-15, -1], [12, -14], [0, 10], [22, -8], [-16, 2], [28, -6], [-9, 15], [-18, 0], [26, -4], [-8, 12], [-12, 8], [5, 11], [42, -4], [-12, -10], [15, -15], [-2, -14], [-26, 4], [-22, 16], [6, 4], [-12, -6], [1, 9], [-12, -22], [-10, -8], [-22, -4], [-14, -12], [-3, 7], [-11, -7], [22, 4], [4, 12], [-32, 6], [-6, -8], [4, 14], [39, 5], [-26, 16], [6, 2], [5, -19], [-19, 19], [-17, 7], [18, 18], [-32, -4], [-14, -2], [-20, 14], [-32, 0], [9, -7], [-38, 12], [-25, 5], [30, 4], [28, -2], [47, 1], [-38, -2], [20, -18], [-26, -12], [0, 22], [4, -30], [-8, -16], [42, 6], [32, -10], [-4, 12], [-14, 12], [-2, 0], [-48, 0], [-3, 15], [22, 16], [-25, -15], [2, -28], [44, -2], [-8, -10], [35, 9], [40, 4], [42, 8], [-33, 3], [2, 10], [46, -6], [-26, -4], [9, -11], [2, 20], [6, -24], [-39, -7], [36, -10], [14, 2], [8, 22], [-6, 4], [49, 3], [0, 12], [-2, 2], [-28, -18], [14, 16], [32, 2], [-18, 24], [-20, 16], [39, 3], [18, 28], [42, -16], [12, -30], [-6, -8], [-25, -15], [-23, 5], [-48, -4], [34, 14], [-29, -11], [30, -8], [-45, 9], [0, -14], [-2, -24], [-20, -4], [-6, 36], [-15, -25], [45, 13], [30, -8], [-24, 6], [0, -4], [3, 41], [-30, 0], [-16, -22], [0, -30], [-34, 2], [10, -30], [11, 19], [-45, -3], [26, -2], [0, -6], [-24, 26], [10, -18], [-5, -9], [42, -4], [11, -11], [12, -8], [-6, 36], [-12, 36], [-5, 13], [-30, -8], [-4, 22], [35, -7], [-12, -10], [-10, 0], [-35, -13], [6, -38], [6, 8], [4, 2], [28, 4], [-15, -35], [-26, -4], [46, 0], [-29, 9], [23, -13], [-20, 0], [10, -8], [-20, 10], [44, -18], [-52, -16], [-18, -16], [-15, -19], [-22, 20], [16, 0], [-1, 11], [0, 10], [-48, 12], [-58, -12], [-17, -23], [59, -5], [-32, -20], [26, 2], [6, 28], [63, 5], [-6, 2], [-33, -11], [-24, -8], [21, 1], [18, -2], [12, 10], [16, 38], [-4, -16], [-42, -2], [10, 0], [-19, 21], [-60, -6], [-58, 4], [2, 4], [18, -32], [10, -10], [18, -36], [4, -34], [-1, -29], [-14, -8], [-30, -28], [16, -16], [71, -5], [-10, -32], [-4, -10], [27, 27], [56, -12], [36, -20], [-34, -32], [33, -17], [-36, -24], [-6, 24], [-35, 5], [3, -7], [-2, 28], [-72, -2], [30, 22], [-14, -8], [54, 14], [26, 4], [-46, 0], [-2, -4], [-37, 17], [30, 28], [-2, -34], [-8, 10], [5, 1], [-11, -25], [24, 10], [18, -8], [10, -4], [52, -12], [-25, 35], [10, 24], [-21, 3], [-70, -12], [26, 16], [2, -14], [21, 15], [-26, 8], [62, 0], [66, -4], [-26, -6], [-42, 12], [-12, 14], [22, 12], [-19, -7], [54, 10], [44, -22], [-2, -28], [46, -24], [30, 16], [1, -1], [-12, -8], [32, 14], [40, 12], [58, -8], [0, 26], [27, 5], [28, 30], [10, -24], [15, -7], [-48, -10], [-55, 1], [20, 30], [56, 2], [-49, -7], [-6, -34], [22, -4], [68, 10], [51, 17], [50, -20], [38, 2], [40, -6], [32, -26], [-10, 2], [-2, 24], [-6, 22], [22, -12], [8, 16], [-34, -4], [67, -7], [-54, -12], [16, 18], [41, -29], [-50, -14], [10, 44], [-23, -43], [9, 3], [-2, -10], [-18, 4], [72, 6], [-21, 15], [18, -30], [10, 0], [12, 20], [-34, 24], [-22, -12], [-34, 4], [-22, 24], [36, 2], [19, 5], [-30, 0], [30, 28], [-54, 10], [-68, -6], [14, -8], [-53, -9], [4, -6], [-34, 24], [34, -20], [-75, 1], [30, 8], [-16, -10], [-36, 4], [52, -8], [43, -31], [84, -2], [39, -15], [38, 8], [-80, 14], [33, 27], [-60, -2], [-30, -44], [2, 36], [11, 15], [6, 42], [58, 20], [72, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6400_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6400_2_a_cj();".
// 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_6400_2_a_cj(:prec:=2)
    chi := MakeCharacter_6400_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_6400_2_a_cj();".
// 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_6400_2_a_cj( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6400_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, -2, 1]>,<7,R![-2, -2, 1]>,<11,R![2, 1]>,<13,R![-12, 0, 1]>,<17,R![-12, 0, 1]>,<29,R![-48, 0, 1]>,<31,R![-8, -4, 1]>],Snew);
    return Vf;
end function;