
// Make newform 6400.2.a.cd in Magma, downloaded from the LMFDB on 25 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_cd();".
// 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_cd();".
// 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], [-3, -1], [0, 2], [0, 2], [0, 2], [2, 0], [-3, 3], [0, 0], [6, -2], [-6, 0], [-6, -2], [-5, 3], [-3, 3], [0, -6], [6, 0], [-6, 4], [5, -3], [6, 6], [-4, 6], [12, 0], [3, -1], [6, 4], [-4, -6], [12, 0], [-3, 3], [15, -1], [6, 4], [6, 4], [-9, -3], [0, -6], [6, -4], [10, 0], [-18, 0], [6, 2], [6, 4], [-11, -3], [3, -3], [-6, 0], [6, -8], [0, 4], [-6, 6], [4, 6], [0, 6], [0, 4], [4, 6], [-3, 7], [3, -9], [12, -4], [-12, -6], [0, 12], [-10, 6], [12, -2], [-6, 0], [15, -3], [-6, -12], [6, 6], [6, -8], [18, -6], [5, 9], [-30, 0], [29, -3], [-18, -6], [22, 0], [12, -6], [16, 6], [14, 0], [9, -11], [12, 12], [-6, -12], [-12, -12], [-9, -11], [6, 8], [2, 0], [9, 3], [-6, 12], [-24, 2], [-18, -8], [-14, -6], [-6, -4], [6, 0], [-18, 6], [-16, 6], [12, -4], [9, 5], [-6, -2], [2, -12], [-12, 0], [-21, -3], [3, 15], [0, 0], [-27, 7], [12, -6], [-2, -24], [-3, 3], [-12, -12], [-18, 8], [-11, -15], [-12, -16], [7, 3], [-6, -12], [21, -3], [18, -14], [-20, 6], [2, 0], [9, 13], [-6, -4], [12, 0], [10, -6], [-3, 3], [24, 6], [24, 6], [-14, -12], [18, -2], [6, 6], [-5, -9], [21, -9], [12, 18], [-6, 20], [-6, -24], [8, -18], [-12, 18], [3, -5], [-16, -6], [6, 12], [-24, -8], [-24, 12], [-15, 11], [-18, 12], [-2, 0], [-21, 9], [30, -6], [-30, -4], [-6, -16], [14, 12], [36, -6], [-11, -3], [-12, -6], [6, 24], [-28, 6], [30, -12], [9, 23], [-3, 1], [-30, 12], [-12, -12], [0, -2], [-30, -8], [-10, 24], [15, -15], [-6, -8], [-18, -6], [-11, -3], [-15, -21], [-17, -9], [6, -18], [12, -16], [18, 6], [-20, -6], [0, -12], [-51, -3], [18, -4], [3, -3], [12, 26], [12, 10], [27, 9], [-6, 22], [36, 6], [-2, 12], [24, 6], [-30, -16], [-24, 20], [42, 6], [-8, 18], [-36, -4], [-6, 20], [-8, -6], [0, -12], [9, -17], [-6, -16], [-9, 5], [-12, 14], [12, -10], [24, -22], [3, -3], [24, 0], [-18, 16], [-1, 3], [-22, -6], [-42, -6], [-2, -12], [21, 1], [-20, 6], [-6, -24], [-27, 5], [-12, 18], [-26, -6], [48, 6], [18, -28], [45, 3], [0, 24], [6, 18], [-12, 6], [-2, 0], [-30, 12], [-30, 0], [-24, -20], [39, 3], [42, -12], [-16, -18], [28, -18], [-18, 0], [27, 1], [-15, 29], [12, 0], [-38, -18], [3, -7], [6, 8], [-33, -15], [-24, -6], [-30, 8], [-12, 28], [18, -4], [9, 27], [-15, 1], [18, -8], [48, 2], [-60, 0], [-9, -7], [48, -10], [-24, -14], [-22, 12], [6, -14], [-6, 10], [-17, -9], [3, -15], [10, 6], [0, -6], [18, 28], [18, 6], [3, -17], [52, 6], [39, 13], [36, 4], [-54, -4], [36, 12], [3, -23], [48, 6], [-50, 0], [27, 21], [36, -18], [6, -16], [-15, 27], [-10, -6], [42, 0], [-30, 8], [-12, 8], [-7, 21], [-6, -36], [14, 0], [3, 13], [3, 31], [-12, 12], [6, -40], [12, -10], [-2, 0], [36, 12], [6, 0], [-35, 9], [6, 12], [48, 4], [-13, -33], [16, -18], [36, 16], [-2, 12], [3, -23], [-21, -5], [-48, -8], [10, -6], [36, 10], [-57, -15], [-18, 6], [-45, 21], [72, -4], [13, 9], [-6, -18], [-44, -18], [24, 6], [-48, 12], [30, 6], [-18, 24], [-15, 9], [-12, -10], [36, -22], [-42, 20], [6, 24], [-30, -10], [6, 12], [-18, 0], [43, 15], [-62, 0], [30, 12], [24, -8], [39, 11], [-64, -6], [-20, 18], [-45, 3], [24, 32], [12, -36], [-6, 0], [9, 3], [-12, -12], [-30, -8], [-11, -3], [39, 9], [-10, -12], [54, 4], [42, -6], [26, 24], [30, -10], [28, -18], [-14, -24], [-6, -12], [3, -19], [-18, 12], [-2, -18], [2, -12], [-23, 21], [-75, 3], [-24, -6], [54, 16], [-18, 12], [-36, -4], [63, 11], [16, 6], [-45, -21], [6, -28], [-6, -24], [-14, 42], [45, 3], [42, -16], [30, -16], [30, -36], [-42, 10], [42, 12], [30, -24], [-30, 4], [53, -15], [6, -30], [12, -6], [-4, -30], [38, 0], [-30, 0], [57, -13], [-36, 20], [48, 2], [12, 0], [24, -2], [-72, 6], [-57, -3], [12, 22], [18, -40], [63, -3], [6, 12], [5, -3], [-28, -6], [0, 42], [3, -39], [34, 30], [36, -6], [74, 0], [-69, -3], [-18, 12], [-66, -6], [-24, -38], [78, -4], [-18, 18], [-2, 24], [-54, 14], [22, -12], [24, 36], [-18, -4], [15, -15], [66, 4], [-22, -36], [-3, -21], [6, -14], [6, -12], [29, 9], [9, -33], [-10, 30], [-30, -12], [6, -28], [55, 27], [-42, 6], [-22, 24], [-36, 20], [-66, 8], [-4, 6], [-6, 12], [-18, 0], [-12, -34], [3, 41], [-78, 8], [-6, -12], [-18, -18], [60, 26], [6, 24], [43, -9], [-30, 16], [-10, 48], [-18, -12], [9, 5], [32, 18], [32, 42], [-12, -24], [48, -12], [39, -23], [12, -14], [3, -39], [-6, 24], [24, -2], [-15, -9], [54, -8], [58, -12], [-18, 12], [3, -9], [-42, 26], [20, -18], [-12, 0]];
    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_cd();".
// 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_cd(: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_cd();".
// 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_cd( : 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![6, 6, 1]>,<11,R![-12, 0, 1]>,<13,R![-12, 0, 1]>,<17,R![-12, 0, 1]>,<29,R![0, 1]>,<31,R![24, -12, 1]>],Snew);
    return Vf;
end function;