
// Make newform 6400.2.a.ch in Magma, downloaded from the LMFDB on 24 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_ch();".
// 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_ch();".
// 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 := [-6, 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], [0, 0], [-3, -1], [0, 0], [-3, 2], [-1, 3], [0, 0], [0, 0], [0, 0], [0, 0], [-3, 4], [10, 0], [0, 0], [0, 0], [6, 0], [0, 0], [-7, 3], [0, 0], [-1, -6], [0, 0], [9, -1], [9, -2], [-10, 0], [0, 0], [0, 0], [3, -7], [0, 0], [9, 4], [0, 0], [18, 0], [3, 8], [-11, 3], [0, 0], [0, 0], [0, 0], [1, -9], [0, 0], [0, 0], [-9, 7], [0, 0], [0, 0], [-11, -6], [0, 0], [0, 0], [7, 9], [0, 0], [30, 0], [0, 0], [-30, 0], [0, 0], [-13, -6], [-3, -11], [30, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-18, 0], [11, -9], [0, 0], [17, -3], [0, 0], [10, 0], [0, 0], [13, -9], [-7, -12], [3, 13], [0, 0], [30, 0], [0, 0], [0, 0], [0, 0], [-19, -3], [0, 0], [0, 0], [0, 0], [3, -14], [-11, -12], [-9, -13], [0, 0], [0, 0], [-19, 6], [0, 0], [-21, -1], [-21, -2], [-13, 12], [0, 0], [0, 0], [30, 0], [0, 0], [0, 0], [-42, 0], [-14, 0], [0, 0], [0, 0], [-3, -16], [19, 9], [0, 0], [-23, -3], [0, 0], [30, 0], [21, -8], [22, 0], [17, 12], [3, -17], [9, -16], [0, 0], [-23, -6], [0, 0], [0, 0], [-30, 0], [-26, 0], [0, 0], [42, 0], [-50, 0], [0, 0], [0, 0], [-9, 17], [0, 0], [-10, 0], [0, 0], [-21, -11], [23, -9], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [34, 0], [0, 0], [0, 0], [0, 0], [-27, -4], [11, -18], [0, 0], [-50, 0], [0, 0], [6, 0], [38, 0], [0, 0], [0, 0], [27, 7], [0, 0], [0, 0], [0, 0], [27, -8], [29, 3], [0, 0], [0, 0], [18, 0], [1, 21], [0, 0], [10, 0], [0, 0], [0, 0], [-54, 0], [-17, 18], [0, 0], [30, 0], [-21, -16], [0, 0], [-27, 11], [-3, 22], [0, 0], [0, 0], [0, 0], [19, 18], [0, 0], [21, 17], [0, 0], [0, 0], [31, 6], [0, 0], [9, -22], [13, 21], [0, 0], [0, 0], [0, 0], [0, 0], [-33, -1], [0, 0], [-33, 2], [0, 0], [0, 0], [0, 0], [-31, 9], [29, -12], [0, 0], [1, -24], [-21, 19], [-17, 21], [0, 0], [33, -7], [-30, 0], [7, 24], [0, 0], [33, 8], [0, 0], [0, 0], [0, 0], [0, 0], [31, 12], [6, 0], [0, 0], [0, 0], [30, 0], [-66, 0], [-58, 0], [70, 0], [0, 0], [0, 0], [27, 17], [0, 0], [13, -24], [0, 0], [3, 26], [0, 0], [0, 0], [0, 0], [0, 0], [-21, -22], [0, 0], [33, 13], [0, 0], [-9, 26], [0, 0], [0, 0], [-27, -19], [0, 0], [-1, -27], [0, 0], [33, -14], [10, 0], [0, 0], [-29, -18], [0, 0], [21, -23], [0, 0], [-39, 1], [37, 9], [0, 0], [0, 0], [-39, -4], [0, 0], [0, 0], [-78, 0], [-11, -27], [0, 0], [0, 0], [-33, -16], [0, 0], [-31, 18], [0, 0], [39, -7], [0, 0], [-13, 27], [0, 0], [-37, -12], [0, 0], [33, -17], [0, 0], [0, 0], [-27, -22], [-41, 3], [0, 0], [78, 0], [-29, 21], [0, 0], [0, 0], [17, 27], [-41, -6], [0, 0], [-50, 0], [0, 0], [27, -23], [0, 0], [-2, 0], [-33, 19], [0, 0], [0, 0], [0, 0], [0, 0], [43, 3], [0, 0], [70, 0], [0, 0], [0, 0], [66, 0], [0, 0], [30, 0], [39, 14], [-3, -31], [0, 0], [0, 0], [0, 0], [0, 0], [54, 0], [-23, 27], [-29, -24], [0, 0], [0, 0], [9, -31], [-43, 9], [37, -18], [90, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [27, -26], [-50, 0], [0, 0], [-34, 0], [39, -17], [0, 0], [-31, 24], [-9, 32], [82, 0], [43, -12], [0, 0], [0, 0], [90, 0], [-38, 0], [-1, 33], [10, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-39, -19], [37, -21], [90, 0], [0, 0], [30, 0], [-47, 6], [0, 0], [0, 0], [27, -28], [0, 0], [0, 0], [0, 0], [66, 0], [0, 0], [-13, -33], [0, 0], [0, 0], [47, 9], [-70, 0], [0, 0], [0, 0], [0, 0], [-9, -34], [0, 0], [-27, -29], [-90, 0], [0, 0], [0, 0], [33, 26], [0, 0], [6, 0], [17, -33], [-49, 6], [0, 0], [0, 0], [37, 24], [18, 0], [-19, -33], [0, 0], [0, 0], [0, 0], [0, 0], [39, 23], [0, 0], [1, 36], [51, 2], [10, 0], [0, 0], [51, -4], [0, 0], [33, 28], [-94, 0], [0, 0], [0, 0], [0, 0], [-70, 0], [0, 0], [-49, 12], [0, 0], [-51, -7], [-23, -33], [0, 0], [11, 36], [0, 0], [-51, 8], [-43, -21], [0, 0], [0, 0], [21, -34], [0, 0], [27, 32], [0, 0], [0, 0], [0, 0], [-102, 0], [-50, 0], [-9, 37], [41, -24], [0, 0], [51, 11], [98, 0], [47, -18], [0, 0], [0, 0], [0, 0], [-3, -38], [0, 0], [0, 0], [0, 0], [0, 0], [-51, 13], [19, -36], [0, 0], [30, 0], [9, 38], [53, -9], [0, 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_ch();".
// 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_ch(: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_ch();".
// 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_ch( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6400_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-5, -2, 1]>,<7,R![0, 1]>,<11,R![3, 6, 1]>,<13,R![0, 1]>,<17,R![-15, 6, 1]>,<29,R![0, 1]>,<31,R![0, 1]>],Snew);
    return Vf;
end function;