
// Make newform 6930.2.a.by in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6930_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_6930_2_a_by();".
// 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_6930_2_a_by();".
// 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 := [-8, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-1, 2]];
    Rf_basisdens := [1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_6930_a();"
function MakeCharacter_6930_a()
    N := 6930;
    order := 1;
    char_gens := [1541, 1387, 2971, 2521];
    v := [1, 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_6930_a_Hecke(Kf)
    return MakeCharacter_6930_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], [0, 0], [-1, 0], [-1, 0], [1, 0], [1, 1], [1, 1], [0, 0], [1, -1], [-2, 0], [0, 0], [3, -1], [-6, 0], [4, 0], [4, 0], [6, 0], [-4, 0], [6, 0], [-7, -1], [0, 0], [6, 0], [12, 0], [4, 0], [-1, -1], [4, 2], [-6, 0], [-2, -2], [4, 0], [9, 1], [5, 1], [8, 0], [2, -2], [1, -3], [10, -2], [-4, 2], [-4, 0], [6, 0], [9, -1], [-3, 3], [-12, 2], [4, 0], [7, -1], [-6, -2], [2, 4], [-12, 2], [4, -4], [19, 1], [10, 2], [-4, 0], [21, 1], [-22, 0], [-1, -3], [14, 0], [4, 0], [-12, -2], [14, 2], [-12, -2], [8, 0], [8, 2], [3, 3], [-7, 3], [6, 0], [1, 3], [1, -1], [6, 0], [-6, 4], [-20, 0], [4, 2], [20, 0], [2, -4], [20, -2], [13, -1], [-10, -2], [-2, -4], [8, -4], [-20, 0], [16, 2], [-4, -6], [10, 4], [12, 2], [14, -2], [-6, -4], [19, 1], [-16, -2], [16, 0], [-12, -4], [14, 0], [-6, -4], [12, -2], [-26, 2], [-4, 0], [8, 0], [-22, -2], [0, -4], [-14, 2], [-23, -1], [-28, -2], [3, -5], [11, 1], [-1, 3], [-2, 6], [0, -2], [6, -2], [21, 1], [-35, -1], [-10, 0], [-28, 0], [-17, 3], [10, 6], [4, -6], [-12, -4], [-20, -2], [25, -3], [15, -3], [4, 4], [26, 4], [-4, 0], [-20, 0], [0, -2], [16, 4], [-29, 3], [20, 2], [12, -6], [12, 4], [-1, -3], [-26, 0], [-6, 4], [-5, 5], [-22, 2], [-7, 1], [37, -1], [0, 0], [-28, 4], [23, -5], [-6, -8], [20, 2], [-20, 6], [-23, 3], [-36, -2], [9, -3], [-16, 0], [-22, 4], [8, 0], [-24, 4], [11, 3], [-23, -1], [-27, -3], [-21, -1], [41, 3], [-15, -1], [-16, 2], [15, -1], [-7, -9], [-5, 5], [7, 1], [-12, -4], [-18, -2], [21, 1], [8, -2], [-6, -8], [30, 2], [28, -2], [4, -4], [-6, 2], [-9, -1], [32, 4], [-12, 4], [31, 3], [-30, 0], [12, -2], [18, -6], [-1, -1], [-23, 3], [6, 0], [2, 6], [3, 3], [37, -1], [0, -6], [-10, -6], [34, -4], [-22, 6], [-10, 2], [-29, -1], [1, 9], [-27, -5], [42, 0], [-6, -8], [20, 0], [-2, -4], [-24, 0], [46, 0], [8, 4], [-10, -2], [30, 4], [0, 4], [-6, 8], [-46, 0], [35, -1], [-45, -1], [3, 5], [8, -2], [-28, 4], [-34, 0], [16, -2], [-16, -4], [24, -6], [-4, -4], [-32, -4], [-5, 3], [-15, -5], [30, 0], [-14, 8], [0, 8], [-24, -4], [2, -2], [-24, -6], [0, -8], [-14, 8], [-27, -5], [-28, -6], [21, 5], [-16, -8], [62, 0], [-40, 0], [-30, 2], [21, 5], [26, 0], [36, -4], [-24, 0], [-16, -4], [21, -3], [30, 2], [4, -8], [-5, 3], [22, -2], [1, -9], [-6, 0], [-6, -4], [48, -4], [-35, 3], [-32, 0], [-16, 8], [14, 6], [-2, 0], [39, -5], [24, 0], [-58, -2], [-42, 6], [7, 5], [-26, 2], [0, 2], [14, 4], [-9, 1], [20, 6], [30, -8], [6, -10], [-6, 4], [23, -3], [-24, 2], [22, 0], [52, 4], [-16, -4], [-46, -4], [-3, 5], [-46, 0], [-7, 3], [20, 6], [-7, -11], [22, -10], [-2, 8], [-21, 3], [-11, 3], [-4, -6], [-66, -2], [-54, 4], [26, 2], [14, 2], [35, -1], [-10, 12], [30, -6], [21, 3], [-8, 0], [-21, 5], [-18, 8], [-31, -5], [-8, 0], [-10, -4], [10, 12], [22, 2], [-4, -2], [18, 0], [-12, 8], [-18, -4], [-6, 6], [16, 2], [12, -2], [6, -14], [-18, -4], [-18, 2], [-21, -7], [-46, 4], [-48, -2], [-36, 0], [60, -4], [10, -6], [34, 0], [18, 6], [1, -3], [12, 4], [10, 0], [-25, 1], [54, 4], [15, -1], [11, -3], [30, -6], [86, 0], [50, 2], [67, 1], [-68, 2], [-13, 3], [24, 0], [-40, -6], [-20, -2], [74, -2], [-12, -10], [-2, -4], [-10, -10], [0, -4], [0, 0], [18, 4], [-49, -5], [-56, 6], [-40, 8], [10, -14], [47, 5], [20, 0], [-62, 4], [55, -5], [-20, -10], [-18, -6], [53, -3], [13, -7], [-34, 4], [68, -4], [18, 4], [-38, 10], [85, 1], [-37, -3], [-50, 2], [52, -2], [-28, -4], [10, 8], [-20, -2], [-30, -2], [19, 7], [3, -9], [-45, -3], [26, -2], [20, -2], [-5, -11], [32, -6], [64, 2], [-37, -3], [-54, 2], [52, 0], [62, -4], [-14, 4], [-4, -4], [16, -6], [0, -4], [29, -9], [-12, 12], [34, -4], [28, -4], [37, -3], [-42, -2], [-53, 1], [-8, 6], [60, 6], [-16, -10], [-2, -12], [13, 1], [8, 0], [50, 8], [38, -6], [-12, -8], [-16, 0], [38, 0], [-44, 8], [-30, 2], [8, -6], [-2, 4], [-40, -4], [14, 6], [-32, -8], [70, 0], [28, 12], [27, 3], [12, 8], [16, 14], [-28, -6], [-57, 7], [-4, 4], [-12, -2], [-20, 6], [-24, 8], [-23, 9], [33, 5], [72, -4], [4, 16], [-4, 2], [-74, 0], [-12, 12], [-19, -9], [28, 2], [60, 6], [35, 9], [-38, -6], [-78, 0], [13, -13], [24, -6], [-11, -11], [71, 1], [-42, -6], [-18, -8], [-28, 6], [-48, 8], [25, -11], [35, 9], [46, 2], [-14, 8], [36, 8], [22, 6], [-32, -4], [-38, 4], [-45, -5], [32, -6], [-28, -14], [35, -11], [-20, -4], [-36, -8], [60, 2], [21, 5], [-55, 3], [14, 0], [48, 2], [-37, -7], [11, -11], [-52, 6], [-34, -8], [44, 0], [15, -7], [12, -8], [24, 2], [20, -2], [-30, -8], [49, -7], [-14, -2], [-25, -13], [-3, 1], [22, -2], [-44, 12], [-38, 2], [34, 4], [-37, -7], [-44, 2], [-2, -2], [24, -4], [61, -7], [33, 11], [8, 0], [42, -2], [31, -7], [-8, -14], [12, -8], [-67, 5], [-2, -4], [-2, 10], [-7, -1], [-64, -2], [8, -2], [-24, -6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6930_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6930_2_a_by();".
// 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_6930_2_a_by(:prec:=2)
    chi := MakeCharacter_6930_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(3449) - 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_6930_2_a_by();".
// 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_6930_2_a_by( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6930_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<13,R![-32, -2, 1]>,<17,R![-32, -2, 1]>,<19,R![0, 1]>,<23,R![-32, -2, 1]>,<29,R![2, 1]>,<31,R![0, 1]>],Snew);
    return Vf;
end function;