
// Make newform 6930.2.a.cj in Magma, downloaded from the LMFDB on 25 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_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_6930_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 := [-1, -3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 2, 0], [-8, -2, 4]];
    Rf_basisdens := [1, 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, 0, 0], [-1, 0, 0], [1, 0, 0], [-1, 0, 0], [0, 1, 0], [-2, 1, 0], [-2, -1, -1], [-2, -2, 1], [-2, -2, 0], [0, 1, 1], [-2, -1, 0], [-2, 1, 1], [-2, -1, -1], [-6, 0, 0], [-2, 1, -1], [-4, -1, -1], [-4, 0, 0], [4, -1, 2], [-2, 4, 0], [-2, -1, -1], [0, 4, 0], [0, 3, -1], [-4, -4, 1], [2, 3, 1], [-6, 0, -2], [0, 1, 1], [-4, -1, -3], [-4, -3, 0], [-4, 2, 1], [4, -1, -1], [-4, 2, 0], [-8, -2, 1], [-10, 3, 1], [2, 2, 0], [8, 0, 0], [2, -3, -3], [-4, -3, 0], [-16, 2, 1], [-10, -4, 0], [-4, 5, 1], [-6, -5, 0], [2, 2, 0], [6, 1, 1], [6, 3, -1], [8, 3, 1], [2, 3, 4], [8, 0, -2], [-8, -5, 1], [-6, 1, 0], [-2, -2, 2], [-12, -3, -2], [-10, -3, 1], [0, -3, 3], [-12, -3, -1], [8, 2, 0], [6, 0, 0], [-4, -6, -2], [4, -1, -3], [18, 3, 0], [-2, 2, 1], [-2, -2, -4], [6, 4, 3], [-2, -5, 2], [10, 2, 4], [-10, -1, 3], [-8, 2, -2], [-10, 0, -2], [4, -5, -3], [-16, -6, 2], [-4, -1, -3], [8, 5, -4], [-8, -1, 3], [-4, 3, -1], [0, 4, -4], [-6, 0, 0], [-10, 5, -1], [26, -3, -1], [0, 0, 0], [-10, 0, -2], [0, -3, 3], [14, 0, 0], [-16, -1, 0], [-6, -5, 1], [8, -3, -3], [0, 3, 5], [4, 4, 0], [2, 9, 1], [14, -1, 1], [20, -6, 0], [-20, 1, 1], [4, 4, 4], [28, 2, 2], [20, -4, -2], [-4, 6, -4], [-8, 10, 1], [-18, 6, 0], [12, 0, -3], [6, 4, 1], [-12, 1, -2], [2, -3, -5], [2, 5, 1], [24, 3, 1], [-18, -3, 4], [2, 3, -2], [18, -2, 0], [-24, -3, 1], [-22, -7, 2], [-14, -8, -2], [22, 2, 4], [20, 6, 0], [8, 9, 3], [-4, 2, -3], [4, 2, 7], [8, -9, 1], [4, -8, -4], [-8, -4, 4], [-6, 6, -2], [-6, -9, -3], [4, -2, 2], [-14, -1, 4], [-2, -10, 4], [10, -2, 4], [4, 7, -5], [12, -8, 1], [6, -6, -4], [2, -6, -2], [-30, 3, 0], [24, 4, -2], [8, 9, -4], [6, -5, -6], [8, -4, 0], [16, 5, 3], [14, 9, -2], [10, 1, -3], [-6, 10, 0], [6, 3, 1], [14, -12, -1], [-14, 7, 1], [-10, -7, 4], [2, 9, -3], [10, 10, -2], [-4, -12, -2], [12, -9, 1], [-18, 1, -6], [-6, -9, -2], [0, 1, -4], [6, 3, 0], [8, 0, 3], [-6, 6, -7], [-4, 9, 3], [8, -4, 1], [20, -3, -4], [12, -6, 3], [16, -1, 0], [38, 2, 2], [8, 0, 8], [-4, 2, 5], [2, -9, -1], [18, -6, 0], [24, -3, 3], [34, 4, 2], [4, -1, -1], [-44, 1, -3], [16, -8, 1], [-6, 0, 0], [40, 5, -1], [12, 1, 2], [-38, 2, 0], [-6, 9, -5], [20, 8, -4], [-22, 3, 0], [-16, -1, -2], [-18, -11, 5], [0, 1, -3], [-8, 4, 1], [-22, -9, 2], [-2, 7, 5], [8, -12, -4], [0, 10, 6], [20, -12, -2], [12, -9, -1], [-26, 5, -2], [22, 1, 0], [18, -6, 1], [26, 2, -6], [20, -9, 3], [8, 12, -8], [-14, -10, 0], [-20, -2, 2], [-10, -6, 0], [20, 8, 4], [-28, 6, 4], [18, 0, 2], [12, 0, 8], [20, -1, 3], [2, -18, 0], [2, 9, 6], [-10, 11, -4], [0, 6, -1], [-14, -8, -2], [-8, -10, 2], [2, -9, 3], [-22, 6, 0], [0, 9, -5], [-10, -13, 5], [-16, -9, -5], [20, -4, 0], [14, -1, -4], [-16, -6, 1], [14, -9, -5], [-6, -6, 0], [-4, 6, 6], [4, 13, -7], [-4, 2, -8], [2, 6, -4], [8, 0, -4], [22, 7, -5], [-22, 8, -1], [26, -1, 5], [-40, -3, -8], [-36, 1, 1], [-24, 0, -4], [-4, -6, 4], [-24, -3, 3], [-40, 9, 0], [-16, -1, -5], [22, -2, 2], [-8, 8, -4], [-4, 2, -4], [6, 1, 0], [26, -9, 1], [-36, -2, 4], [22, 7, 8], [-24, 22, 4], [0, -6, 1], [-18, 4, 2], [-22, 2, 2], [-28, 11, -1], [10, 7, -6], [4, -5, -9], [22, 5, 9], [16, -16, 2], [2, 0, 4], [10, 7, 4], [4, -8, -4], [16, 14, -4], [52, 1, 5], [-22, 9, 2], [24, -18, -4], [-12, -5, 5], [22, -11, 5], [30, 6, 3], [-26, -4, -12], [34, -2, 6], [-12, -6, 8], [22, -4, -4], [2, 18, -5], [-18, 9, 3], [14, -3, -9], [12, 10, -6], [60, 0, 0], [-12, -8, 0], [-36, -5, -2], [8, -1, -5], [-24, 4, 3], [30, -3, 7], [16, 4, 3], [14, -3, 7], [42, 0, 0], [-6, -5, 0], [32, 3, 6], [-18, 7, 5], [-12, -2, 4], [-34, -9, 7], [32, 3, 3], [-16, 5, 3], [4, 11, 0], [-46, 6, 0], [-4, 2, 4], [12, -18, -3], [-24, -5, -11], [-12, -6, -13], [-24, -14, 6], [26, 6, 1], [10, 10, 10], [-10, 6, 0], [34, -8, 0], [24, -5, 3], [-10, 11, 5], [54, 0, -4], [0, 9, -3], [-2, -2, -10], [-20, 1, -3], [-36, 1, -1], [2, 5, 3], [56, -3, 1], [-34, -10, 6], [-4, -7, -3], [-6, -14, -1], [62, 3, -3], [42, 3, -1], [44, 0, 2], [20, 17, -7], [-16, -12, -2], [-46, 0, 4], [16, 7, 5], [-26, 11, -2], [-18, 12, 8], [-12, 1, -7], [-32, 4, 5], [-14, 4, -8], [10, 13, 10], [-16, 15, -4], [64, -2, 0], [2, 9, 1], [8, -13, 5], [4, -5, -4], [-14, -13, 9], [-42, 3, -8], [-50, -7, -3], [6, 10, 2], [-22, 11, -7], [-28, -6, 6], [6, 6, 0], [-30, 4, -4], [16, 8, -6], [16, -10, 6], [28, -20, -4], [2, 8, -8], [-32, 11, 4], [-22, -1, -7], [-48, -8, 4], [8, -7, 1], [46, 5, -2], [-20, -5, -5], [-46, 6, 6], [-6, 3, -8], [14, -6, -2], [-12, -2, 0], [-38, -1, -6], [20, 8, 3], [-26, -2, 0], [8, 6, -10], [22, 1, -1], [-20, 10, 4], [2, -15, 8], [-28, 3, -8], [-20, -2, -4], [6, 9, 11], [-22, 9, -7], [38, -6, -4], [10, -17, -3], [20, -6, 4], [-20, 5, 2], [-34, -1, 0], [10, -5, 12], [-20, 13, -3], [-38, 4, -6], [14, 8, -11], [-40, 15, 5], [-50, 1, -1], [-34, -10, 3], [-4, -16, -8], [52, 2, -6], [2, 18, 6], [-38, 10, 2], [-28, -6, -6], [-10, -12, -2], [8, -19, -3], [-24, 10, 5], [28, 10, 2], [-14, 10, 2], [-16, 9, -3], [-34, 3, 2], [28, 7, 11], [12, 15, 12], [-22, -21, 11], [-62, 1, -1], [10, -1, -9], [10, 4, 12], [-28, 2, -3], [44, -3, 1], [34, 4, -4], [-58, 3, 1], [18, -6, 10], [48, 4, 8], [14, 3, 7], [-82, 3, -1], [-38, 10, 0], [-10, -21, 7], [34, -8, 8], [-12, 0, -4], [8, -12, 2], [-34, 2, 6], [-46, -15, 5], [32, -18, -2], [0, 0, -11], [32, -24, -4], [-42, 18, 2], [-4, -18, 4], [-56, 10, 3], [-40, 21, 5], [0, -9, -3], [6, -3, 3], [8, -12, -4], [-10, -21, 6], [-30, -9, -2], [-12, 4, -16], [-68, 1, 1], [-22, -21, 11], [22, -8, 8], [16, -11, 3], [2, 3, -6], [-26, 8, 2], [-26, 16, -2], [-60, 3, -4], [16, 2, 4], [-8, -11, 9], [8, -4, 7], [54, 3, 5], [-16, -3, 0], [66, 6, -5], [12, 0, -6], [-22, -6, -4], [22, -23, 3], [0, 9, -5], [-6, 15, 2], [-16, -18, 7], [-4, 20, -2], [2, 18, 10], [-20, 4, 4], [-48, -14, 8], [-6, -24, -4], [2, 15, -5], [4, -2, 3], [38, 12, 6], [-14, -10, 0], [-12, 7, 8], [-28, -18, -2], [12, -12, -8], [4, 16, -10], [-8, 11, -10], [-8, -5, 10], [2, -3, 1], [-18, -12, 14], [-22, 0, 1], [-16, -22, -1], [-22, 6, -4], [-12, 22, -6], [-6, -11, -7], [10, 7, -8], [-44, -8, -4], [-70, -1, -3], [-46, 18, 4], [6, 12, -8], [4, 11, -10], [-12, 3, -5], [32, -9, 0], [12, 18, 5], [-16, 0, 10], [-8, -4, 0], [-28, -22, 6], [6, 4, -8], [50, 12, 1], [-58, 12, -2], [12, 16, 6], [24, -18, -2], [-38, 1, -8], [-12, 4, -9], [-36, -8, 12], [36, -15, -7], [50, -1, 8], [-18, 4, -6], [-4, -4, -12], [20, -21, 6], [-42, 0, 4], [32, 15, -7], [60, -6, -7], [10, 19, 1], [34, 17, -5], [-38, 19, 5]];
    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_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_6930_2_a_cj(:prec:=3)
    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_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_6930_2_a_cj( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6930_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<13,R![-8, -12, 0, 1]>,<17,R![-24, 0, 6, 1]>,<19,R![-152, -36, 6, 1]>,<23,R![-456, -72, 6, 1]>,<29,R![-24, -36, 6, 1]>,<31,R![64, -48, 0, 1]>],Snew);
    return Vf;
end function;