
// Make newform 6930.2.a.bt in Magma, downloaded from the LMFDB on 17 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_bt();".
// 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_bt();".
// 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;
    Rf_num := [[1, 0], [0, 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], [0, 1], [0, 1], [0, 0], [-2, 1], [-2, 0], [0, 2], [4, 1], [2, 2], [4, -2], [0, -2], [-6, -2], [-4, 2], [2, -2], [6, 1], [4, 0], [6, -2], [8, 0], [0, -2], [0, -1], [2, -4], [2, -4], [16, 0], [0, 2], [4, 3], [-8, -1], [4, 2], [8, -2], [8, -1], [4, -2], [-6, -2], [8, 0], [2, -6], [6, 1], [6, 1], [2, 2], [0, 6], [-20, 1], [-8, -2], [18, -2], [18, 0], [0, -2], [-6, -5], [4, 6], [0, -2], [-8, 3], [-2, 0], [6, -3], [-14, 2], [12, -2], [10, 2], [-4, 6], [10, 2], [8, -4], [-10, 0], [4, 3], [-2, 1], [14, 0], [-10, 5], [-10, 1], [-6, 8], [10, -2], [-4, 0], [2, 6], [8, -2], [-22, -2], [-14, -2], [-6, -1], [16, 0], [2, -2], [-28, 0], [-8, 6], [6, -8], [-10, 0], [-2, 8], [-6, -2], [0, -4], [-26, 0], [6, 5], [10, -8], [4, -6], [-20, -2], [22, 0], [26, -2], [18, 0], [0, 6], [24, 2], [0, -4], [8, -2], [12, -4], [24, -2], [-2, -3], [26, 2], [-24, -1], [34, 3], [-8, 5], [-8, 0], [18, 4], [4, 8], [8, 9], [6, 5], [18, 8], [0, 6], [4, -9], [24, -6], [-30, 2], [16, 0], [-2, 0], [-8, 3], [-2, 5], [-4, 6], [6, 0], [4, -8], [16, 2], [-18, -4], [12, 8], [12, -3], [-22, 2], [34, 2], [-20, -2], [14, -3], [-2, 0], [6, -4], [-6, 3], [20, -2], [0, -7], [-2, 5], [0, -4], [20, -6], [-12, -11], [18, 6], [18, 6], [-26, 0], [38, 1], [-18, -4], [8, 1], [0, 4], [-26, 4], [36, 4], [0, -6], [20, 9], [-34, 1], [-16, 9], [4, -1], [10, 3], [-18, -7], [6, -4], [0, -5], [-10, -11], [34, -5], [10, -9], [4, 0], [52, -2], [36, -3], [10, -4], [-22, 0], [24, -4], [-6, 10], [-20, -2], [-8, -8], [20, 5], [-24, 6], [-20, -6], [36, -5], [-46, -4], [30, 8], [24, 6], [-12, 5], [-6, -13], [34, 2], [20, -12], [-48, -1], [-2, 5], [30, -4], [-44, -2], [-38, -2], [8, 6], [4, 4], [-36, 1], [24, -3], [-42, -7], [22, 12], [10, 6], [-20, -8], [34, 8], [-24, 4], [50, 0], [28, -8], [-16, 6], [-2, -8], [12, 0], [-26, -8], [-54, -4], [-36, -3], [28, 3], [18, -9], [26, 6], [0, -12], [2, -10], [10, -6], [-28, 2], [-42, -4], [-40, 2], [-4, -4], [-4, -9], [18, -1], [-14, 6], [-34, 8], [-20, -4], [0, -2], [-4, -14], [-18, 6], [-8, 0], [-46, 6], [14, 5], [-42, 0], [28, 3], [44, -2], [-2, -8], [4, -16], [-4, 8], [-4, 7], [6, 4], [12, -8], [24, 0], [44, 0], [-32, 3], [-4, -2], [-16, -4], [-4, -17], [0, -14], [6, -3], [10, -8], [-22, 12], [-40, -2], [-66, 1], [-52, 2], [-8, -4], [-28, 6], [14, -4], [12, -13], [24, 8], [20, 6], [8, -4], [10, -9], [12, -6], [-18, 16], [26, 2], [10, -13], [-18, -10], [-42, -4], [-8, 6], [-10, -16], [-6, -5], [6, -8], [-10, 6], [-40, -8], [-12, 12], [-22, 6], [-8, 9], [6, -8], [2, 15], [-58, 4], [-20, -7], [-48, 8], [2, 4], [-52, 1], [-26, -3], [26, 4], [28, 10], [46, -6], [24, 12], [16, 12], [16, -3], [-14, -8], [16, 2], [14, 13], [-4, 10], [66, -1], [-38, -6], [-10, -11], [44, 8], [2, -4], [-46, 0], [-20, 4], [-18, -4], [30, -12], [0, -18], [6, -20], [44, -4], [-58, -4], [18, -8], [36, 8], [6, -12], [-28, -8], [-14, -1], [-34, -10], [18, -16], [-56, -8], [4, 2], [-16, -6], [-14, -8], [0, 12], [16, 3], [-52, 8], [10, 10], [18, 19], [-42, 8], [12, -1], [-14, 15], [60, 2], [18, -18], [40, -4], [30, -3], [-46, 4], [-36, 3], [0, 0], [-50, -6], [-30, -4], [32, 14], [-14, -6], [-6, -20], [0, -14], [20, -12], [48, -4], [34, 16], [-48, -7], [6, 20], [-16, 4], [60, -8], [-54, -1], [0, 6], [-42, 0], [-36, -5], [54, 2], [-36, 2], [32, -5], [-16, -13], [-54, 8], [16, 0], [26, 2], [-40, -14], [-8, 3], [-46, 11], [28, -14], [62, -4], [32, 4], [-38, 0], [6, 0], [-28, 6], [56, -3], [44, 15], [-46, 7], [12, -20], [18, 2], [-22, -5], [14, 0], [62, 8], [-6, -9], [24, -18], [-84, 4], [-14, 16], [38, 0], [-40, 0], [-42, 2], [24, 10], [50, -5], [0, -4], [30, -12], [-20, 6], [-24, 7], [24, 8], [22, -11], [10, 4], [-34, -8], [2, 12], [6, -12], [48, 11], [-4, -26], [30, -8], [-28, -18], [8, -10], [40, -8], [18, -6], [12, -22], [-36, -12], [2, -12], [14, -4], [-60, -4], [48, -6], [76, 0], [26, -6], [16, -20], [0, 11], [-4, -16], [34, -14], [46, 4], [-12, 21], [68, 2], [2, -10], [10, -4], [64, 4], [32, 19], [-32, 9], [20, -12], [-36, -10], [-38, 0], [10, 0], [4, 2], [-10, 21], [-54, 14], [-38, -10], [-26, -7], [28, -18], [-2, -12], [14, 9], [22, 12], [0, 17], [-54, -5], [-72, -2], [58, 0], [6, 12], [60, -10], [-24, -11], [38, 9], [-20, 18], [66, -4], [52, 16], [24, -6], [80, -6], [42, 6], [-48, 7], [-34, -6], [-6, 26], [18, -9], [16, -16], [4, 20], [2, -6], [-36, -9], [58, -9], [34, 6], [-70, 2], [-70, 7], [74, 3], [54, 2], [34, -10], [-28, -6], [2, -13], [-20, -8], [-26, -8], [46, 2], [-26, -4], [36, -21], [28, -8], [20, 15], [0, 31], [32, 10], [32, -4], [-8, 2], [-50, 12], [34, 19], [2, -10], [4, -10], [-60, -8], [0, -23], [42, -9], [16, 16], [-40, 16], [-6, 11], [-34, 2], [12, 24], [-40, 5], [-106, 0], [-4, -24], [46, -15], [-18, -20], [-70, 4], [30, 12]];
    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_bt();".
// 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_bt(: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_bt();".
// 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_bt( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6930_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<13,R![-12, 0, 1]>,<17,R![-12, 0, 1]>,<19,R![0, 1]>,<23,R![-8, 4, 1]>,<29,R![2, 1]>,<31,R![-48, 0, 1]>],Snew);
    return Vf;
end function;