
// Make newform 3024.2.cz.b in Magma, downloaded from the LMFDB on 26 July 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_3024_cz();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3024_cz_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_3024_2_cz_b();".
// 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_3024_2_cz_b();".
// 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, -1, 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_3024_cz();"
function MakeCharacter_3024_cz()
    N := 3024;
    order := 6;
    char_gens := [1135, 757, 785, 2593];
    v := [3, 6, 2, 5];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3024_cz_Hecke();"
function MakeCharacter_3024_cz_Hecke( : Kf := false)
    N := 3024;
    order := 6;
    char_gens := [1135, 757, 785, 2593];
    char_values := [[-1, 0], [1, 0], [0, -1], [0, 1]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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]];
    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], [0, 0], [0, 0], [2, -3], [-4, 2], [-2, 1], [-1, -1], [0, 5], [2, 2], [3, -3], [-1, 0], [0, -7], [2, -1], [-1, -1], [-9, 0], [-9, 9], [-15, 0], [1, -2], [-9, 18], [6, -12], [1, 1], [-1, 2], [-9, 9], [-2, 1], [-1, -1], [16, -8], [-8, 8], [-2, 1], [-11, 11], [0, -9], [-6, 12], [-12, 12], [0, -18], [0, -5], [-6, 6], [-20, 10], [7, -14], [18, -9], [0, 21], [11, -22], [-7, -7], [-8, 16], [1, -2], [-17, 0], [-6, 0], [-1, 1], [-14, 7], [5, -5], [0, -12], [8, 8], [0, -21], [-26, 13], [-24, 12], [0, 0], [16, 16], [36, -18], [-13, -13], [0, 1], [0, 14], [15, -15], [31, 0], [-30, 15], [-8, 0], [-3, 0], [17, -34], [21, 0], [3, -6], [0, -11], [13, -26], [15, 15], [24, -12], [14, -7], [0, 8], [10, -10], [14, -28], [-24, 24], [0, 18], [18, -9], [30, -30], [-17, 34], [0, -15], [5, -5], [-15, -15], [8, -16], [25, 0], [-15, 30], [-18, 0], [-1, 0], [13, 13], [42, -21], [0, 15], [0, 0], [-9, -9], [-10, 5], [-2, -2], [-24, 0], [20, 20], [-13, -13], [0, 13], [0, -43], [-5, -5], [-33, 33], [21, 0], [-3, 0], [23, -46], [-15, -15], [27, -27], [-10, 5], [21, -42], [-21, -21], [32, -32], [-35, 35], [0, 27], [0, 28], [18, -36], [0, 6], [0, -13], [-15, 15], [30, -30], [13, 13], [-5, 10], [-7, 7], [-1, 2], [25, 25], [5, 0], [-18, 0], [-37, 0], [0, 27], [-35, 35], [12, 12], [-9, -9], [14, -7], [10, 10], [2, 0], [28, 28], [6, -3], [19, 19], [-13, 0], [-38, 19], [-9, 9], [-52, 0], [33, 0], [7, -14], [10, -20], [9, 9], [27, -27], [31, 31], [8, -4], [-4, 4], [30, -15], [46, -46], [12, -24], [-18, 36], [12, -12], [-44, 22], [33, 33], [26, -13], [-5, 10], [-28, 56], [17, -34], [-3, 6], [18, 0], [18, -9], [0, 15], [-51, 0], [0, 12], [-21, -21], [64, -32], [-50, 0], [24, 24], [-68, 34], [-42, 21], [6, 6], [0, 26], [-5, 0], [-14, 7], [3, 3], [-9, 9], [56, 0], [-11, 22], [-1, 2], [39, -39], [0, 5], [30, -15], [-19, 38], [-72, 36], [-11, 11], [0, -44], [50, -50], [-3, 3], [42, -21], [-15, -15], [-6, 3], [3, -6], [1, 1], [-23, 46], [-5, 0], [-17, 0], [29, 29], [0, -9], [-39, 0], [-31, 31], [-8, -8], [-32, 16], [12, 0], [-17, -17], [0, 29], [30, 30], [3, -3], [-17, 0], [0, -47], [34, -17], [-29, -29], [-69, 0], [21, 0], [33, -66], [7, -14], [72, -36], [-58, 29], [0, -69], [-50, 50], [0, 31], [30, -30], [-14, 7], [0, -27], [53, -53], [-9, 18], [-19, -19], [5, 0], [-7, 14], [23, 0], [11, -11], [-14, 7], [-3, 0], [-19, 19], [0, -48], [-36, -36], [0, 39], [-26, 13], [-72, 0], [-68, 34], [0, 73], [-65, 0], [0, -43], [-30, 15], [-69, 0], [8, 0], [57, 0], [-17, 34], [14, -28], [0, -23], [-34, 17], [-35, 70], [15, 15], [-40, 20], [-26, 13], [5, -5], [0, -28], [8, -16], [74, -37], [60, -30], [-27, -27], [15, -30], [-32, 64], [-15, 30], [-55, 0], [78, 0], [1, 1], [-86, 43], [-39, 0], [-8, -8], [-9, -9], [24, -12], [-42, -42], [62, -31], [0, 37], [-26, -26], [0, -34], [0, 53], [-57, 0], [33, 0], [39, -78], [39, -39], [-9, -9], [4, -4], [6, -3], [-83, 83], [0, -21], [0, 64], [-12, 24], [0, -30], [-27, 27], [6, -6], [0, -63], [1, -1], [25, -50], [-35, 0], [-27, -27], [0, 15], [-35, 35], [28, 28], [0, -69], [11, 11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3024_cz_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3024_2_cz_b();".
// 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_3024_2_cz_b(:prec:=2)
    chi := MakeCharacter_3024_cz();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_3024_2_cz_b();".
// 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_3024_2_cz_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3024_cz();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<11,R![12, 6, 1]>],Snew);
    return Vf;
end function;