
// Make newform 3024.2.r.j in Magma, downloaded from the LMFDB on 21 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_3024_r();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3024_r_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_r_j();".
// 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_r_j();".
// 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, -12, 19, -15, 10, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [2, -1, 1, 0, 0, 0], [6, -18, 5, -8, 1, -1], [3, -5, 6, -2, 1, 0], [9, -21, 19, -16, 5, -2], [-9, 30, -22, 19, -5, 2]];
    Rf_basisdens := [1, 1, 3, 1, 3, 3];
    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_3024_r();"
function MakeCharacter_3024_r()
    N := 3024;
    order := 3;
    char_gens := [1135, 757, 785, 2593];
    v := [3, 3, 1, 3];
    // 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_r_Hecke();"
function MakeCharacter_3024_r_Hecke( : Kf := false)
    N := 3024;
    order := 3;
    char_gens := [1135, 757, 785, 2593];
    char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0], [1, 0, 0, 0, 0, 0]];
    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], [0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 1], [-1, 0, 0, 0, 1, 0], [-1, 0, 2, 0, 1, 1], [0, 2, -2, -2, -1, -2], [-1, -2, 0, 1, 0, 0], [0, 2, 0, 1, 0, 0], [0, 1, 1, 1, -4, -1], [1, 0, 1, 0, -1, 3], [0, 3, 0, 0, 0, -3], [-2, -1, 0, -2, 0, 0], [0, -1, -2, -2, -1, 1], [0, 0, 4, 0, 0, 1], [-8, 0, 1, 0, 8, -2], [-2, 1, 0, -1, 0, 0], [0, -1, -1, -1, -11, 1], [-4, 0, -1, 0, 4, -7], [0, 2, 1, 1, 3, -2], [3, -5, 0, -3, 0, 0], [0, 2, 0, 1, 0, 0], [-4, 0, -1, 0, 4, -4], [-7, 0, -1, 0, 7, -2], [-5, 1, 0, -4, 0, 0], [1, 0, 5, 0, -1, -1], [4, 0, 1, 0, -4, 0], [0, -3, 3, 3, -4, 3], [9, 6, 0, 0, 0, 0], [4, 6, 0, 3, 0, 0], [0, 2, 6, 6, 0, -2], [-3, -3, 0, -6, 0, 0], [0, 3, 1, 1, -1, -3], [-8, 0, 4, 0, 8, -3], [0, -2, -1, -1, 2, 2], [0, 3, 0, 0, 3, -3], [1, 0, 3, 0, -1, -3], [0, 0, 3, 3, 4, 0], [-3, 2, 0, 1, 0, 0], [0, -4, 2, 2, 1, 4], [6, 0, 6, 0, -6, -1], [0, 0, 0, -3, 0, 0], [-6, 3, 0, -6, 0, 0], [-8, 0, -2, 0, 8, -5], [0, 1, 5, 5, 5, -1], [5, 7, 0, 1, 0, 0], [-4, -5, 0, -7, 0, 0], [0, 4, -4, -4, -15, -4], [9, 0, 4, 0, -9, 4], [-19, 0, -1, 0, 19, -1], [0, -4, 4, 4, -11, 4], [2, 7, 0, -8, 0, 0], [0, -2, -10, -10, -5, 2], [12, 0, -7, 0, -12, -1], [-7, -4, 0, 7, 0, 0], [0, 4, -10, -10, -2, -4], [2, 0, -1, 0, -2, -10], [-11, 5, 0, -1, 0, 0], [-11, 1, 0, 2, 0, 0], [-11, 0, 3, 0, 11, 0], [-23, 0, 1, 0, 23, 2], [0, 1, 2, 2, 2, -1], [0, -13, 1, 1, -7, 13], [7, 2, 0, 7, 0, 0], [0, 5, -10, -10, -11, -5], [-1, 0, 9, 0, 1, 9], [9, 0, 0, 0, -9, 3], [-1, 0, -3, 0, 1, 6], [0, 3, -9, -9, -15, -3], [0, 8, 7, 7, 8, -8], [24, 0, -2, 0, -24, -2], [0, 0, -3, 0, 0, 9], [9, 8, 0, -9, 0, 0], [6, 0, 4, 0, -6, 7], [0, -5, -4, -4, 12, 5], [-15, -17, 0, 5, 0, 0], [0, -1, 6, 6, -6, 1], [-20, 0, 1, 0, 20, 6], [-6, 3, 0, -6, 0, 0], [0, -7, -2, -2, -19, 7], [0, -4, -2, -2, 6, 4], [0, -5, 5, 5, 10, 5], [2, 0, -7, 0, -2, -4], [-17, -11, 0, 5, 0, 0], [21, -2, 0, -4, 0, 0], [-5, 0, 1, 0, 5, -5], [-12, 0, 6, 0, 12, 8], [21, 4, 0, 3, 0, 0], [4, 0, -1, 0, -4, -10], [-13, 0, -7, 0, 13, 4], [0, 15, -9, -9, 4, -15], [-4, 7, 0, 10, 0, 0], [-14, 0, -5, 0, 14, 0], [-11, 3, 0, 3, 0, 0], [0, 1, 2, 2, 28, -1], [0, 6, 0, 0, 16, -6], [-11, -5, 0, -4, 0, 0], [0, 1, 11, 11, 16, -1], [9, -6, 0, 12, 0, 0], [13, 3, 0, 3, 0, 0], [13, -2, 0, 8, 0, 0], [2, 0, -7, 0, -2, 5], [-1, 9, 0, -11, 0, 0], [0, -19, 6, 6, -21, 19], [-15, 0, 12, 0, 15, 6], [0, -7, 7, 7, 11, 7], [-6, -13, 0, -5, 0, 0], [3, 0, -6, 0, -3, 5], [0, 12, 0, 3, 0, 0], [0, 2, -2, -2, 2, -2], [-23, 0, 0, 0, 23, -3], [0, 0, -6, -6, 3, 0], [-9, 6, 0, 6, 0, 0], [0, -1, -4, -4, -23, 1], [0, 0, -13, 0, 0, -13], [15, -7, 0, -2, 0, 0], [-34, 0, 5, 0, 34, -3], [0, -15, 3, 3, -13, 15], [30, 6, 0, 3, 0, 0], [0, 8, 7, 7, 17, -8], [14, 0, -13, 0, -14, -8], [0, 5, -5, -5, -23, -5], [17, 0, -2, 0, -17, -5], [21, 0, -3, 0, -21, -9], [2, 20, 0, -2, 0, 0], [11, 0, 0, 0, -11, -3], [16, -4, 0, 2, 0, 0], [-6, 0, -10, 0, 6, 2], [5, -7, 0, -11, 0, 0], [-10, 0, -8, 0, 10, -11], [0, 2, 10, 10, 23, -2], [-7, 5, 0, 1, 0, 0], [0, -12, 3, 3, 9, 12], [0, -4, -5, -5, -28, 4], [3, -6, 0, -15, 0, 0], [0, 2, 14, 14, -2, -2], [0, -9, 3, 3, 14, 9], [-21, -8, 0, 0, 0, 0], [0, -3, -12, -12, 12, 3], [0, -4, 1, 1, 11, 4], [17, 18, 0, 1, 0, 0], [-22, 3, 0, -15, 0, 0], [-6, 0, -9, 0, 6, 4], [0, 0, 0, 0, 16, 0], [-5, 19, 0, 5, 0, 0], [-8, 0, 0, 3, 0, 0], [19, 0, -14, 0, -19, -6], [2, 0, -11, 0, -2, -2], [20, 0, -7, 0, -20, -7], [0, 6, 0, 0, -25, -6], [27, -10, 0, 3, 0, 0], [0, -14, -1, -1, -27, 14], [26, 5, 0, -2, 0, 0], [18, 11, 0, 4, 0, 0], [0, 13, -7, -7, 4, -13], [-9, 0, -12, 0, 9, 12], [-1, 0, 5, 0, 1, 17], [-31, -3, 0, -6, 0, 0], [25, 0, 7, 0, -25, 25], [-21, -14, 0, 5, 0, 0], [0, -1, -12, -12, 12, 1], [-34, 0, 5, 0, 34, 9], [2, -8, 0, -8, 0, 0], [0, -2, 5, 5, -15, 2], [-17, -8, 0, -7, 0, 0], [0, 8, -8, -8, 5, -8], [-22, 0, 5, 0, 22, -7], [30, 8, 0, 7, 0, 0], [38, 0, 4, 0, -38, -8], [-9, -8, 0, -13, 0, 0], [0, 22, -7, -7, 28, -22], [-9, 0, 9, 0, 9, -23], [0, -27, 0, 0, -13, 27], [0, 6, 0, 0, -3, -6], [-33, 0, 0, 0, 33, 0], [0, -20, 8, 8, 1, 20], [0, 19, 4, 4, 11, -19], [4, 0, -8, 0, -4, 19], [0, -25, 0, 0, 0, 0], [-30, -13, 0, 1, 0, 0], [-20, 0, -12, 0, 20, -9], [-26, 0, 7, 0, 26, 1], [4, 0, 13, 0, -4, 12], [0, -10, -2, -2, -32, 10], [-18, 4, 0, 9, 0, 0], [0, 5, 17, 17, -2, -5], [44, 0, -7, 0, -44, -4], [-22, 5, 0, -2, 0, 0], [-13, 0, 3, 0, 13, 0], [0, 15, -6, -6, 15, -15], [24, 0, 0, -3, 0, 0], [19, -3, 0, 21, 0, 0], [6, 0, 12, 0, -6, 3], [17, 15, 0, -9, 0, 0], [23, 0, 2, 0, -23, 3], [18, 15, 0, -6, 0, 0], [0, -15, 1, 1, 2, 15], [0, -2, 5, 5, 2, 2], [12, 0, 7, 0, -12, 22], [27, 0, -15, 0, -27, -6], [-17, 16, 0, -10, 0, 0], [0, 8, 10, 10, 5, -8], [-9, 0, 6, 0, 9, -24], [0, 5, 13, 13, 27, -5], [-29, 0, 10, 0, 29, -5], [25, 1, 0, -4, 0, 0], [-23, -5, 0, -16, 0, 0], [2, 10, 0, -10, 0, 0], [0, 7, -7, -7, -26, -7], [16, 0, -2, 0, -16, -14], [0, 23, -8, -8, 7, -23], [-1, 3, 0, -12, 0, 0], [0, 6, -13, -13, 28, -6], [-14, 0, -8, 0, 14, -2], [29, 0, 2, 0, -29, 3], [0, 8, -11, -11, -1, -8], [25, 0, 9, 0, -25, 21], [0, 1, 11, 11, 38, -1], [-18, 0, -12, 0, 18, 7], [31, -10, 0, 5, 0, 0], [0, 17, -7, -7, 31, -17], [0, 21, -15, -15, 23, -21], [0, 0, -12, -12, -2, 0], [0, 12, 0, 0, -24, -12], [0, 19, 0, -16, 0, 0], [0, -14, 2, 2, -14, 14], [13, 0, -3, 0, -13, -12], [-1, 0, 8, 0, 1, -27], [0, -1, 0, -9, 0, 0], [-1, 0, -4, 0, 1, -1], [12, 0, 0, 0, -12, -9], [0, 29, 16, 16, -4, -29], [2, 6, 0, -6, 0, 0], [0, 10, -4, -4, 26, -10], [0, -2, 5, 5, -38, 2], [-36, 0, 10, 0, 36, -8], [-22, 0, -10, 0, 22, -13], [0, -14, -4, -4, 12, 14], [12, 21, 0, -3, 0, 0], [0, -9, 6, 6, 36, 9], [-40, -2, 0, 1, 0, 0], [-17, 0, -8, 0, 17, -9], [15, -13, 0, -11, 0, 0], [0, 3, -6, -6, -13, -3], [-1, -7, 0, 13, 0, 0], [0, -11, 15, 15, -6, 11], [17, 0, 17, 0, -17, 8], [-31, -13, 0, -11, 0, 0], [0, -17, -2, -2, 5, 17], [0, 18, 3, 3, -28, -18], [-21, -12, 0, 12, 0, 0], [0, 27, -12, -12, 5, -27], [-4, -16, 0, 10, 0, 0], [0, -7, 7, 7, -4, 7], [17, 0, 9, 0, -17, 6], [-10, 0, 23, 0, 10, 0], [-27, -20, 0, -3, 0, 0], [-16, -12, 0, 9, 0, 0], [12, 0, 15, 0, -12, 6], [-18, 7, 0, -12, 0, 0], [-9, 16, 0, -4, 0, 0], [6, 0, -25, 0, -6, -7], [13, 0, -11, 0, -13, -4], [0, 11, -2, -2, 7, -11], [-13, -14, 0, 2, 0, 0], [0, 4, -4, -4, 4, -4], [4, 0, 6, 0, -4, 0], [-58, -10, 0, 7, 0, 0], [-12, 0, 12, 0, 12, -13], [0, 7, 2, 2, -37, -7], [0, 15, -3, -3, -30, -15], [0, 18, 15, 15, -21, -18], [18, -22, 0, 1, 0, 0], [19, 0, -14, 0, -19, -8], [0, 10, 5, 5, 3, -10], [0, -1, -5, -5, -47, 1], [7, -8, 0, 8, 0, 0], [0, -5, -5, -5, 14, 5], [-34, 0, -13, 0, 34, -10], [32, 16, 0, -10, 0, 0], [38, 0, 2, 0, -38, -1], [0, 30, 0, 0, 21, -30], [0, -17, -4, -4, 7, 17], [30, 0, 9, 0, -30, 9], [29, 0, -5, 0, -29, 13], [0, 10, -7, -7, 1, -10], [10, 26, 0, -1, 0, 0], [-4, 26, 0, -17, 0, 0], [0, -15, 13, 13, 35, 15], [7, 0, -13, 0, -7, -1], [5, -22, 0, -11, 0, 0], [0, 0, 9, 0, 0, -15], [-2, -20, 0, -4, 0, 0], [0, 12, -3, -3, -36, -12], [0, -16, -1, -1, 16, 16], [-10, 0, -5, 0, 10, 31], [0, 17, -8, -8, -28, -17], [-19, 0, 3, 0, 19, 15], [22, 0, -26, 0, -22, 10], [-48, -9, 0, -12, 0, 0], [-23, 0, 4, 0, 23, 10], [0, -4, 4, 4, -23, 4], [-6, -3, 0, 3, 0, 0], [5, 33, 0, -3, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3024_r_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3024_2_r_j();".
// 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_r_j(:prec:=6)
    chi := MakeCharacter_3024_r();
    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_r_j();".
// 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_r_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3024_r();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, -4, 17, 2, 5, -1, 1]>,<11,R![3481, 1475, 743, 68, 29, 2, 1]>],Snew);
    return Vf;
end function;