
// Make newform 3024.2.r.g in Magma, downloaded from the LMFDB on 25 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_g();".
// 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_g();".
// 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, 0, -1, -1, -2, 0], [-1, 0, 0, 0, 1, 0], [1, 0, 1, 0, -1, 2], [0, 0, 0, 0, -1, 0], [4, 1, 0, -1, 0, 0], [1, -1, 0, 1, 0, 0], [0, -1, -2, -2, -1, 1], [0, 0, 0, 0, 0, -1], [0, -2, -1, -1, -2, 2], [0, 0, 0, 3, 0, 0], [0, 0, 1, 1, -7, 0], [0, 0, 1, 0, 0, 4], [3, 0, -3, 0, -3, -3], [5, -1, 0, -2, 0, 0], [0, 1, 2, 2, 4, -1], [1, 0, 2, 0, -1, -1], [0, 7, -1, -1, 2, -7], [2, -1, 0, -2, 0, 0], [-1, -1, 0, -5, 0, 0], [3, 0, 1, 0, -3, -5], [4, 0, 1, 0, -4, 1], [-1, -6, 0, 1, 0, 0], [-2, 0, -2, 0, 2, -5], [-3, 0, -3, 0, 3, 1], [0, -1, -2, -2, -11, 1], [1, 0, 0, -4, 0, 0], [-9, -3, 0, -3, 0, 0], [0, -6, -4, -4, -2, 6], [-5, -2, 0, -1, 0, 0], [0, -1, -4, -4, -8, 1], [-8, 0, 1, 0, 8, -2], [0, 1, -1, -1, 1, -1], [0, 6, -3, -3, 3, -6], [0, 0, 4, 0, 0, -5], [0, 1, -1, -1, -3, -1], [0, 7, 0, -1, 0, 0], [0, -4, -4, -4, 1, 4], [2, 0, -1, 0, -2, -6], [-1, 3, 0, -5, 0, 0], [14, -2, 0, 5, 0, 0], [-6, 0, -3, 0, 6, -2], [0, 9, 0, 0, 10, -9], [-2, -7, 0, 2, 0, 0], [3, 9, 0, -6, 0, 0], [0, 0, 6, 6, 1, 0], [-5, 0, -6, 0, 5, -6], [6, 0, 0, 0, -6, 3], [0, -10, 4, 4, -9, 10], [6, -6, 0, -3, 0, 0], [0, -2, 2, 2, -11, 2], [3, 0, -10, 0, -3, -7], [2, -5, 0, 7, 0, 0], [0, 2, 2, 2, -8, -2], [-3, 0, 3, 0, 3, -5], [14, -1, 0, 4, 0, 0], [11, -6, 0, -3, 0, 0], [2, 0, 3, 0, -2, 3], [12, 0, -9, 0, -12, 1], [0, -4, 7, 7, 6, 4], [0, -11, 0, 0, -12, 11], [12, -5, 0, 5, 0, 0], [0, -6, -3, -3, 3, 6], [8, 0, 4, 0, -8, 7], [-15, 0, 3, 0, 15, -6], [-8, 0, -1, 0, 8, 5], [0, -5, 8, 8, 12, 5], [0, 5, -3, -3, -15, -5], [-10, 0, 0, 0, 10, 6], [-21, 0, 0, 0, 21, 5], [14, -3, 0, 7, 0, 0], [16, 0, -11, 0, -16, -2], [0, 2, 1, 1, -12, -2], [4, 7, 0, 2, 0, 0], [0, -4, 1, 1, 2, 4], [15, 0, 3, 0, -15, -3], [-2, 10, 0, 5, 0, 0], [0, -4, 1, 1, -13, 4], [0, -6, 6, 6, 16, 6], [0, -3, 6, 6, -15, 3], [17, 0, -11, 0, -17, 1], [30, 3, 0, 0, 0, 0], [-21, -10, 0, -2, 0, 0], [-10, 0, 2, 0, 10, 5], [-10, 0, 2, 0, 10, -10], [0, -3, 0, 3, 0, 0], [9, 0, 1, 0, -9, 7], [22, 0, 1, 0, -22, 11], [0, 1, 2, 2, 17, -1], [-8, 4, 0, -7, 0, 0], [-17, 0, 1, 0, 17, 3], [-2, -7, 0, 4, 0, 0], [0, 6, 5, 5, -2, -6], [0, -12, 6, 6, -20, 12], [9, 12, 0, -3, 0, 0], [0, 9, -8, -8, 11, -9], [19, 10, 0, 2, 0, 0], [-24, 1, 0, 2, 0, 0], [-7, 0, 0, 6, 0, 0], [3, 0, -4, 0, -3, 5], [-24, 1, 0, -12, 0, 0], [0, 14, 5, 5, -11, -14], [-3, 0, -6, 0, 3, -4], [0, 1, 14, 14, 12, -1], [1, -3, 0, 6, 0, 0], [-27, 0, 3, 0, 27, -2], [13, 13, 0, -1, 0, 0], [0, 2, 4, 4, -16, -2], [-7, 0, -1, 0, 7, -10], [0, 8, 4, 4, -5, -8], [25, 6, 0, 0, 0, 0], [0, 2, 1, 1, -7, -2], [15, 0, -10, 0, -15, 5], [-19, -14, 0, -1, 0, 0], [-11, 0, 10, 0, 11, 7], [0, 1, -4, -4, -6, -1], [-3, -11, 0, 15, 0, 0], [0, 1, 11, 11, -8, -1], [1, 0, 1, 0, -1, 7], [0, -9, -12, -12, 2, 9], [-1, 0, -3, 0, 1, 12], [10, 0, -2, 0, -10, 7], [-8, -14, 0, -10, 0, 0], [-29, 0, -7, 0, 29, -4], [-10, 14, 0, 10, 0, 0], [0, 0, -12, 0, 0, 6], [-14, 7, 0, 2, 0, 0], [20, 0, 3, 0, -20, 6], [0, 12, 0, 0, -9, -12], [4, 19, 0, -4, 0, 0], [0, 9, -9, -9, 18, -9], [0, 5, -11, -11, 9, -5], [-24, 7, 0, -1, 0, 0], [0, -10, 4, 4, -4, 10], [0, -7, -14, -14, -15, 7], [5, -14, 0, 4, 0, 0], [0, -14, -7, -7, 2, 14], [0, -7, 9, 9, 24, 7], [-6, -9, 0, -9, 0, 0], [13, 9, 0, 6, 0, 0], [-25, 0, -1, 0, 25, 3], [0, 0, 0, 0, -36, 0], [-22, -13, 0, 10, 0, 0], [9, -11, 0, -1, 0, 0], [7, 0, -14, 0, -7, -16], [19, 0, -9, 0, -19, 9], [-33, 0, 0, 0, 33, -7], [0, 2, 16, 16, 1, -2], [-12, -15, 0, -3, 0, 0], [0, 9, -3, -3, 20, -9], [-2, 10, 0, -1, 0, 0], [18, 8, 0, -11, 0, 0], [0, 9, -4, -4, 31, -9], [31, 0, 2, 0, -31, 8], [-8, 0, 10, 0, 8, 13], [23, 10, 0, 11, 0, 0], [4, 0, -14, 0, -4, 5], [12, -3, 0, 3, 0, 0], [0, -12, -5, -5, -22, 12], [23, 0, 8, 0, -23, 13], [18, 8, 0, -12, 0, 0], [0, -21, -9, -9, -8, 21], [-2, 13, 0, 5, 0, 0], [0, 16, -4, -4, -11, -16], [15, 0, -18, 0, -15, 3], [7, -3, 0, -3, 0, 0], [30, 0, 4, 0, -30, 16], [26, 13, 0, -1, 0, 0], [0, -1, 11, 11, 19, 1], [-4, 0, -10, 0, 4, -19], [0, -2, 5, 5, 7, 2], [0, -4, 16, 16, 5, 4], [-21, 0, 8, 0, 21, -4], [0, -2, -10, -10, -21, 2], [0, -2, -21, -21, -9, 2], [16, 0, 3, 0, -16, 0], [12, -14, 0, 9, 0, 0], [-11, -11, 0, 14, 0, 0], [2, 0, 9, 0, -2, -12], [17, 0, -6, 0, -17, 3], [-33, 0, -9, 0, 33, 9], [0, 10, 8, 8, 26, -10], [-3, -1, 0, 9, 0, 0], [0, 3, -18, -18, -27, -3], [1, 0, 1, 0, -1, -7], [-24, 8, 0, 1, 0, 0], [-6, 0, 11, 0, 6, 5], [0, 4, -1, -1, -51, -4], [-23, -9, 0, -1, 0, 0], [-14, -27, 0, 18, 0, 0], [14, 0, -1, 0, -14, 8], [-2, -15, 0, 12, 0, 0], [17, 0, 11, 0, -17, -16], [20, -2, 0, 7, 0, 0], [0, 23, 10, 10, -1, -23], [0, -17, -1, -1, -1, 17], [9, 0, 9, 0, -9, 3], [-2, 0, -17, 0, 2, -17], [-41, 4, 0, -16, 0, 0], [0, -2, -6, -6, 3, 2], [23, 0, -24, 0, -23, 0], [0, 17, -14, -14, 2, -17], [-7, 0, -9, 0, 7, 12], [-7, -12, 0, -5, 0, 0], [-11, 6, 0, -13, 0, 0], [-44, -2, 0, -4, 0, 0], [0, 3, 0, 0, -27, -3], [6, 0, 12, 0, -6, 18], [0, -4, -11, -11, 15, 4], [11, -10, 0, -11, 0, 0], [0, 5, -11, -11, 29, -5], [-14, 0, -6, 0, 14, -24], [47, 0, -7, 0, -47, -6], [0, -3, 5, 5, 34, 3], [-22, 0, 0, 0, 22, 3], [0, 1, -4, -4, -21, -1], [0, 0, -15, 0, 0, 20], [-18, -1, 0, -9, 0, 0], [0, 21, -12, -12, -12, -21], [0, -11, -4, -4, 20, 11], [0, -8, 20, 20, -10, 8], [0, -28, -4, -4, -20, 28], [-12, -4, 0, 1, 0, 0], [0, -6, -4, -4, 16, 6], [4, 0, -15, 0, -4, -15], [23, 0, 11, 0, -23, 10], [-7, 21, 0, -14, 0, 0], [-25, 0, 3, 0, 25, -24], [32, 0, -1, 0, -32, 20], [0, -2, 11, 11, -12, 2], [-24, -38, 0, 8, 0, 0], [0, 4, 14, 14, 18, -4], [0, 11, 15, 15, 15, -11], [30, 0, -6, 0, -30, -6], [-10, 0, -13, 0, 10, -28], [0, -16, -2, -2, 16, 16], [-7, 25, 0, 4, 0, 0], [0, -6, 3, 3, 30, 6], [-39, -5, 0, 9, 0, 0], [11, 0, 5, 0, -11, -4], [24, 3, 0, -12, 0, 0], [0, 14, 19, 19, 17, -14], [-4, 1, 0, 8, 0, 0], [0, -19, -8, -8, -19, 19], [22, 0, 13, 0, -22, 11], [-30, 7, 0, -28, 0, 0], [0, 6, -25, -25, -17, -6], [0, -3, 15, 15, 1, 3], [3, 0, 0, 6, 0, 0], [0, -18, 21, 21, -11, 18], [-10, 2, 0, -2, 0, 0], [0, 13, -12, -12, 21, -13], [8, 0, 3, 0, -8, 3], [33, 0, -9, 0, -33, -13], [12, 7, 0, -15, 0, 0], [-39, -17, 0, -7, 0, 0], [-47, 0, -5, 0, 47, 7], [8, 20, 0, 7, 0, 0], [17, 6, 0, -12, 0, 0], [-7, 0, 18, 0, 7, 3], [18, 0, -27, 0, -18, -7], [0, -12, 9, 9, 13, 12], [23, 12, 0, -12, 0, 0], [0, -24, 16, 16, -16, 24], [62, 0, -6, 0, -62, 6], [-19, 5, 0, -11, 0, 0], [2, 0, 5, 0, -2, -8], [0, -2, -7, -7, -31, 2], [0, 1, -14, -14, -13, -1], [0, -17, 17, 17, -18, 17], [-11, -9, 0, 9, 0, 0], [11, 0, 18, 0, -11, -12], [0, -1, 7, 7, 22, 1], [0, -17, 2, 2, -6, 17], [9, 26, 0, 10, 0, 0], [0, -19, 2, 2, 31, 19], [27, 0, 9, 0, -27, -21], [6, 26, 0, -2, 0, 0], [18, 0, 3, 0, -18, 8], [0, 10, 16, 16, -25, -10], [0, 20, -17, -17, -25, -20], [35, 0, 14, 0, -35, -5], [4, 0, -22, 0, -4, 5], [0, -11, -1, -1, 48, 11], [47, 5, 0, -5, 0, 0], [-5, 9, 0, 9, 0, 0], [0, 5, -8, -8, 44, -5], [16, 0, 8, 0, -16, 23], [-2, -11, 0, -1, 0, 0], [23, 0, 2, 0, -23, 7], [-2, 12, 0, 8, 0, 0], [0, 15, -27, -27, -3, -15], [0, -11, -7, -7, -5, 11], [-13, 0, 18, 0, 13, 9], [0, 28, -23, -23, 2, -28], [6, 0, -16, 0, -6, -1], [-36, 0, 0, 0, 36, -26], [-50, -10, 0, -1, 0, 0], [23, 0, 12, 0, -23, 18], [0, 22, 8, 8, 31, -22], [-7, -1, 0, 22, 0, 0], [-12, 19, 0, 20, 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_g();".
// 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_g(: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_g();".
// 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_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3024_r();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![121, -22, 59, 32, 23, 5, 1]>,<11,R![2209, -893, 455, -56, 23, -2, 1]>],Snew);
    return Vf;
end function;