LMFDB - Newform orbit 1620.1.bp.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1620,1,Mod(79,1620)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1620.79"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1620, base_ring=CyclotomicField(54)) chi = DirichletCharacter(H, H._module([27, 28, 27])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1620.bp (of order $54$, degree $18$, minimal)

Newform invariants

comment:select newform

sage:traces = [18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.808485320465$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\Q(\zeta_{54})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{9} + 1 $ x^18 - x^9 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{27}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{27} - \cdots)$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

The $q$-expansion and trace form are shown below.

$f(q)$	$=$	$ q - \zeta_{54}^{25} q^{2} + \zeta_{54}^{26} q^{3} - \zeta_{54}^{23} q^{4} + \zeta_{54}^{8} q^{5} + \zeta_{54}^{24} q^{6} + (\zeta_{54}^{12} - \zeta_{54}^{5}) q^{7} - \zeta_{54}^{21} q^{8} - \zeta_{54}^{25} q^{9} + \cdots + (\zeta_{54}^{22} + \cdots + \zeta_{54}^{8}) q^{98} +O(q^{100}) $ q - z^25 * q^2 + z^26 * q^3 - z^23 * q^4 + z^8 * q^5 + z^24 * q^6 + (z^12 - z^5) * q^7 - z^21 * q^8 - z^25 * q^9 + z^6 * q^10 + z^22 * q^12 + (z^10 - z^3) * q^14 - z^7 * q^15 - z^19 * q^16 - z^23 * q^18 + z^4 * q^20 + (-z^11 + z^4) * q^21 + (z^10 + 1) * q^23 + z^20 * q^24 + z^16 * q^25 + z^24 * q^27 + (z^8 - z) * q^28 + (-z^21 + z^20) * q^29 - z^5 * q^30 - z^17 * q^32 + (z^20 - z^13) * q^35 - z^21 * q^36 + z^2 * q^40 + (z^18 - z^13) * q^41 + (-z^9 + z^2) * q^42 + (z^16 + z^4) * q^43 + z^6 * q^45 + (-z^25 + z^8) * q^46 + (-z^15 - z^11) * q^47 + z^18 * q^48 + (z^24 - z^17 + z^10) * q^49 + z^14 * q^50 + z^22 * q^54 + (z^26 + z^6) * q^56 + (-z^19 + z^18) * q^58 - z^3 * q^60 + (-z^5 - z^3) * q^61 + (z^10 - z^3) * q^63 - z^15 * q^64 + (-z^13 + 1) * q^67 + (z^26 - z^9) * q^69 + (z^18 - z^11) * q^70 - z^19 * q^72 - z^15 * q^75 + q^80 - z^23 * q^81 + (z^16 - z^11) * q^82 + (z^14 - z^9) * q^83 + (-z^7 + 1) * q^84 + (z^14 + z^2) * q^86 + (z^20 - z^19) * q^87 + (-z^23 - z^19) * q^89 + z^4 * q^90 + (-z^23 + z^6) * q^92 + (-z^13 - z^9) * q^94 + z^16 * q^96 + (z^22 - z^15 + z^8) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 18 q^{23} - 9 q^{41} - 9 q^{42} - 9 q^{48} - 9 q^{58} + 18 q^{67} - 9 q^{69} - 9 q^{70} + 18 q^{80} - 9 q^{83} + 18 q^{84} - 9 q^{94}+O(q^{100}) $ 18 * q + 18 * q^23 - 9 * q^41 - 9 * q^42 - 9 * q^48 - 9 * q^58 + 18 * q^67 - 9 * q^69 - 9 * q^70 + 18 * q^80 - 9 * q^83 + 18 * q^84 - 9 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$-1$	$-1$	$-\zeta_{54}^{23}$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

79.1

0.686242	−	0.727374i
−0.893633	−	0.448799i
0.993238	−	0.116093i
0.993238	+	0.116093i
0.835488	+	0.549509i
−0.973045	+	0.230616i
0.286803	+	0.957990i
0.835488	−	0.549509i
−0.396080	+	0.918216i
−0.597159	+	0.802123i
−0.893633	+	0.448799i
0.286803	−	0.957990i
−0.973045	−	0.230616i
0.0581448	+	0.998308i
−0.597159	−	0.802123i
−0.396080	−	0.918216i
0.0581448	−	0.998308i
0.686242	+	0.727374i

−0.0581448

0.998308i

−0.686242

−

0.727374i

−0.993238

−

0.116093i

0.973045

−

0.230616i

0.766044

−

0.642788i

−0.342534

−

0.460103i

0.173648

−

0.984808i

−0.0581448

0.998308i

0.173648

0.984808i

139.1

0.597159

−

0.802123i

0.893633

−

0.448799i

−0.286803

−

0.957990i

−0.835488

−

0.549509i

0.173648

−

0.984808i

0.0798028

0.0845860i

−0.939693

−

0.342020i

0.597159

−

0.802123i

−0.939693

0.342020i

259.1

0.973045

0.230616i

−0.993238

−

0.116093i

0.893633

0.448799i

0.597159

−

0.802123i

−0.939693

−

0.342020i

−0.661840

−

0.435299i

0.766044

0.642788i

0.973045

0.230616i

0.766044

−

0.642788i

319.1

0.973045

−

0.230616i

−0.993238

0.116093i

0.893633

−

0.448799i

0.597159

0.802123i

−0.939693

0.342020i

−0.661840

0.435299i

0.766044

−

0.642788i

0.973045

−

0.230616i

0.766044

0.642788i

439.1

0.396080

−

0.918216i

−0.835488

0.549509i

−0.686242

−

0.727374i

−0.0581448

−

0.998308i

0.173648

0.984808i

1.73909

0.412172i

−0.939693

0.342020i

0.396080

−

0.918216i

−0.939693

−

0.342020i

499.1

0.893633

0.448799i

0.973045

0.230616i

0.597159

0.802123i

−0.286803

−

0.957990i

0.766044

0.642788i

−0.543613

−

1.26024i

0.173648

0.984808i

0.893633

0.448799i

0.173648

−

0.984808i

619.1

−0.835488

−

0.549509i

−0.286803

0.957990i

0.396080

0.918216i

−0.686242

−

0.727374i

0.766044

−

0.642788i

−1.93293

0.225927i

0.173648

−

0.984808i

−0.835488

−

0.549509i

0.173648

0.984808i

679.1

0.396080

0.918216i

−0.835488

−

0.549509i

−0.686242

0.727374i

−0.0581448

0.998308i

0.173648

−

0.984808i

1.73909

−

0.412172i

−0.939693

−

0.342020i

0.396080

0.918216i

−0.939693

0.342020i

799.1

−0.686242

0.727374i

0.396080

0.918216i

−0.0581448

−

0.998308i

−0.993238

−

0.116093i

−0.939693

−

0.342020i

1.06728

−

0.536009i

0.766044

0.642788i

−0.686242

0.727374i

0.766044

−

0.642788i

859.1

−0.286803

0.957990i

0.597159

0.802123i

−0.835488

−

0.549509i

0.396080

−

0.918216i

−0.939693

0.342020i

0.115503

1.98312i

0.766044

−

0.642788i

−0.286803

0.957990i

0.766044

0.642788i

979.1

0.597159

0.802123i

0.893633

0.448799i

−0.286803

0.957990i

−0.835488

0.549509i

0.173648

0.984808i

0.0798028

−

0.0845860i

−0.939693

0.342020i

0.597159

0.802123i

−0.939693

−

0.342020i

1039.1

−0.835488

0.549509i

−0.286803

−

0.957990i

0.396080

−

0.918216i

−0.686242

0.727374i

0.766044

0.642788i

−1.93293

−

0.225927i

0.173648

0.984808i

−0.835488

0.549509i

0.173648

−

0.984808i

1159.1

0.893633

−

0.448799i

0.973045

−

0.230616i

0.597159

−

0.802123i

−0.286803

0.957990i

0.766044

−

0.642788i

−0.543613

1.26024i

0.173648

−

0.984808i

0.893633

−

0.448799i

0.173648

0.984808i

1219.1

−0.993238

−

0.116093i

−0.0581448

0.998308i

0.973045

0.230616i

0.893633

−

0.448799i

0.173648

−

0.984808i

0.479241

−

1.60078i

−0.939693

−

0.342020i

−0.993238

−

0.116093i

−0.939693

0.342020i

1339.1

−0.286803

−

0.957990i

0.597159

−

0.802123i

−0.835488

0.549509i

0.396080

0.918216i

−0.939693

−

0.342020i

0.115503

−

1.98312i

0.766044

0.642788i

−0.286803

−

0.957990i

0.766044

−

0.642788i

1399.1

−0.686242

−

0.727374i

0.396080

−

0.918216i

−0.0581448

0.998308i

−0.993238

0.116093i

−0.939693

0.342020i

1.06728

0.536009i

0.766044

−

0.642788i

−0.686242

−

0.727374i

0.766044

0.642788i

1519.1

−0.993238

0.116093i

−0.0581448

−

0.998308i

0.973045

−

0.230616i

0.893633

0.448799i

0.173648

0.984808i

0.479241

1.60078i

−0.939693

0.342020i

−0.993238

0.116093i

−0.939693

−

0.342020i

1579.1

−0.0581448

−

0.998308i

−0.686242

0.727374i

−0.993238

0.116093i

0.973045

0.230616i

0.766044

0.642788i

−0.342534

0.460103i

0.173648

0.984808i

−0.0581448

−

0.998308i

0.173648

−

0.984808i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	CM by $\Q(\sqrt{-5}) $
81.g	even	27	1	inner
1620.bp	odd	54	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1620.1.bp.b	yes	18
4.b	odd	2	1		1620.1.bp.a	✓	18
5.b	even	2	1		1620.1.bp.a	✓	18
20.d	odd	2	1	CM	1620.1.bp.b	yes	18
81.g	even	27	1	inner	1620.1.bp.b	yes	18
324.n	odd	54	1		1620.1.bp.a	✓	18
405.t	even	54	1		1620.1.bp.a	✓	18
1620.bp	odd	54	1	inner	1620.1.bp.b	yes	18

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1620.1.bp.a	✓	18	4.b	odd	2	1
1620.1.bp.a	✓	18	5.b	even	2	1
1620.1.bp.a	✓	18	324.n	odd	54	1
1620.1.bp.a	✓	18	405.t	even	54	1
1620.1.bp.b	yes	18	1.a	even	1	1	trivial
1620.1.bp.b	yes	18	20.d	odd	2	1	CM
1620.1.bp.b	yes	18	81.g	even	27	1	inner
1620.1.bp.b	yes	18	1620.bp	odd	54	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{18} + 3 T_{7}^{15} - 18 T_{7}^{14} + 6 T_{7}^{12} - 9 T_{7}^{11} + 99 T_{7}^{10} + 8 T_{7}^{9} + \cdots + 1 $ T7^18 + 3*T7^15 - 18*T7^14 + 6*T7^12 - 9*T7^11 + 99*T7^10 + 8*T7^9 + 135*T7^8 - 135*T7^7 - 159*T7^6 + 63*T7^5 + 297*T7^4 + 159*T7^3 + 45*T7^2 - 9*T7 + 1 acting on $S_{1}^{\mathrm{new}}(1620, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{18} + T^{9} + 1 $ T^18 + T^9 + 1
$3$	$ T^{18} + T^{9} + 1 $ T^18 + T^9 + 1
$5$	$ T^{18} + T^{9} + 1 $ T^18 + T^9 + 1
$7$	$ T^{18} + 3 T^{15} + \cdots + 1 $ T^18 + 3T^15 - 18T^14 + 6T^12 - 9T^11 + 99T^10 + 8T^9 + 135T^8 - 135T^7 - 159T^6 + 63T^5 + 297T^4 + 159T^3 + 45T^2 - 9T + 1
$11$	$ T^{18} $ T^18
$13$	$ T^{18} $ T^18
$17$	$ T^{18} $ T^18
$19$	$ T^{18} $ T^18
$23$	$ T^{18} - 18 T^{17} + \cdots + 1 $ T^18 - 18T^17 + 153T^16 - 816T^15 + 3060T^14 - 8568T^13 + 18564T^12 - 31824T^11 + 43758T^10 - 48619T^9 + 43749T^8 - 31788T^7 + 18480T^6 - 8442T^5 + 2934T^4 - 732T^3 + 117T^2 - 9*T + 1
$29$	$ T^{18} + 3 T^{15} + \cdots + 1 $ T^18 + 3T^15 - 18T^14 + 6T^12 - 9T^11 + 99T^10 + 8T^9 + 135T^8 - 135T^7 - 159T^6 + 63T^5 + 297T^4 + 159T^3 + 45T^2 - 9T + 1
$31$	$ T^{18} $ T^18
$37$	$ T^{18} $ T^18
$41$	$ T^{18} + 9 T^{17} + \cdots + 1 $ T^18 + 9T^17 + 45T^16 + 156T^15 + 414T^14 + 882T^13 + 1554T^12 + 2304T^11 + 2907T^10 + 3140T^9 + 2898T^8 + 2232T^7 + 1470T^6 + 1008T^5 + 666T^4 + 240T^3 + 9T^2 - 9*T + 1
$43$	$ T^{18} + 9 T^{15} + \cdots + 1 $ T^18 + 9T^15 + 72T^12 + 323T^9 + 549T^6 - 36*T^3 + 1
$47$	$ T^{18} + 3 T^{15} + \cdots + 1 $ T^18 + 3T^15 + 18T^13 + 6T^12 - 18T^11 - 72T^10 + 8T^9 + 297T^8 - 135T^7 - 150T^6 - 135T^5 + 72T^4 + 168T^3 + 90T^2 + 9T + 1
$53$	$ T^{18} $ T^18
$59$	$ T^{18} $ T^18
$61$	$ T^{18} + 3 T^{15} + \cdots + 1 $ T^18 + 3T^15 + 9T^14 + 6T^12 - 63T^11 + 72T^10 + 8T^9 - 108T^8 - 216T^7 + 84T^6 + 63T^5 + 297T^4 + 159T^3 + 99T^2 + 18T + 1
$67$	$ T^{18} - 18 T^{17} + \cdots + 1 $ T^18 - 18T^17 + 153T^16 - 816T^15 + 3060T^14 - 8568T^13 + 18564T^12 - 31824T^11 + 43758T^10 - 48619T^9 + 43749T^8 - 31788T^7 + 18480T^6 - 8442T^5 + 2934T^4 - 732T^3 + 117T^2 - 9*T + 1
$71$	$ T^{18} $ T^18
$73$	$ T^{18} $ T^18
$79$	$ T^{18} $ T^18
$83$	$ T^{18} + 9 T^{17} + \cdots + 1 $ T^18 + 9T^17 + 45T^16 + 156T^15 + 414T^14 + 882T^13 + 1554T^12 + 2304T^11 + 2907T^10 + 3140T^9 + 2898T^8 + 2232T^7 + 1470T^6 + 1008T^5 + 666T^4 + 240T^3 + 9T^2 - 9*T + 1
$89$	$ T^{18} + 9 T^{13} + \cdots + 1 $ T^18 + 9T^13 + 30T^12 + 27T^11 + 9T^10 + 2T^9 + 81T^8 + 540T^7 + 1386T^6 + 1782T^5 + 1278T^4 + 516T^3 + 108T^2 + 9*T + 1
$97$	$ T^{18} $ T^18
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Elliptic curves over $\Q$ over number fields
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Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1620.bp (of order \(54\), degree \(18\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{54}^{25} q^{2} + \zeta_{54}^{26} q^{3} - \zeta_{54}^{23} q^{4} + \zeta_{54}^{8} q^{5} + \zeta_{54}^{24} q^{6} + (\zeta_{54}^{12} - \zeta_{54}^{5}) q^{7} - \zeta_{54}^{21} q^{8} - \zeta_{54}^{25} q^{9} + \cdots + (\zeta_{54}^{22} + \cdots + \zeta_{54}^{8}) q^{98} +O(q^{100}) \) q - z^25 * q^2 + z^26 * q^3 - z^23 * q^4 + z^8 * q^5 + z^24 * q^6 + (z^12 - z^5) * q^7 - z^21 * q^8 - z^25 * q^9 + z^6 * q^10 + z^22 * q^12 + (z^10 - z^3) * q^14 - z^7 * q^15 - z^19 * q^16 - z^23 * q^18 + z^4 * q^20 + (-z^11 + z^4) * q^21 + (z^10 + 1) * q^23 + z^20 * q^24 + z^16 * q^25 + z^24 * q^27 + (z^8 - z) * q^28 + (-z^21 + z^20) * q^29 - z^5 * q^30 - z^17 * q^32 + (z^20 - z^13) * q^35 - z^21 * q^36 + z^2 * q^40 + (z^18 - z^13) * q^41 + (-z^9 + z^2) * q^42 + (z^16 + z^4) * q^43 + z^6 * q^45 + (-z^25 + z^8) * q^46 + (-z^15 - z^11) * q^47 + z^18 * q^48 + (z^24 - z^17 + z^10) * q^49 + z^14 * q^50 + z^22 * q^54 + (z^26 + z^6) * q^56 + (-z^19 + z^18) * q^58 - z^3 * q^60 + (-z^5 - z^3) * q^61 + (z^10 - z^3) * q^63 - z^15 * q^64 + (-z^13 + 1) * q^67 + (z^26 - z^9) * q^69 + (z^18 - z^11) * q^70 - z^19 * q^72 - z^15 * q^75 + q^80 - z^23 * q^81 + (z^16 - z^11) * q^82 + (z^14 - z^9) * q^83 + (-z^7 + 1) * q^84 + (z^14 + z^2) * q^86 + (z^20 - z^19) * q^87 + (-z^23 - z^19) * q^89 + z^4 * q^90 + (-z^23 + z^6) * q^92 + (-z^13 - z^9) * q^94 + z^16 * q^96 + (z^22 - z^15 + z^8) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 18 q^{23} - 9 q^{41} - 9 q^{42} - 9 q^{48} - 9 q^{58} + 18 q^{67} - 9 q^{69} - 9 q^{70} + 18 q^{80} - 9 q^{83} + 18 q^{84} - 9 q^{94}+O(q^{100}) \) 18 * q + 18 * q^23 - 9 * q^41 - 9 * q^42 - 9 * q^48 - 9 * q^58 + 18 * q^67 - 9 * q^69 - 9 * q^70 + 18 * q^80 - 9 * q^83 + 18 * q^84 - 9 * q^94

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Newspace parameters

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Character values

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Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

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Label	1620.1.bp.b

Level	$1620$
Weight	$1$
Character orbit	1620.bp
Analytic conductor	$0.808$
Analytic rank	$0$
Dimension	$18$
Projective image	$D_{27}$
CM discriminant	-20
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.808485320465\)
Analytic rank:	\(0\)
Dimension:	\(18\)
Coefficient field:	\(\Q(\zeta_{54})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{18} - x^{9} + 1 \) x^18 - x^9 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{27}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{27} - \cdots)\)

\(n\)	\(811\)	\(1297\)	\(1541\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-\zeta_{54}^{23}\)

\(n\):		e.g. 2-40 or 80-90
Embeddings:		e.g. 1-3 or 79.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{18} + T^{9} + 1 \) T^18 + T^9 + 1
$3$	\( T^{18} + T^{9} + 1 \) T^18 + T^9 + 1
$5$	\( T^{18} + T^{9} + 1 \) T^18 + T^9 + 1
$7$	\( T^{18} + 3 T^{15} + \cdots + 1 \) T^18 + 3T^15 - 18T^14 + 6T^12 - 9T^11 + 99T^10 + 8T^9 + 135T^8 - 135T^7 - 159T^6 + 63T^5 + 297T^4 + 159T^3 + 45T^2 - 9T + 1
$11$	\( T^{18} \) T^18
$13$	\( T^{18} \) T^18
$17$	\( T^{18} \) T^18
$19$	\( T^{18} \) T^18
$23$	\( T^{18} - 18 T^{17} + \cdots + 1 \) T^18 - 18T^17 + 153T^16 - 816T^15 + 3060T^14 - 8568T^13 + 18564T^12 - 31824T^11 + 43758T^10 - 48619T^9 + 43749T^8 - 31788T^7 + 18480T^6 - 8442T^5 + 2934T^4 - 732T^3 + 117T^2 - 9*T + 1
$29$	\( T^{18} + 3 T^{15} + \cdots + 1 \) T^18 + 3T^15 - 18T^14 + 6T^12 - 9T^11 + 99T^10 + 8T^9 + 135T^8 - 135T^7 - 159T^6 + 63T^5 + 297T^4 + 159T^3 + 45T^2 - 9T + 1
$31$	\( T^{18} \) T^18
$37$	\( T^{18} \) T^18
$41$	\( T^{18} + 9 T^{17} + \cdots + 1 \) T^18 + 9T^17 + 45T^16 + 156T^15 + 414T^14 + 882T^13 + 1554T^12 + 2304T^11 + 2907T^10 + 3140T^9 + 2898T^8 + 2232T^7 + 1470T^6 + 1008T^5 + 666T^4 + 240T^3 + 9T^2 - 9*T + 1
$43$	\( T^{18} + 9 T^{15} + \cdots + 1 \) T^18 + 9T^15 + 72T^12 + 323T^9 + 549T^6 - 36*T^3 + 1
$47$	\( T^{18} + 3 T^{15} + \cdots + 1 \) T^18 + 3T^15 + 18T^13 + 6T^12 - 18T^11 - 72T^10 + 8T^9 + 297T^8 - 135T^7 - 150T^6 - 135T^5 + 72T^4 + 168T^3 + 90T^2 + 9T + 1
$53$	\( T^{18} \) T^18
$59$	\( T^{18} \) T^18
$61$	\( T^{18} + 3 T^{15} + \cdots + 1 \) T^18 + 3T^15 + 9T^14 + 6T^12 - 63T^11 + 72T^10 + 8T^9 - 108T^8 - 216T^7 + 84T^6 + 63T^5 + 297T^4 + 159T^3 + 99T^2 + 18T + 1
$67$	\( T^{18} - 18 T^{17} + \cdots + 1 \) T^18 - 18T^17 + 153T^16 - 816T^15 + 3060T^14 - 8568T^13 + 18564T^12 - 31824T^11 + 43758T^10 - 48619T^9 + 43749T^8 - 31788T^7 + 18480T^6 - 8442T^5 + 2934T^4 - 732T^3 + 117T^2 - 9*T + 1
$71$	\( T^{18} \) T^18
$73$	\( T^{18} \) T^18
$79$	\( T^{18} \) T^18
$83$	\( T^{18} + 9 T^{17} + \cdots + 1 \) T^18 + 9T^17 + 45T^16 + 156T^15 + 414T^14 + 882T^13 + 1554T^12 + 2304T^11 + 2907T^10 + 3140T^9 + 2898T^8 + 2232T^7 + 1470T^6 + 1008T^5 + 666T^4 + 240T^3 + 9T^2 - 9*T + 1
$89$	\( T^{18} + 9 T^{13} + \cdots + 1 \) T^18 + 9T^13 + 30T^12 + 27T^11 + 9T^10 + 2T^9 + 81T^8 + 540T^7 + 1386T^6 + 1782T^5 + 1278T^4 + 516T^3 + 108T^2 + 9*T + 1
$97$	\( T^{18} \) T^18
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