LMFDB - Newform orbit 2312.1.x.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2312,1,Mod(35,2312)] 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("2312.35"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2312, base_ring=CyclotomicField(34)) chi = DirichletCharacter(H, H._module([17, 17, 14])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2312.x (of order $34$, degree $16$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$1.15383830921$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{34})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + \cdots + 1 $ x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{17}$
Projective field:	Galois closure of 17.1.39726964294200827244451673677113820402444926976.1

$q$-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_{34} q^{2} + ( - \zeta_{34}^{7} - \zeta_{34}^{5}) q^{3} + \zeta_{34}^{2} q^{4} + (\zeta_{34}^{8} + \zeta_{34}^{6}) q^{6} - \zeta_{34}^{3} q^{8} + (\zeta_{34}^{14} + \cdots + \zeta_{34}^{10}) q^{9} + \cdots + (2 \zeta_{34}^{16} + \cdots + 2 \zeta_{34}^{12}) q^{99} +O(q^{100}) $ q - z * q^2 + (-z^7 - z^5) * q^3 + z^2 * q^4 + (z^8 + z^6) * q^6 - z^3 * q^8 + (z^14 + z^12 + z^10) * q^9 + 2z^2 q^11 + (-z^9 - z^7) * q^12 + z^4 * q^16 - z * q^17 + (-z^15 - z^13 - z^11) * q^18 + (z^8 - z^7) * q^19 - 2z^3 q^22 + (z^10 + z^8) * q^24 - z^11 * q^25 + (-z^15 + z^4 + z^2 + 1) * q^27 - z^5 * q^32 + (-2z^9 - 2z^7) * q^33 + z^2 * q^34 + (z^16 + z^14 + z^12) * q^36 + (-z^9 + z^8) * q^38 + (-z^11 - z) * q^41 + (-z^15 - z^11) * q^43 + 2z^4 q^44 + (-z^11 - z^9) * q^48 - z^7 * q^49 + z^12 * q^50 + (z^8 + z^6) * q^51 + (z^16 - z^5 - z^3 - z) * q^54 + (-z^15 + z^14 - z^13 + z^12) * q^57 + (-z^13 - z^7) * q^59 + z^6 * q^64 + (2z^10 + 2z^8) * q^66 + (-z^15 + 1) * q^67 - z^3 * q^68 + (-z^15 - z^13 + 1) * q^72 + (-z^15 + z^10) * q^73 + (z^16 - z) * q^75 + (z^10 - z^9) * q^76 + (-z^11 - z^9 - z^7 - z^5 - z^3) * q^81 + (z^12 + z^2) * q^82 + (z^16 + z^6) * q^83 + (z^16 + z^12) * q^86 - 2z^5 q^88 + (z^10 - z) * q^89 + (z^12 + z^10) * q^96 + (-z^5 + z^2) * q^97 + z^8 * q^98 + (2z^16 + 2z^14 + 2z^12) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{6} - q^{8} - 3 q^{9} - 2 q^{11} - 2 q^{12} - q^{16} - q^{17} - 3 q^{18} - 2 q^{19} - 2 q^{22} - 2 q^{24} - q^{25} + 13 q^{27} - q^{32} - 4 q^{33} - q^{34} - 3 q^{36}+ \cdots - 6 q^{99}+O(q^{100}) $ 16 * q - q^2 - 2 * q^3 - q^4 - 2 * q^6 - q^8 - 3 * q^9 - 2 * q^11 - 2 * q^12 - q^16 - q^17 - 3 * q^18 - 2 * q^19 - 2 * q^22 - 2 * q^24 - q^25 + 13 * q^27 - q^32 - 4 * q^33 - q^34 - 3 * q^36 - 2 * q^38 - 2 * q^41 - 2 * q^43 - 2 * q^44 - 2 * q^48 - q^49 - q^50 - 2 * q^51 - 4 * q^54 - 4 * q^57 - 2 * q^59 - q^64 - 4 * q^66 + 15 * q^67 - q^68 + 14 * q^72 - 2 * q^73 - 2 * q^75 - 2 * q^76 - 5 * q^81 - 2 * q^82 - 2 * q^83 - 2 * q^86 - 2 * q^88 - 2 * q^89 - 2 * q^96 - 2 * q^97 - q^98 - 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2312\mathbb{Z}\right)^\times$.

$n$	$1157$	$1735$	$1737$
$\chi(n)$	$-1$	$-1$	$\zeta_{34}^{12}$

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} $

35.1

−0.739009	−	0.673696i
0.850217	+	0.526432i
−0.932472	−	0.361242i
0.982973	+	0.183750i
0.982973	−	0.183750i
−0.932472	+	0.361242i
0.850217	−	0.526432i
−0.739009	+	0.673696i
0.602635	−	0.798017i
−0.445738	+	0.895163i
0.273663	−	0.961826i
−0.0922684	+	0.995734i
−0.0922684	−	0.995734i
0.273663	+	0.961826i
−0.445738	−	0.895163i
0.602635	+	0.798017i

0.739009

0.673696i

−0.404479

−

1.42160i

0.0922684

0.995734i

0.658809

−

1.32307i

−0.602635

0.798017i

−1.00711

0.623578i

171.1

−0.850217

−

0.526432i

1.67148

0.312454i

0.445738

0.895163i

−1.25664

−

1.14558i

0.0922684

−

0.995734i

1.76375

0.683280i

307.1

0.932472

0.361242i

−1.12388

1.48826i

0.739009

0.673696i

−1.58561

0.981767i

0.445738

0.895163i

−0.678142

−

2.38342i

443.1

−0.982973

−

0.183750i

−0.876298

−

1.75984i

0.932472

0.361242i

0.538007

1.89090i

−0.850217

−

0.526432i

−1.72651

2.28628i

715.1

−0.982973

0.183750i

−0.876298

1.75984i

0.932472

−

0.361242i

0.538007

−

1.89090i

−0.850217

0.526432i

−1.72651

−

2.28628i

851.1

0.932472

−

0.361242i

−1.12388

−

1.48826i

0.739009

−

0.673696i

−1.58561

−

0.981767i

0.445738

−

0.895163i

−0.678142

2.38342i

987.1

−0.850217

0.526432i

1.67148

−

0.312454i

0.445738

−

0.895163i

−1.25664

1.14558i

0.0922684

0.995734i

1.76375

−

0.683280i

1123.1

0.739009

−

0.673696i

−0.404479

1.42160i

0.0922684

−

0.995734i

0.658809

1.32307i

−0.602635

−

0.798017i

−1.00711

−

0.623578i

1259.1

−0.602635

0.798017i

−0.890705

−

0.811985i

−0.273663

−

0.961826i

1.18475

−

0.221468i

0.932472

0.361242i

0.0417675

0.450743i

1395.1

0.445738

−

0.895163i

0.831277

−

0.322039i

−0.602635

−

0.798017i

0.0822551

−

0.887674i

−0.982973

0.183750i

−0.151696

0.138289i

1531.1

−0.273663

0.961826i

−0.0505009

0.544991i

−0.850217

−

0.526432i

−0.510366

−

0.197717i

0.739009

−

0.673696i

0.688508

0.128704i

1667.1

0.0922684

−

0.995734i

−0.156896

−

0.0971461i

−0.982973

−

0.183750i

−0.111208

0.147263i

−0.273663

0.961826i

−0.430559

−

0.864680i

1803.1

0.0922684

0.995734i

−0.156896

0.0971461i

−0.982973

0.183750i

−0.111208

−

0.147263i

−0.273663

−

0.961826i

−0.430559

0.864680i

1939.1

−0.273663

−

0.961826i

−0.0505009

−

0.544991i

−0.850217

0.526432i

−0.510366

0.197717i

0.739009

0.673696i

0.688508

−

0.128704i

2075.1

0.445738

0.895163i

0.831277

0.322039i

−0.602635

0.798017i

0.0822551

0.887674i

−0.982973

−

0.183750i

−0.151696

−

0.138289i

2211.1

−0.602635

−

0.798017i

−0.890705

0.811985i

−0.273663

0.961826i

1.18475

0.221468i

0.932472

−

0.361242i

0.0417675

−

0.450743i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
289.f	even	17	1	inner
2312.x	odd	34	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2312.1.x.a	✓	16
8.d	odd	2	1	CM	2312.1.x.a	✓	16
289.f	even	17	1	inner	2312.1.x.a	✓	16
2312.x	odd	34	1	inner	2312.1.x.a	✓	16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2312.1.x.a	✓	16	1.a	even	1	1	trivial
2312.1.x.a	✓	16	8.d	odd	2	1	CM
2312.1.x.a	✓	16	289.f	even	17	1	inner
2312.1.x.a	✓	16	2312.x	odd	34	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(2312, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} + T^{15} + \cdots + 1 $ T^16 + T^15 + T^14 + T^13 + T^12 + T^11 + T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$3$	$ T^{16} + 2 T^{15} + \cdots + 1 $ T^16 + 2T^15 + 4T^14 - 9T^13 - 18T^12 - 36T^11 + 30T^10 + 60T^9 + 120T^8 - 15T^7 - 47T^6 - 94T^5 + 50T^4 + 15T^3 + 30T^2 + 9*T + 1
$5$	$ T^{16} $ T^16
$7$	$ T^{16} $ T^16
$11$	$ T^{16} + 2 T^{15} + \cdots + 65536 $ T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 + 32T^11 + 64T^10 + 128T^9 + 256T^8 + 512T^7 + 1024T^6 + 2048T^5 + 4096T^4 + 8192T^3 + 16384T^2 + 32768*T + 65536
$13$	$ T^{16} $ T^16
$17$	$ T^{16} + T^{15} + \cdots + 1 $ T^16 + T^15 + T^14 + T^13 + T^12 + T^11 + T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$19$	$ T^{16} + 2 T^{15} + \cdots + 1 $ T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 + 32T^11 + 64T^10 + 77T^9 + 154T^8 + 70T^7 + 140T^6 + 25T^5 + 33T^4 - 36T^3 + 13T^2 + 9*T + 1
$23$	$ T^{16} $ T^16
$29$	$ T^{16} $ T^16
$31$	$ T^{16} $ T^16
$37$	$ T^{16} $ T^16
$41$	$ T^{16} + 2 T^{15} + \cdots + 1 $ T^16 + 2T^15 + 4T^14 + 8T^13 - T^12 - 2T^11 - 4T^10 - 25T^9 + 35T^8 + 70T^7 + 140T^6 + 59T^5 - T^4 - 2T^3 + 30T^2 - 8*T + 1
$43$	$ T^{16} + 2 T^{15} + \cdots + 1 $ T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 + 32T^11 + 64T^10 + 77T^9 + 154T^8 + 70T^7 + 140T^6 + 25T^5 + 33T^4 - 36T^3 + 13T^2 + 9*T + 1
$47$	$ T^{16} $ T^16
$53$	$ T^{16} $ T^16
$59$	$ T^{16} + 2 T^{15} + \cdots + 1 $ T^16 + 2T^15 + 4T^14 + 8T^13 - T^12 - 2T^11 - 4T^10 - 25T^9 + 35T^8 + 70T^7 + 140T^6 + 59T^5 - T^4 - 2T^3 + 30T^2 - 8*T + 1
$61$	$ T^{16} $ T^16
$67$	$ T^{16} - 15 T^{15} + \cdots + 1 $ T^16 - 15T^15 + 106T^14 - 468T^13 + 1444T^12 - 3300T^11 + 5776T^10 - 7896T^9 + 8518T^8 - 7274T^7 + 4900T^6 - 2576T^5 + 1036T^4 - 308T^3 + 64T^2 - 8*T + 1
$71$	$ T^{16} $ T^16
$73$	$ T^{16} + 2 T^{15} + \cdots + 1 $ T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 + 32T^11 + 47T^10 + 9T^9 + T^8 + 2T^7 + 4T^6 + 8T^5 + 67T^4 - 104T^3 + 47T^2 - 8T + 1
$79$	$ T^{16} $ T^16
$83$	$ T^{16} + 2 T^{15} + \cdots + 1 $ T^16 + 2T^15 + 4T^14 + 8T^13 - T^12 - 2T^11 - 4T^10 - 25T^9 + 35T^8 + 70T^7 + 140T^6 + 59T^5 - T^4 - 2T^3 + 30T^2 - 8*T + 1
$89$	$ T^{16} + 2 T^{15} + \cdots + 1 $ T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 + 32T^11 + 47T^10 + 9T^9 + T^8 + 2T^7 + 4T^6 + 8T^5 + 67T^4 - 104T^3 + 47T^2 - 8T + 1
$97$	$ T^{16} + 2 T^{15} + \cdots + 1 $ T^16 + 2T^15 + 4T^14 + 8T^13 - T^12 - 2T^11 - 4T^10 - 25T^9 + 35T^8 + 70T^7 + 140T^6 + 59T^5 - T^4 - 2T^3 + 30T^2 - 8*T + 1
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2312.x (of order \(34\), degree \(16\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{34} q^{2} + ( - \zeta_{34}^{7} - \zeta_{34}^{5}) q^{3} + \zeta_{34}^{2} q^{4} + (\zeta_{34}^{8} + \zeta_{34}^{6}) q^{6} - \zeta_{34}^{3} q^{8} + (\zeta_{34}^{14} + \cdots + \zeta_{34}^{10}) q^{9} + \cdots + (2 \zeta_{34}^{16} + \cdots + 2 \zeta_{34}^{12}) q^{99} +O(q^{100}) \) q - z * q^2 + (-z^7 - z^5) * q^3 + z^2 * q^4 + (z^8 + z^6) * q^6 - z^3 * q^8 + (z^14 + z^12 + z^10) * q^9 + 2z^2 q^11 + (-z^9 - z^7) * q^12 + z^4 * q^16 - z * q^17 + (-z^15 - z^13 - z^11) * q^18 + (z^8 - z^7) * q^19 - 2z^3 q^22 + (z^10 + z^8) * q^24 - z^11 * q^25 + (-z^15 + z^4 + z^2 + 1) * q^27 - z^5 * q^32 + (-2z^9 - 2z^7) * q^33 + z^2 * q^34 + (z^16 + z^14 + z^12) * q^36 + (-z^9 + z^8) * q^38 + (-z^11 - z) * q^41 + (-z^15 - z^11) * q^43 + 2z^4 q^44 + (-z^11 - z^9) * q^48 - z^7 * q^49 + z^12 * q^50 + (z^8 + z^6) * q^51 + (z^16 - z^5 - z^3 - z) * q^54 + (-z^15 + z^14 - z^13 + z^12) * q^57 + (-z^13 - z^7) * q^59 + z^6 * q^64 + (2z^10 + 2z^8) * q^66 + (-z^15 + 1) * q^67 - z^3 * q^68 + (-z^15 - z^13 + 1) * q^72 + (-z^15 + z^10) * q^73 + (z^16 - z) * q^75 + (z^10 - z^9) * q^76 + (-z^11 - z^9 - z^7 - z^5 - z^3) * q^81 + (z^12 + z^2) * q^82 + (z^16 + z^6) * q^83 + (z^16 + z^12) * q^86 - 2z^5 q^88 + (z^10 - z) * q^89 + (z^12 + z^10) * q^96 + (-z^5 + z^2) * q^97 + z^8 * q^98 + (2z^16 + 2z^14 + 2z^12) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{6} - q^{8} - 3 q^{9} - 2 q^{11} - 2 q^{12} - q^{16} - q^{17} - 3 q^{18} - 2 q^{19} - 2 q^{22} - 2 q^{24} - q^{25} + 13 q^{27} - q^{32} - 4 q^{33} - q^{34} - 3 q^{36}+ \cdots - 6 q^{99}+O(q^{100}) \) 16 * q - q^2 - 2 * q^3 - q^4 - 2 * q^6 - q^8 - 3 * q^9 - 2 * q^11 - 2 * q^12 - q^16 - q^17 - 3 * q^18 - 2 * q^19 - 2 * q^22 - 2 * q^24 - q^25 + 13 * q^27 - q^32 - 4 * q^33 - q^34 - 3 * q^36 - 2 * q^38 - 2 * q^41 - 2 * q^43 - 2 * q^44 - 2 * q^48 - q^49 - q^50 - 2 * q^51 - 4 * q^54 - 4 * q^57 - 2 * q^59 - q^64 - 4 * q^66 + 15 * q^67 - q^68 + 14 * q^72 - 2 * q^73 - 2 * q^75 - 2 * q^76 - 5 * q^81 - 2 * q^82 - 2 * q^83 - 2 * q^86 - 2 * q^88 - 2 * q^89 - 2 * q^96 - 2 * q^97 - q^98 - 6 * q^99

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2312.1.x.a

Level	$2312$
Weight	$1$
Character orbit	2312.x
Analytic conductor	$1.154$
Analytic rank	$0$
Dimension	$16$
Projective image	$D_{17}$
CM discriminant	-8
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.15383830921\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\Q(\zeta_{34})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + \cdots + 1 \) x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{17}\)
Projective field:	Galois closure of 17.1.39726964294200827244451673677113820402444926976.1

\(n\)	\(1157\)	\(1735\)	\(1737\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(\zeta_{34}^{12}\)

\(n\):		e.g. 2-40 or 80-90
Embeddings:		e.g. 1-3 or 35.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
289.f	even	17	1	inner
2312.x	odd	34	1	inner

$p$	$F_p(T)$
$2$	\( T^{16} + T^{15} + \cdots + 1 \) T^16 + T^15 + T^14 + T^13 + T^12 + T^11 + T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$3$	\( T^{16} + 2 T^{15} + \cdots + 1 \) T^16 + 2T^15 + 4T^14 - 9T^13 - 18T^12 - 36T^11 + 30T^10 + 60T^9 + 120T^8 - 15T^7 - 47T^6 - 94T^5 + 50T^4 + 15T^3 + 30T^2 + 9*T + 1
$5$	\( T^{16} \) T^16
$7$	\( T^{16} \) T^16
$11$	\( T^{16} + 2 T^{15} + \cdots + 65536 \) T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 + 32T^11 + 64T^10 + 128T^9 + 256T^8 + 512T^7 + 1024T^6 + 2048T^5 + 4096T^4 + 8192T^3 + 16384T^2 + 32768*T + 65536
$13$	\( T^{16} \) T^16
$17$	\( T^{16} + T^{15} + \cdots + 1 \) T^16 + T^15 + T^14 + T^13 + T^12 + T^11 + T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$19$	\( T^{16} + 2 T^{15} + \cdots + 1 \) T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 + 32T^11 + 64T^10 + 77T^9 + 154T^8 + 70T^7 + 140T^6 + 25T^5 + 33T^4 - 36T^3 + 13T^2 + 9*T + 1
$23$	\( T^{16} \) T^16
$29$	\( T^{16} \) T^16
$31$	\( T^{16} \) T^16
$37$	\( T^{16} \) T^16
$41$	\( T^{16} + 2 T^{15} + \cdots + 1 \) T^16 + 2T^15 + 4T^14 + 8T^13 - T^12 - 2T^11 - 4T^10 - 25T^9 + 35T^8 + 70T^7 + 140T^6 + 59T^5 - T^4 - 2T^3 + 30T^2 - 8*T + 1
$43$	\( T^{16} + 2 T^{15} + \cdots + 1 \) T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 + 32T^11 + 64T^10 + 77T^9 + 154T^8 + 70T^7 + 140T^6 + 25T^5 + 33T^4 - 36T^3 + 13T^2 + 9*T + 1
$47$	\( T^{16} \) T^16
$53$	\( T^{16} \) T^16
$59$	\( T^{16} + 2 T^{15} + \cdots + 1 \) T^16 + 2T^15 + 4T^14 + 8T^13 - T^12 - 2T^11 - 4T^10 - 25T^9 + 35T^8 + 70T^7 + 140T^6 + 59T^5 - T^4 - 2T^3 + 30T^2 - 8*T + 1
$61$	\( T^{16} \) T^16
$67$	\( T^{16} - 15 T^{15} + \cdots + 1 \) T^16 - 15T^15 + 106T^14 - 468T^13 + 1444T^12 - 3300T^11 + 5776T^10 - 7896T^9 + 8518T^8 - 7274T^7 + 4900T^6 - 2576T^5 + 1036T^4 - 308T^3 + 64T^2 - 8*T + 1
$71$	\( T^{16} \) T^16
$73$	\( T^{16} + 2 T^{15} + \cdots + 1 \) T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 + 32T^11 + 47T^10 + 9T^9 + T^8 + 2T^7 + 4T^6 + 8T^5 + 67T^4 - 104T^3 + 47T^2 - 8T + 1
$79$	\( T^{16} \) T^16
$83$	\( T^{16} + 2 T^{15} + \cdots + 1 \) T^16 + 2T^15 + 4T^14 + 8T^13 - T^12 - 2T^11 - 4T^10 - 25T^9 + 35T^8 + 70T^7 + 140T^6 + 59T^5 - T^4 - 2T^3 + 30T^2 - 8*T + 1
$89$	\( T^{16} + 2 T^{15} + \cdots + 1 \) T^16 + 2T^15 + 4T^14 + 8T^13 + 16T^12 + 32T^11 + 47T^10 + 9T^9 + T^8 + 2T^7 + 4T^6 + 8T^5 + 67T^4 - 104T^3 + 47T^2 - 8T + 1
$97$	\( T^{16} + 2 T^{15} + \cdots + 1 \) T^16 + 2T^15 + 4T^14 + 8T^13 - T^12 - 2T^11 - 4T^10 - 25T^9 + 35T^8 + 70T^7 + 140T^6 + 59T^5 - T^4 - 2T^3 + 30T^2 - 8*T + 1
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