LMFDB - Newform orbit 2952.1.et.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2952,1,Mod(11,2952)] 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("2952.11"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2952, base_ring=CyclotomicField(120)) chi = DirichletCharacter(H, H._module([60, 60, 20, 9])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 2952 = 2^{3} \cdot 3^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2952.et (of order $120$, degree $32$, minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$1.47323991729$
Analytic rank:	$0$
Dimension:	$32$
Coefficient field:	$\Q(\zeta_{120})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{32} + x^{28} - x^{20} - x^{16} - x^{12} + x^{4} + 1 $ x^32 + x^28 - x^20 - x^16 - x^12 + x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{120}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{120} - \cdots)$

$q$-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_{120}^{29} q^{2} - \zeta_{120}^{44} q^{3} + \zeta_{120}^{58} q^{4} + \zeta_{120}^{13} q^{6} - \zeta_{120}^{27} q^{8} - \zeta_{120}^{28} q^{9} + ( - \zeta_{120}^{34} + \zeta_{120}^{15}) q^{11} + \cdots + ( - \zeta_{120}^{43} - \zeta_{120}^{2}) q^{99} +O(q^{100}) $ q + z^29 * q^2 - z^44 * q^3 + z^58 * q^4 + z^13 * q^6 - z^27 * q^8 - z^28 * q^9 + (-z^34 + z^15) * q^11 + z^42 * q^12 - z^56 * q^16 + (-z^32 - z^7) * q^17 - z^57 * q^18 + (z^50 - z^37) * q^19 + (z^44 + z^3) * q^22 - z^11 * q^24 - z^26 * q^25 - z^12 * q^27 + z^25 * q^32 + (-z^59 - z^18) * q^33 + (-z^36 + z) * q^34 + z^26 * q^36 + (-z^19 + z^6) * q^38 + z^23 * q^41 + (z^33 - z^25) * q^43 + (z^32 - z^13) * q^44 - z^40 * q^48 + z^37 * q^49 - z^55 * q^50 + (z^51 - z^16) * q^51 - z^41 * q^54 + (z^34 - z^21) * q^57 + (-z^39 - z^29) * q^59 + z^54 * q^64 + (-z^47 + z^28) * q^66 + (-z^51 + z^20) * q^67 + (z^30 + z^5) * q^68 + z^55 * q^72 + (z^19 + z^11) * q^73 - z^10 * q^75 + (-z^48 + z^35) * q^76 + z^56 * q^81 + z^52 * q^82 + (-z^38 + z^2) * q^83 + (-z^54 - z^2) * q^86 + (-z^42 - z) * q^88 + (-z^33 - z^24) * q^89 + z^9 * q^96 + (z^22 + z^9) * q^97 - z^6 * q^98 + (-z^43 - z^2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 4 q^{3} + 4 q^{9} - 4 q^{16} - 4 q^{17} - 4 q^{22} - 8 q^{27} - 8 q^{34} + 4 q^{44} + 16 q^{48} - 4 q^{51} - 4 q^{66} + 16 q^{67} + 8 q^{76} + 4 q^{81} - 4 q^{82} + 8 q^{89}+O(q^{100}) $ 32 * q + 4 * q^3 + 4 * q^9 - 4 * q^16 - 4 * q^17 - 4 * q^22 - 8 * q^27 - 8 * q^34 + 4 * q^44 + 16 * q^48 - 4 * q^51 - 4 * q^66 + 16 * q^67 + 8 * q^76 + 4 * q^81 - 4 * q^82 + 8 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2952\mathbb{Z}\right)^\times$.

$n$	$1441$	$1477$	$2215$	$2297$
$\chi(n)$	$\zeta_{120}^{3}$	$-1$	$-1$	$-\zeta_{120}^{40}$

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} $

11.1

−0.358368	−	0.933580i
0.0523360	+	0.998630i
0.933580	+	0.358368i
0.777146	−	0.629320i
−0.838671	+	0.544639i
0.629320	−	0.777146i
−0.838671	−	0.544639i
−0.998630	+	0.0523360i
−0.933580	+	0.358368i
−0.629320	+	0.777146i
0.544639	−	0.838671i
−0.629320	−	0.777146i
0.838671	−	0.544639i
−0.933580	−	0.358368i
0.629320	+	0.777146i
−0.544639	+	0.838671i
−0.0523360	+	0.998630i
0.544639	+	0.838671i
0.933580	−	0.358368i
0.777146	+	0.629320i

0.933580

0.358368i

0.913545

−

0.406737i

0.743145

0.669131i

0.998630

−

0.0523360i

0.453990

0.891007i

0.669131

−

0.743145i

227.1

0.998630

0.0523360i

0.669131

0.743145i

0.994522

0.104528i

0.629320

0.777146i

0.987688

0.156434i

−0.104528

0.994522i

275.1

−0.358368

−

0.933580i

0.913545

0.406737i

−0.743145

0.669131i

0.0523360

−

0.998630i

0.891007

0.453990i

0.669131

0.743145i

299.1

0.629320

−

0.777146i

−0.104528

−

0.994522i

−0.207912

−

0.978148i

−0.838671

−

0.544639i

−0.891007

−

0.453990i

−0.978148

0.207912i

347.1

0.544639

−

0.838671i

−0.978148

0.207912i

−0.406737

−

0.913545i

−0.358368

0.933580i

−0.987688

−

0.156434i

0.913545

−

0.406737i

563.1

0.777146

−

0.629320i

−0.104528

0.994522i

0.207912

−

0.978148i

0.544639

0.838671i

−0.453990

−

0.891007i

−0.978148

−

0.207912i

587.1

0.544639

0.838671i

−0.978148

−

0.207912i

−0.406737

0.913545i

−0.358368

−

0.933580i

−0.987688

0.156434i

0.913545

0.406737i

731.1

−0.0523360

0.998630i

0.669131

0.743145i

−0.994522

−

0.104528i

−0.777146

0.629320i

0.156434

−

0.987688i

−0.104528

0.994522i

803.1

0.358368

−

0.933580i

0.913545

−

0.406737i

−0.743145

−

0.669131i

−0.0523360

−

0.998630i

−0.891007

0.453990i

0.669131

−

0.743145i

995.1

−0.777146

0.629320i

−0.104528

0.994522i

0.207912

−

0.978148i

−0.544639

−

0.838671i

0.453990

0.891007i

−0.978148

−

0.207912i

1019.1

−0.838671

0.544639i

−0.978148

−

0.207912i

0.406737

−

0.913545i

0.933580

−

0.358368i

0.156434

0.987688i

0.913545

0.406737i

1163.1

−0.777146

−

0.629320i

−0.104528

−

0.994522i

0.207912

0.978148i

−0.544639

0.838671i

0.453990

−

0.891007i

−0.978148

0.207912i

1211.1

−0.544639

0.838671i

−0.978148

0.207912i

−0.406737

−

0.913545i

0.358368

−

0.933580i

0.987688

0.156434i

0.913545

−

0.406737i

1283.1

0.358368

0.933580i

0.913545

0.406737i

−0.743145

0.669131i

−0.0523360

0.998630i

−0.891007

−

0.453990i

0.669131

0.743145i

1379.1

0.777146

0.629320i

−0.104528

−

0.994522i

0.207912

0.978148i

0.544639

−

0.838671i

−0.453990

0.891007i

−0.978148

0.207912i

1523.1

0.838671

−

0.544639i

−0.978148

−

0.207912i

0.406737

−

0.913545i

−0.933580

0.358368i

−0.156434

−

0.987688i

0.913545

0.406737i

1571.1

−0.998630

0.0523360i

0.669131

−

0.743145i

0.994522

−

0.104528i

−0.629320

0.777146i

−0.987688

0.156434i

−0.104528

−

0.994522i

1715.1

−0.838671

−

0.544639i

−0.978148

0.207912i

0.406737

0.913545i

0.933580

0.358368i

0.156434

−

0.987688i

0.913545

−

0.406737i

1739.1

−0.358368

0.933580i

0.913545

−

0.406737i

−0.743145

−

0.669131i

0.0523360

0.998630i

0.891007

−

0.453990i

0.669131

−

0.743145i

1787.1

0.629320

0.777146i

−0.104528

0.994522i

−0.207912

0.978148i

−0.838671

0.544639i

−0.891007

0.453990i

−0.978148

−

0.207912i

See all 32 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
369.bf	even	120	1	inner
2952.et	odd	120	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2952.1.et.b	yes	32
8.d	odd	2	1	CM	2952.1.et.b	yes	32
9.d	odd	6	1		2952.1.et.a	✓	32
41.h	odd	40	1		2952.1.et.a	✓	32
72.l	even	6	1		2952.1.et.a	✓	32
328.bd	even	40	1		2952.1.et.a	✓	32
369.bf	even	120	1	inner	2952.1.et.b	yes	32
2952.et	odd	120	1	inner	2952.1.et.b	yes	32

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2952.1.et.a	✓	32	9.d	odd	6	1
2952.1.et.a	✓	32	41.h	odd	40	1
2952.1.et.a	✓	32	72.l	even	6	1
2952.1.et.a	✓	32	328.bd	even	40	1
2952.1.et.b	yes	32	1.a	even	1	1	trivial
2952.1.et.b	yes	32	8.d	odd	2	1	CM
2952.1.et.b	yes	32	369.bf	even	120	1	inner
2952.1.et.b	yes	32	2952.et	odd	120	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{11}^{32} + 2 T_{11}^{30} + 4 T_{11}^{29} + 9 T_{11}^{28} + 24 T_{11}^{27} + 12 T_{11}^{26} + 16 T_{11}^{25} + \cdots + 1 $ T11^32 + 2*T11^30 + 4*T11^29 + 9*T11^28 + 24*T11^27 + 12*T11^26 + 16*T11^25 + 127*T11^24 - 24*T11^23 + 322*T11^22 + 388*T11^21 - 88*T11^20 + 2936*T11^19 + 1184*T11^18 + 7616*T11^17 + 13971*T11^16 - 1536*T11^15 + 8346*T11^14 - 8264*T11^13 - 23988*T11^12 + 15120*T11^11 + 1460*T11^10 - 1532*T11^9 + 13374*T11^8 + 784*T11^7 + 3416*T11^6 - 3820*T11^5 + 100*T11^4 + 840*T11^3 + 304*T11^2 - 24*T11 + 1 acting on $S_{1}^{\mathrm{new}}(2952, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{32} + T^{28} + \cdots + 1 $ T^32 + T^28 - T^20 - T^16 - T^12 + T^4 + 1
$3$	$ (T^{8} - T^{7} + T^{5} + \cdots + 1)^{4} $ (T^8 - T^7 + T^5 - T^4 + T^3 - T + 1)^4
$5$	$ T^{32} $ T^32
$7$	$ T^{32} $ T^32
$11$	$ T^{32} + 2 T^{30} + \cdots + 1 $ T^32 + 2T^30 + 4T^29 + 9T^28 + 24T^27 + 12T^26 + 16T^25 + 127T^24 - 24T^23 + 322T^22 + 388T^21 - 88T^20 + 2936T^19 + 1184T^18 + 7616T^17 + 13971T^16 - 1536T^15 + 8346T^14 - 8264T^13 - 23988T^12 + 15120T^11 + 1460T^10 - 1532T^9 + 13374T^8 + 784T^7 + 3416T^6 - 3820T^5 + 100T^4 + 840T^3 + 304T^2 - 24T + 1
$13$	$ T^{32} $ T^32
$17$	$ T^{32} + 4 T^{31} + \cdots + 1 $ T^32 + 4T^31 + 6T^30 - 14T^28 - 56T^27 - 134T^26 - 160T^25 - 45T^24 + 48T^23 + 46T^22 + 304T^21 + 1648T^20 + 5028T^19 + 8472T^18 + 8312T^17 + 4809T^16 + 7756T^15 + 30120T^14 + 41952T^13 + 29392T^12 + 16424T^11 + 8194T^10 + 45600T^9 + 67827T^8 + 29620T^7 + 9670T^6 + 2648T^5 + 658T^4 + 180T^3 + 42T^2 + 8T + 1
$19$	$ T^{32} - 8 T^{30} + \cdots + 1 $ T^32 - 8T^30 + 4T^29 + 37T^28 - 40T^27 - 122T^26 + 184T^25 + 216T^24 - 672T^23 - 74T^22 + 1712T^21 - 1043T^20 - 1728T^19 + 6420T^18 - 5024T^17 - 8809T^16 + 16008T^15 - 19256T^14 + 10012T^13 + 24869T^12 - 36712T^11 + 36448T^10 - 29220T^9 + 19158T^8 - 6312T^7 + 10158T^6 - 4852T^5 + 4187T^4 - 1216T^3 + 298T^2 + 16T + 1
$23$	$ T^{32} $ T^32
$29$	$ T^{32} $ T^32
$31$	$ T^{32} $ T^32
$37$	$ T^{32} $ T^32
$41$	$ T^{32} + T^{28} + \cdots + 1 $ T^32 + T^28 - T^20 - T^16 - T^12 + T^4 + 1
$43$	$ T^{32} + 4 T^{28} + \cdots + 1 $ T^32 + 4T^28 + 250T^24 + 1706T^20 + 6999T^16 + 4546T^12 + 835T^8 - 61*T^4 + 1
$47$	$ T^{32} $ T^32
$53$	$ T^{32} $ T^32
$59$	$ T^{32} + 4 T^{30} + \cdots + 1 $ T^32 + 4T^30 + T^28 - 52T^26 - 208T^24 - 776T^22 - 193T^20 + 10088T^18 + 40351T^16 + 10088T^14 - 193T^12 - 776T^10 - 208T^8 - 52T^6 + T^4 + 4*T^2 + 1
$61$	$ T^{32} $ T^32
$67$	$ T^{32} - 16 T^{31} + \cdots + 1 $ T^32 - 16T^31 + 136T^30 - 800T^29 + 3618T^28 - 13300T^27 + 41130T^26 - 109512T^25 + 255288T^24 - 527676T^23 + 976746T^22 - 1631928T^21 + 2476556T^20 - 3430376T^19 + 4353008T^18 - 5074656T^17 + 5446721T^16 - 5391148T^15 + 4925106T^14 - 4151096T^13 + 3224214T^12 - 2310692T^11 + 1533262T^10 - 932968T^9 + 505869T^8 - 246052T^7 + 117554T^6 - 55904T^5 + 21709T^4 - 5512T^3 + 762T^2 - 44*T + 1
$71$	$ T^{32} $ T^32
$73$	$ (T^{16} + 29 T^{12} + \cdots + 1)^{2} $ (T^16 + 29T^12 + 246T^8 + 524*T^4 + 1)^2
$79$	$ T^{32} $ T^32
$83$	$ (T^{8} - 3 T^{6} + 8 T^{4} + \cdots + 1)^{4} $ (T^8 - 3T^6 + 8T^4 - 3*T^2 + 1)^4
$89$	$ (T^{16} - 4 T^{15} + \cdots + 1)^{2} $ (T^16 - 4T^15 + 10T^14 - 20T^13 + 34T^12 - 44T^11 + 82T^10 - 160T^9 + 256T^8 - 304T^7 + 162T^6 + 72T^5 - 131T^4 + 32T^3 + 46T^2 + 8*T + 1)^2
$97$	$ T^{32} + 2 T^{30} + \cdots + 1 $ T^32 + 2T^30 + 16T^29 - T^28 + 16T^27 + 162T^26 - 56T^25 + 112T^24 + 944T^23 - 418T^22 + 212T^21 + 4672T^20 - 2376T^19 + 1304T^18 + 13524T^17 - 2614T^16 - 8824T^15 + 35076T^14 - 9736T^13 - 19493T^12 + 29760T^11 - 2820T^10 - 12648T^9 + 5124T^8 + 9096T^7 + 2956T^6 + 1140T^5 + 2710T^4 + 280T^3 + 314T^2 - 36*T + 1
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Classical	Maass
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
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Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2952 = 2^{3} \cdot 3^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2952.et (of order \(120\), degree \(32\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{120}^{29} q^{2} - \zeta_{120}^{44} q^{3} + \zeta_{120}^{58} q^{4} + \zeta_{120}^{13} q^{6} - \zeta_{120}^{27} q^{8} - \zeta_{120}^{28} q^{9} + ( - \zeta_{120}^{34} + \zeta_{120}^{15}) q^{11} + \cdots + ( - \zeta_{120}^{43} - \zeta_{120}^{2}) q^{99} +O(q^{100}) \) q + z^29 * q^2 - z^44 * q^3 + z^58 * q^4 + z^13 * q^6 - z^27 * q^8 - z^28 * q^9 + (-z^34 + z^15) * q^11 + z^42 * q^12 - z^56 * q^16 + (-z^32 - z^7) * q^17 - z^57 * q^18 + (z^50 - z^37) * q^19 + (z^44 + z^3) * q^22 - z^11 * q^24 - z^26 * q^25 - z^12 * q^27 + z^25 * q^32 + (-z^59 - z^18) * q^33 + (-z^36 + z) * q^34 + z^26 * q^36 + (-z^19 + z^6) * q^38 + z^23 * q^41 + (z^33 - z^25) * q^43 + (z^32 - z^13) * q^44 - z^40 * q^48 + z^37 * q^49 - z^55 * q^50 + (z^51 - z^16) * q^51 - z^41 * q^54 + (z^34 - z^21) * q^57 + (-z^39 - z^29) * q^59 + z^54 * q^64 + (-z^47 + z^28) * q^66 + (-z^51 + z^20) * q^67 + (z^30 + z^5) * q^68 + z^55 * q^72 + (z^19 + z^11) * q^73 - z^10 * q^75 + (-z^48 + z^35) * q^76 + z^56 * q^81 + z^52 * q^82 + (-z^38 + z^2) * q^83 + (-z^54 - z^2) * q^86 + (-z^42 - z) * q^88 + (-z^33 - z^24) * q^89 + z^9 * q^96 + (z^22 + z^9) * q^97 - z^6 * q^98 + (-z^43 - z^2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 4 q^{3} + 4 q^{9} - 4 q^{16} - 4 q^{17} - 4 q^{22} - 8 q^{27} - 8 q^{34} + 4 q^{44} + 16 q^{48} - 4 q^{51} - 4 q^{66} + 16 q^{67} + 8 q^{76} + 4 q^{81} - 4 q^{82} + 8 q^{89}+O(q^{100}) \) 32 * q + 4 * q^3 + 4 * q^9 - 4 * q^16 - 4 * q^17 - 4 * q^22 - 8 * q^27 - 8 * q^34 + 4 * q^44 + 16 * q^48 - 4 * q^51 - 4 * q^66 + 16 * q^67 + 8 * q^76 + 4 * q^81 - 4 * q^82 + 8 * q^89

Introduction

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Modular forms

Varieties

Fields

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Database

Properties

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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	2952.1.et.b

Level	$2952$
Weight	$1$
Character orbit	2952.et
Analytic conductor	$1.473$
Analytic rank	$0$
Dimension	$32$
Projective image	$D_{120}$
CM discriminant	-8
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.47323991729\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Coefficient field:	\(\Q(\zeta_{120})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{32} + x^{28} - x^{20} - x^{16} - x^{12} + x^{4} + 1 \) x^32 + x^28 - x^20 - x^16 - x^12 + x^4 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{120}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{120} - \cdots)\)

\(n\)	\(1441\)	\(1477\)	\(2215\)	\(2297\)
\(\chi(n)\)	\(\zeta_{120}^{3}\)	\(-1\)	\(-1\)	\(-\zeta_{120}^{40}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 11.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
369.bf	even	120	1	inner
2952.et	odd	120	1	inner

$p$	$F_p(T)$
$2$	\( T^{32} + T^{28} + \cdots + 1 \) T^32 + T^28 - T^20 - T^16 - T^12 + T^4 + 1
$3$	\( (T^{8} - T^{7} + T^{5} + \cdots + 1)^{4} \) (T^8 - T^7 + T^5 - T^4 + T^3 - T + 1)^4
$5$	\( T^{32} \) T^32
$7$	\( T^{32} \) T^32
$11$	\( T^{32} + 2 T^{30} + \cdots + 1 \) T^32 + 2T^30 + 4T^29 + 9T^28 + 24T^27 + 12T^26 + 16T^25 + 127T^24 - 24T^23 + 322T^22 + 388T^21 - 88T^20 + 2936T^19 + 1184T^18 + 7616T^17 + 13971T^16 - 1536T^15 + 8346T^14 - 8264T^13 - 23988T^12 + 15120T^11 + 1460T^10 - 1532T^9 + 13374T^8 + 784T^7 + 3416T^6 - 3820T^5 + 100T^4 + 840T^3 + 304T^2 - 24T + 1
$13$	\( T^{32} \) T^32
$17$	\( T^{32} + 4 T^{31} + \cdots + 1 \) T^32 + 4T^31 + 6T^30 - 14T^28 - 56T^27 - 134T^26 - 160T^25 - 45T^24 + 48T^23 + 46T^22 + 304T^21 + 1648T^20 + 5028T^19 + 8472T^18 + 8312T^17 + 4809T^16 + 7756T^15 + 30120T^14 + 41952T^13 + 29392T^12 + 16424T^11 + 8194T^10 + 45600T^9 + 67827T^8 + 29620T^7 + 9670T^6 + 2648T^5 + 658T^4 + 180T^3 + 42T^2 + 8T + 1
$19$	\( T^{32} - 8 T^{30} + \cdots + 1 \) T^32 - 8T^30 + 4T^29 + 37T^28 - 40T^27 - 122T^26 + 184T^25 + 216T^24 - 672T^23 - 74T^22 + 1712T^21 - 1043T^20 - 1728T^19 + 6420T^18 - 5024T^17 - 8809T^16 + 16008T^15 - 19256T^14 + 10012T^13 + 24869T^12 - 36712T^11 + 36448T^10 - 29220T^9 + 19158T^8 - 6312T^7 + 10158T^6 - 4852T^5 + 4187T^4 - 1216T^3 + 298T^2 + 16T + 1
$23$	\( T^{32} \) T^32
$29$	\( T^{32} \) T^32
$31$	\( T^{32} \) T^32
$37$	\( T^{32} \) T^32
$41$	\( T^{32} + T^{28} + \cdots + 1 \) T^32 + T^28 - T^20 - T^16 - T^12 + T^4 + 1
$43$	\( T^{32} + 4 T^{28} + \cdots + 1 \) T^32 + 4T^28 + 250T^24 + 1706T^20 + 6999T^16 + 4546T^12 + 835T^8 - 61*T^4 + 1
$47$	\( T^{32} \) T^32
$53$	\( T^{32} \) T^32
$59$	\( T^{32} + 4 T^{30} + \cdots + 1 \) T^32 + 4T^30 + T^28 - 52T^26 - 208T^24 - 776T^22 - 193T^20 + 10088T^18 + 40351T^16 + 10088T^14 - 193T^12 - 776T^10 - 208T^8 - 52T^6 + T^4 + 4*T^2 + 1
$61$	\( T^{32} \) T^32
$67$	\( T^{32} - 16 T^{31} + \cdots + 1 \) T^32 - 16T^31 + 136T^30 - 800T^29 + 3618T^28 - 13300T^27 + 41130T^26 - 109512T^25 + 255288T^24 - 527676T^23 + 976746T^22 - 1631928T^21 + 2476556T^20 - 3430376T^19 + 4353008T^18 - 5074656T^17 + 5446721T^16 - 5391148T^15 + 4925106T^14 - 4151096T^13 + 3224214T^12 - 2310692T^11 + 1533262T^10 - 932968T^9 + 505869T^8 - 246052T^7 + 117554T^6 - 55904T^5 + 21709T^4 - 5512T^3 + 762T^2 - 44*T + 1
$71$	\( T^{32} \) T^32
$73$	\( (T^{16} + 29 T^{12} + \cdots + 1)^{2} \) (T^16 + 29T^12 + 246T^8 + 524*T^4 + 1)^2
$79$	\( T^{32} \) T^32
$83$	\( (T^{8} - 3 T^{6} + 8 T^{4} + \cdots + 1)^{4} \) (T^8 - 3T^6 + 8T^4 - 3*T^2 + 1)^4
$89$	\( (T^{16} - 4 T^{15} + \cdots + 1)^{2} \) (T^16 - 4T^15 + 10T^14 - 20T^13 + 34T^12 - 44T^11 + 82T^10 - 160T^9 + 256T^8 - 304T^7 + 162T^6 + 72T^5 - 131T^4 + 32T^3 + 46T^2 + 8*T + 1)^2
$97$	\( T^{32} + 2 T^{30} + \cdots + 1 \) T^32 + 2T^30 + 16T^29 - T^28 + 16T^27 + 162T^26 - 56T^25 + 112T^24 + 944T^23 - 418T^22 + 212T^21 + 4672T^20 - 2376T^19 + 1304T^18 + 13524T^17 - 2614T^16 - 8824T^15 + 35076T^14 - 9736T^13 - 19493T^12 + 29760T^11 - 2820T^10 - 12648T^9 + 5124T^8 + 9096T^7 + 2956T^6 + 1140T^5 + 2710T^4 + 280T^3 + 314T^2 - 36*T + 1
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