LMFDB - Newform orbit 3283.1.cd.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3283,1,Mod(374,3283)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3283, base_ring=CyclotomicField(66))

chi = DirichletCharacter(H, H._module([11, 58]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3283.374");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3283 = 7^{2} \cdot 67 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3283.cd (of order $66$, degree $20$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$1.63843043647$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 $ x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 469)
Projective image:	$D_{33}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{33} - \cdots)$

$q$-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_{66}^{31} - \zeta_{66}^{7}) q^{2} + ( - \zeta_{66}^{29} + \cdots - \zeta_{66}^{5}) q^{4}+ \cdots + \zeta_{66}^{8} q^{9}+O(q^{10}) $ q + (-z^31 - z^7) * q^2 + (-z^29 + z^14 - z^5) * q^4 + (-z^27 - z^21 + z^12 + z^3) * q^8 + z^8 * q^9	$ q + ( - \zeta_{66}^{31} - \zeta_{66}^{7}) q^{2} + ( - \zeta_{66}^{29} + \cdots - \zeta_{66}^{5}) q^{4}+ \cdots + (\zeta_{66}^{32} - \zeta_{66}^{15}) q^{99}+O(q^{100}) $ q + (-z^31 - z^7) * q^2 + (-z^29 + z^14 - z^5) * q^4 + (-z^27 - z^21 + z^12 + z^3) * q^8 + z^8 * q^9 + (z^24 - z^7) * q^11 + (z^28 - z^25 - z^19 + z^10 + z) * q^16 + (-z^15 + z^6) * q^18 + (-z^31 + z^22 + z^14 - z^5) * q^22 + (z^6 + z^2) * q^23 - z^31 * q^25 + (z^8 - z^3) * q^29 + (z^32 + z^26 - z^23 - z^17 + z^8 + z^2) * q^32 + (z^22 - z^13 + z^4) * q^36 + (-z^23 - z^21) * q^37 + (-z^11 - z) * q^43 + (-z^29 - z^21 + z^20 + z^12 - z^5 - z^3) * q^44 + (-z^13 - z^9 + z^4 + 1) * q^46 + (-z^29 - z^5) * q^50 + (z^20 + z^12) * q^53 + (-z^15 + z^10 + z^6 - z) * q^58 + (z^30 + z^24 - z^21 - z^15 - z^9 + z^6 - 1) * q^64 - z^21 * q^67 + (z^26 + z^6) * q^71 + (-z^29 + z^20 - z^11 + z^2) * q^72 + (z^30 + z^28 - z^21 - z^19) * q^74 + (-z^27 + z^24) * q^79 + z^16 * q^81 + (z^32 + z^18 - z^9 + z^8) * q^86 + (z^28 - z^27 - z^19 + z^18 + z^12 + z^10 - z^3 - z) * q^88 + (-z^31 + z^20 + z^16 - z^11 - z^7 + z^2) * q^92 + (z^32 - z^15) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2 q^{2} + 3 q^{4} - 8 q^{8} + q^{9}+O(q^{10}) $ 20 * q + 2 * q^2 + 3 * q^4 - 8 * q^8 + q^9	$ 20 q + 2 q^{2} + 3 q^{4} - 8 q^{8} + q^{9} - q^{11} + 5 q^{16} - 4 q^{18} - 7 q^{22} - q^{23} + q^{25} - q^{29} + 6 q^{32} - 8 q^{36} - q^{37} - 9 q^{43} - 3 q^{44} + 20 q^{46} + 2 q^{50} - q^{53} - 2 q^{58} + 8 q^{64} - 2 q^{67} - q^{71} - 7 q^{72} - 2 q^{74} - 4 q^{79} + q^{81} - 2 q^{86} - 4 q^{88} - 5 q^{92} - q^{99}+O(q^{100}) $ 20 * q + 2 * q^2 + 3 * q^4 - 8 * q^8 + q^9 - q^11 + 5 * q^16 - 4 * q^18 - 7 * q^22 - q^23 + q^25 - q^29 + 6 * q^32 - 8 * q^36 - q^37 - 9 * q^43 - 3 * q^44 + 20 * q^46 + 2 * q^50 - q^53 - 2 * q^58 + 8 * q^64 - 2 * q^67 - q^71 - 7 * q^72 - 2 * q^74 - 4 * q^79 + q^81 - 2 * q^86 - 4 * q^88 - 5 * q^92 - q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3283\mathbb{Z}\right)^\times$.

$n$	$738$	$1275$
$\chi(n)$	$\zeta_{66}^{11}$	$\zeta_{66}^{16}$

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} $

374.1

0.0475819	+	0.998867i
0.928368	−	0.371662i
0.723734	−	0.690079i
0.981929	−	0.189251i
0.580057	−	0.814576i
0.235759	−	0.971812i
−0.327068	−	0.945001i
0.235759	+	0.971812i
0.981929	+	0.189251i
0.928368	+	0.371662i
−0.888835	−	0.458227i
−0.786053	+	0.618159i
0.723734	+	0.690079i
−0.995472	−	0.0950560i
−0.995472	+	0.0950560i
0.580057	+	0.814576i
−0.327068	+	0.945001i
−0.888835	+	0.458227i
0.0475819	−	0.998867i
−0.786053	−	0.618159i

−1.32254

−

1.04006i

0.431635

1.77922i

0.580699

−

1.27155i

0.928368

−

0.371662i

607.1

−0.165101

0.231852i

0.300571

0.868442i

−0.524075

−

0.153882i

−0.995472

−

0.0950560i

619.1

0.627639

1.81344i

−2.10859

1.65822i

−2.71616

−

1.74557i

0.981929

0.189251i

668.1

1.16413

−

0.600149i

0.414955

−

0.582723i

−0.0530529

0.368991i

0.0475819

−

0.998867i

705.1

0.601300

0.573338i

−0.0147371

−

0.309371i

0.712591

−

0.822373i

0.235759

−

0.971812i

1011.1

−1.88431

0.363170i

2.49035

−

0.996987i

−2.71616

1.74557i

−0.327068

0.945001i

1158.1

−0.0623191

−

1.30824i

−0.712131

0.0680003i

−0.0530529

−

0.368991i

−0.888835

−

0.458227i

1195.1

−1.88431

−

0.363170i

2.49035

0.996987i

−2.71616

−

1.74557i

−0.327068

−

0.945001i

1440.1

1.16413

0.600149i

0.414955

0.582723i

−0.0530529

−

0.368991i

0.0475819

0.998867i

1893.1

−0.165101

−

0.231852i

0.300571

−

0.868442i

−0.524075

0.153882i

−0.995472

0.0950560i

1979.1

1.56199

−

0.625325i

1.32503

−

1.26342i

0.580699

−

1.27155i

−0.786053

−

0.618159i

2371.1

0.283341

−

0.0270558i

−0.902379

0.173919i

−0.524075

0.153882i

0.580057

0.814576i

2567.1

0.627639

−

1.81344i

−2.10859

−

1.65822i

−2.71616

1.74557i

0.981929

−

0.189251i

2824.1

0.195876

−

0.807410i

0.275291

0.141923i

0.712591

−

0.822373i

0.723734

0.690079i

2861.1

0.195876

0.807410i

0.275291

−

0.141923i

0.712591

0.822373i

0.723734

−

0.690079i

2971.1

0.601300

−

0.573338i

−0.0147371

0.309371i

0.712591

0.822373i

0.235759

0.971812i

3008.1

−0.0623191

1.30824i

−0.712131

−

0.0680003i

−0.0530529

0.368991i

−0.888835

0.458227i

3069.1

1.56199

0.625325i

1.32503

1.26342i

0.580699

1.27155i

−0.786053

0.618159i

3204.1

−1.32254

1.04006i

0.431635

−

1.77922i

0.580699

1.27155i

0.928368

0.371662i

3265.1

0.283341

0.0270558i

−0.902379

−

0.173919i

−0.524075

−

0.153882i

0.580057

−

0.814576i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by $\Q(\sqrt{-7}) $
469.bb	even	33	1	inner
469.bj	odd	66	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3283.1.cd.a		20
7.b	odd	2	1	CM	3283.1.cd.a		20
7.c	even	3	1		469.1.bl.a	✓	20
7.c	even	3	1		3283.1.bw.a		20
7.d	odd	6	1		469.1.bl.a	✓	20
7.d	odd	6	1		3283.1.bw.a		20
67.g	even	33	1		3283.1.bw.a		20
469.y	even	33	1		469.1.bl.a	✓	20
469.bb	even	33	1	inner	3283.1.cd.a		20
469.bc	odd	66	1		469.1.bl.a	✓	20
469.bj	odd	66	1	inner	3283.1.cd.a		20
469.bl	odd	66	1		3283.1.bw.a		20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
469.1.bl.a	✓	20	7.c	even	3	1
469.1.bl.a	✓	20	7.d	odd	6	1
469.1.bl.a	✓	20	469.y	even	33	1
469.1.bl.a	✓	20	469.bc	odd	66	1
3283.1.bw.a		20	7.c	even	3	1
3283.1.bw.a		20	7.d	odd	6	1
3283.1.bw.a		20	67.g	even	33	1
3283.1.bw.a		20	469.bl	odd	66	1
3283.1.cd.a		20	1.a	even	1	1	trivial
3283.1.cd.a		20	7.b	odd	2	1	CM
3283.1.cd.a		20	469.bb	even	33	1	inner
3283.1.cd.a		20	469.bj	odd	66	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(3283, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{20} - 2 T^{19} + \cdots + 1 $ T^20 - 2T^19 + 8T^17 - 16T^16 + 42T^14 - 29T^13 - 33T^12 + 94T^11 - 45T^10 - 284T^9 + 869T^8 - 1207T^7 + 1116T^6 - 671T^5 + 266T^4 - 48T^3 + 11T^2 - 6*T + 1
$3$	$ T^{20} $ T^20
$5$	$ T^{20} $ T^20
$7$	$ T^{20} $ T^20
$11$	$ T^{20} + T^{19} + \cdots + 1 $ T^20 + T^19 + 10T^17 + 10T^16 + 78T^14 + 56T^13 - 22T^12 + 241T^11 + 153T^10 - 89T^9 + 605T^8 + 1299T^7 + 793T^6 + 220T^5 + 274T^4 - 100T^3 + 121T^2 - 21T + 1
$13$	$ T^{20} $ T^20
$17$	$ T^{20} $ T^20
$19$	$ T^{20} $ T^20
$23$	$ T^{20} + T^{19} + \cdots + 1 $ T^20 + T^19 - T^17 - T^16 + 22T^15 + 23T^14 + 34T^13 + 22T^12 - T^11 + 450T^10 + 450T^9 + 1452T^8 + 760T^7 + 881T^6 - 12T^4 + 230T^3 + 154T^2 + 23*T + 1
$29$	$ T^{20} + T^{19} + \cdots + 1 $ T^20 + T^19 + 11T^18 + 10T^17 + 76T^16 + 66T^15 + 320T^14 + 254T^13 + 968T^12 + 714T^11 + 1935T^10 + 1220T^9 + 2673T^8 + 1431T^7 + 2025T^6 + 550T^5 + 703T^4 + 230T^3 + 132T^2 + 12T + 1
$31$	$ T^{20} $ T^20
$37$	$ T^{20} + T^{19} + \cdots + 1 $ T^20 + T^19 + 11T^18 + 10T^17 + 76T^16 + 66T^15 + 320T^14 + 254T^13 + 968T^12 + 714T^11 + 1935T^10 + 1220T^9 + 2673T^8 + 1431T^7 + 2025T^6 + 550T^5 + 703T^4 + 230T^3 + 132T^2 + 12T + 1
$41$	$ T^{20} $ T^20
$43$	$ T^{20} + 9 T^{19} + \cdots + 1 $ T^20 + 9T^19 + 47T^18 + 172T^17 + 489T^16 + 1127T^15 + 2174T^14 + 3567T^13 + 5047T^12 + 6194T^11 + 6633T^10 + 6194T^9 + 5069T^8 + 3578T^7 + 1965T^6 + 874T^5 + 742T^4 + 381T^3 + 36T^2 - 13*T + 1
$47$	$ T^{20} $ T^20
$53$	$ T^{20} + T^{19} + \cdots + 1 $ T^20 + T^19 + 10T^17 + 10T^16 + 78T^14 + 56T^13 - 22T^12 + 241T^11 + 153T^10 - 89T^9 + 605T^8 + 1299T^7 + 793T^6 + 220T^5 + 274T^4 - 100T^3 + 121T^2 - 21T + 1
$59$	$ T^{20} $ T^20
$61$	$ T^{20} $ T^20
$67$	$ (T^{10} + T^{9} + T^{8} + \cdots + 1)^{2} $ (T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1)^2
$71$	$ T^{20} + T^{19} + \cdots + 1 $ T^20 + T^19 - T^17 + 10T^16 + 22T^15 + 12T^14 - 10T^13 + 77T^12 + 362T^11 + 395T^10 + 32T^9 - 121T^8 - 274T^7 - 395T^6 - 22T^5 + 835T^4 - 595T^3 + 143T^2 - 10T + 1
$73$	$ T^{20} $ T^20
$79$	$ (T^{10} + 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} $ (T^10 + 2T^9 + 4T^8 + 8T^7 + 5T^6 - T^5 - 2T^4 - 4T^3 + 14T^2 - 5T + 1)^2
$83$	$ T^{20} $ T^20
$89$	$ T^{20} $ T^20
$97$	$ T^{20} $ T^20
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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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3283 = 7^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3283.cd (of order \(66\), degree \(20\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{66}^{31} - \zeta_{66}^{7}) q^{2} + ( - \zeta_{66}^{29} + \cdots - \zeta_{66}^{5}) q^{4}+ \cdots + \zeta_{66}^{8} q^{9}+O(q^{10}) \) q + (-z^31 - z^7) * q^2 + (-z^29 + z^14 - z^5) * q^4 + (-z^27 - z^21 + z^12 + z^3) * q^8 + z^8 * q^9	\( q + ( - \zeta_{66}^{31} - \zeta_{66}^{7}) q^{2} + ( - \zeta_{66}^{29} + \cdots - \zeta_{66}^{5}) q^{4}+ \cdots + (\zeta_{66}^{32} - \zeta_{66}^{15}) q^{99}+O(q^{100}) \) q + (-z^31 - z^7) * q^2 + (-z^29 + z^14 - z^5) * q^4 + (-z^27 - z^21 + z^12 + z^3) * q^8 + z^8 * q^9 + (z^24 - z^7) * q^11 + (z^28 - z^25 - z^19 + z^10 + z) * q^16 + (-z^15 + z^6) * q^18 + (-z^31 + z^22 + z^14 - z^5) * q^22 + (z^6 + z^2) * q^23 - z^31 * q^25 + (z^8 - z^3) * q^29 + (z^32 + z^26 - z^23 - z^17 + z^8 + z^2) * q^32 + (z^22 - z^13 + z^4) * q^36 + (-z^23 - z^21) * q^37 + (-z^11 - z) * q^43 + (-z^29 - z^21 + z^20 + z^12 - z^5 - z^3) * q^44 + (-z^13 - z^9 + z^4 + 1) * q^46 + (-z^29 - z^5) * q^50 + (z^20 + z^12) * q^53 + (-z^15 + z^10 + z^6 - z) * q^58 + (z^30 + z^24 - z^21 - z^15 - z^9 + z^6 - 1) * q^64 - z^21 * q^67 + (z^26 + z^6) * q^71 + (-z^29 + z^20 - z^11 + z^2) * q^72 + (z^30 + z^28 - z^21 - z^19) * q^74 + (-z^27 + z^24) * q^79 + z^16 * q^81 + (z^32 + z^18 - z^9 + z^8) * q^86 + (z^28 - z^27 - z^19 + z^18 + z^12 + z^10 - z^3 - z) * q^88 + (-z^31 + z^20 + z^16 - z^11 - z^7 + z^2) * q^92 + (z^32 - z^15) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{2} + 3 q^{4} - 8 q^{8} + q^{9}+O(q^{10}) \) 20 * q + 2 * q^2 + 3 * q^4 - 8 * q^8 + q^9	\( 20 q + 2 q^{2} + 3 q^{4} - 8 q^{8} + q^{9} - q^{11} + 5 q^{16} - 4 q^{18} - 7 q^{22} - q^{23} + q^{25} - q^{29} + 6 q^{32} - 8 q^{36} - q^{37} - 9 q^{43} - 3 q^{44} + 20 q^{46} + 2 q^{50} - q^{53} - 2 q^{58} + 8 q^{64} - 2 q^{67} - q^{71} - 7 q^{72} - 2 q^{74} - 4 q^{79} + q^{81} - 2 q^{86} - 4 q^{88} - 5 q^{92} - q^{99}+O(q^{100}) \) 20 * q + 2 * q^2 + 3 * q^4 - 8 * q^8 + q^9 - q^11 + 5 * q^16 - 4 * q^18 - 7 * q^22 - q^23 + q^25 - q^29 + 6 * q^32 - 8 * q^36 - q^37 - 9 * q^43 - 3 * q^44 + 20 * q^46 + 2 * q^50 - q^53 - 2 * q^58 + 8 * q^64 - 2 * q^67 - q^71 - 7 * q^72 - 2 * q^74 - 4 * q^79 + q^81 - 2 * q^86 - 4 * q^88 - 5 * q^92 - q^99

Introduction

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Modular forms

Varieties

Fields

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Groups

Database

Properties

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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	3283.1.cd.a

Level	$3283$
Weight	$1$
Character orbit	3283.cd
Analytic conductor	$1.638$
Analytic rank	$0$
Dimension	$20$
Projective image	$D_{33}$
CM discriminant	-7
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.63843043647\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Coefficient field:	\(\Q(\zeta_{33})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \) x^20 - x^19 + x^17 - x^16 + x^14 - x^13 + x^11 - x^10 + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 469)
Projective image:	\(D_{33}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{33} - \cdots)\)

\(n\)	\(738\)	\(1275\)
\(\chi(n)\)	\(\zeta_{66}^{11}\)	\(\zeta_{66}^{16}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 374.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)
469.bb	even	33	1	inner
469.bj	odd	66	1	inner

$p$	$F_p(T)$
$2$	\( T^{20} - 2 T^{19} + \cdots + 1 \) T^20 - 2T^19 + 8T^17 - 16T^16 + 42T^14 - 29T^13 - 33T^12 + 94T^11 - 45T^10 - 284T^9 + 869T^8 - 1207T^7 + 1116T^6 - 671T^5 + 266T^4 - 48T^3 + 11T^2 - 6*T + 1
$3$	\( T^{20} \) T^20
$5$	\( T^{20} \) T^20
$7$	\( T^{20} \) T^20
$11$	\( T^{20} + T^{19} + \cdots + 1 \) T^20 + T^19 + 10T^17 + 10T^16 + 78T^14 + 56T^13 - 22T^12 + 241T^11 + 153T^10 - 89T^9 + 605T^8 + 1299T^7 + 793T^6 + 220T^5 + 274T^4 - 100T^3 + 121T^2 - 21T + 1
$13$	\( T^{20} \) T^20
$17$	\( T^{20} \) T^20
$19$	\( T^{20} \) T^20
$23$	\( T^{20} + T^{19} + \cdots + 1 \) T^20 + T^19 - T^17 - T^16 + 22T^15 + 23T^14 + 34T^13 + 22T^12 - T^11 + 450T^10 + 450T^9 + 1452T^8 + 760T^7 + 881T^6 - 12T^4 + 230T^3 + 154T^2 + 23*T + 1
$29$	\( T^{20} + T^{19} + \cdots + 1 \) T^20 + T^19 + 11T^18 + 10T^17 + 76T^16 + 66T^15 + 320T^14 + 254T^13 + 968T^12 + 714T^11 + 1935T^10 + 1220T^9 + 2673T^8 + 1431T^7 + 2025T^6 + 550T^5 + 703T^4 + 230T^3 + 132T^2 + 12T + 1
$31$	\( T^{20} \) T^20
$37$	\( T^{20} + T^{19} + \cdots + 1 \) T^20 + T^19 + 11T^18 + 10T^17 + 76T^16 + 66T^15 + 320T^14 + 254T^13 + 968T^12 + 714T^11 + 1935T^10 + 1220T^9 + 2673T^8 + 1431T^7 + 2025T^6 + 550T^5 + 703T^4 + 230T^3 + 132T^2 + 12T + 1
$41$	\( T^{20} \) T^20
$43$	\( T^{20} + 9 T^{19} + \cdots + 1 \) T^20 + 9T^19 + 47T^18 + 172T^17 + 489T^16 + 1127T^15 + 2174T^14 + 3567T^13 + 5047T^12 + 6194T^11 + 6633T^10 + 6194T^9 + 5069T^8 + 3578T^7 + 1965T^6 + 874T^5 + 742T^4 + 381T^3 + 36T^2 - 13*T + 1
$47$	\( T^{20} \) T^20
$53$	\( T^{20} + T^{19} + \cdots + 1 \) T^20 + T^19 + 10T^17 + 10T^16 + 78T^14 + 56T^13 - 22T^12 + 241T^11 + 153T^10 - 89T^9 + 605T^8 + 1299T^7 + 793T^6 + 220T^5 + 274T^4 - 100T^3 + 121T^2 - 21T + 1
$59$	\( T^{20} \) T^20
$61$	\( T^{20} \) T^20
$67$	\( (T^{10} + T^{9} + T^{8} + \cdots + 1)^{2} \) (T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1)^2
$71$	\( T^{20} + T^{19} + \cdots + 1 \) T^20 + T^19 - T^17 + 10T^16 + 22T^15 + 12T^14 - 10T^13 + 77T^12 + 362T^11 + 395T^10 + 32T^9 - 121T^8 - 274T^7 - 395T^6 - 22T^5 + 835T^4 - 595T^3 + 143T^2 - 10T + 1
$73$	\( T^{20} \) T^20
$79$	\( (T^{10} + 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} \) (T^10 + 2T^9 + 4T^8 + 8T^7 + 5T^6 - T^5 - 2T^4 - 4T^3 + 14T^2 - 5T + 1)^2
$83$	\( T^{20} \) T^20
$89$	\( T^{20} \) T^20
$97$	\( T^{20} \) T^20
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