LMFDB - Newform orbit 2717.1.db.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2717,1,Mod(142,2717)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2717, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 9, 8]))

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

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

chi := DirichletCharacter("2717.142");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2717 = 11 \cdot 13 \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2717.db (of order $18$, degree $6$, 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:	$1.35595963932$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$4$ over $\Q(\zeta_{18})$
Coefficient field:	$\Q(\zeta_{45})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} - x^{21} + x^{15} - x^{12} + x^{9} - x^{3} + 1 $ x^24 - x^21 + x^15 - x^12 + x^9 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{45}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{45} - \cdots)$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2717\mathbb{Z}\right)^\times$.

$n$	$210$	$287$	$2224$
$\chi(n)$	$-1$	$-\zeta_{90}^{5}$	$-1$

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} $

142.1

−0.241922	+	0.970296i
−0.997564	+	0.0697565i
0.848048	+	0.529919i
−0.374607	−	0.927184i
0.438371	−	0.898794i
0.990268	+	0.139173i
−0.719340	−	0.694658i
−0.882948	+	0.469472i
0.438371	+	0.898794i
0.990268	−	0.139173i
−0.719340	+	0.694658i
−0.882948	−	0.469472i
−0.615661	+	0.788011i
0.559193	+	0.829038i
0.961262	−	0.275637i
0.0348995	−	0.999391i
−0.615661	−	0.788011i
0.559193	−	0.829038i
0.961262	+	0.275637i
0.0348995	+	0.999391i

−0.333843

−

1.89332i

0.856733

−

0.718885i

−2.53350

0.922119i

−1.64709

−

1.38207i

−0.615661

1.06636i

1.63039

2.82392i

0.0435487

−

0.246977i

142.2

−0.194206

−

1.10140i

1.47274

−

1.23577i

−0.235663

0.0857741i

−1.64709

−

1.38207i

0.0348995

−

0.0604477i

−0.418955

−

0.725651i

0.468172

−

2.65514i

142.3

−0.0121205

−

0.0687386i

−0.943248

0.791479i

0.935115

−

0.340354i

0.0658378

0.0552444i

0.961262

−

1.66495i

−0.0696290

−

0.120601i

0.0896296

−

0.508315i

142.4

0.213817

1.21262i

0.0534691

−

0.0448659i

−0.485028

0.176536i

0.0658378

0.0552444i

0.559193

−

0.968551i

0.297884

0.515950i

−0.172802

0.980010i

1429.1

−1.87481

−

0.682374i

−0.0840186

−

0.476493i

2.28322

1.91585i

−0.167628

0.950665i

0.848048

−

1.46886i

−1.97571

−

3.42203i

0.719706

−

0.261952i

1429.2

−0.704030

−

0.256246i

0.294524

1.67033i

−0.336048

−

0.281978i

0.220661

−

1.25143i

−0.997564

1.72783i

0.538939

0.933469i

−1.76356

0.641884i

1429.3

−0.454664

−

0.165484i

−0.346450

−

1.96482i

−0.586710

−

0.492308i

−0.167628

0.950665i

−0.374607

0.648838i

0.427209

0.739947i

−2.80079

1.01940i

1429.4

1.59381

0.580099i

−0.130100

−

0.737831i

1.43767

1.20635i

0.220661

−

1.25143i

−0.241922

0.419021i

0.743520

1.28781i

0.412224

−

0.150037i

1715.1

−1.87481

0.682374i

−0.0840186

0.476493i

2.28322

−

1.91585i

−0.167628

−

0.950665i

0.848048

1.46886i

−1.97571

3.42203i

0.719706

0.261952i

1715.2

−0.704030

0.256246i

0.294524

−

1.67033i

−0.336048

0.281978i

0.220661

1.25143i

−0.997564

−

1.72783i

0.538939

−

0.933469i

−1.76356

−

0.641884i

1715.3

−0.454664

0.165484i

−0.346450

1.96482i

−0.586710

0.492308i

−0.167628

−

0.950665i

−0.374607

−

0.648838i

0.427209

−

0.739947i

−2.80079

−

1.01940i

1715.4

1.59381

−

0.580099i

−0.130100

0.737831i

1.43767

−

1.20635i

0.220661

1.25143i

−0.241922

−

0.419021i

0.743520

−

1.28781i

0.412224

0.150037i

2001.1

−1.51718

1.27306i

1.65940

0.603972i

0.507491

−

2.87812i

−3.28650

1.19619i

0.438371

−

0.759281i

1.90381

3.29750i

1.62278

1.36167i

2001.2

−0.671624

0.563559i

1.35192

0.492057i

−0.0401688

0.227809i

−1.18528

0.431408i

−0.882948

1.52931i

−0.539776

−

0.934920i

0.819514

0.687654i

2001.3

1.10209

−

0.924765i

−0.823868

−

0.299864i

0.185769

−

1.05355i

−1.18528

0.431408i

0.990268

−

1.71519i

−0.0502092

−

0.0869649i

−0.177204

−

0.148692i

2001.4

1.35275

−

1.13510i

−1.86110

−

0.677383i

0.367854

−

2.08620i

−3.28650

1.19619i

−0.719340

1.24593i

−0.987476

−

1.71036i

2.23878

1.87856i

2144.1

−1.51718

−

1.27306i

1.65940

−

0.603972i

0.507491

2.87812i

−3.28650

−

1.19619i

0.438371

0.759281i

1.90381

−

3.29750i

1.62278

−

1.36167i

2144.2

−0.671624

−

0.563559i

1.35192

−

0.492057i

−0.0401688

−

0.227809i

−1.18528

−

0.431408i

−0.882948

−

1.52931i

−0.539776

0.934920i

0.819514

−

0.687654i

2144.3

1.10209

0.924765i

−0.823868

0.299864i

0.185769

1.05355i

−1.18528

−

0.431408i

0.990268

1.71519i

−0.0502092

0.0869649i

−0.177204

0.148692i

2144.4

1.35275

1.13510i

−1.86110

0.677383i

0.367854

2.08620i

−3.28650

−

1.19619i

−0.719340

−

1.24593i

−0.987476

1.71036i

2.23878

−

1.87856i

See all 24 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
143.d	odd	2	1	CM by $\Q(\sqrt{-143}) $
19.e	even	9	1	inner
2717.db	odd	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2717.1.db.c	✓	24
11.b	odd	2	1		2717.1.db.d	yes	24
13.b	even	2	1		2717.1.db.d	yes	24
19.e	even	9	1	inner	2717.1.db.c	✓	24
143.d	odd	2	1	CM	2717.1.db.c	✓	24
209.q	odd	18	1		2717.1.db.d	yes	24
247.bn	even	18	1		2717.1.db.d	yes	24
2717.db	odd	18	1	inner	2717.1.db.c	✓	24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2717.1.db.c	✓	24	1.a	even	1	1	trivial
2717.1.db.c	✓	24	19.e	even	9	1	inner
2717.1.db.c	✓	24	143.d	odd	2	1	CM
2717.1.db.c	✓	24	2717.db	odd	18	1	inner
2717.1.db.d	yes	24	11.b	odd	2	1
2717.1.db.d	yes	24	13.b	even	2	1
2717.1.db.d	yes	24	209.q	odd	18	1
2717.1.db.d	yes	24	247.bn	even	18	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{24} + 3 T_{2}^{23} + 3 T_{2}^{22} - T_{2}^{21} + 15 T_{2}^{19} + 68 T_{2}^{18} + 90 T_{2}^{17} + \cdots + 1 $ T2^24 + 3*T2^23 + 3*T2^22 - T2^21 + 15*T2^19 + 68*T2^18 + 90*T2^17 - 18*T2^16 - 94*T2^15 + 303*T2^14 + 465*T2^13 + 991*T2^12 + 1278*T2^11 + 1209*T2^10 + 1669*T2^9 + 3231*T2^8 + 5364*T2^7 + 7256*T2^6 + 7125*T2^5 + 4470*T2^4 + 1579*T2^3 + 261*T2^2 + 12*T2 + 1 acting on $S_{1}^{\mathrm{new}}(2717, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{24} + 3 T^{23} + \cdots + 1 $ T^24 + 3T^23 + 3T^22 - T^21 + 15T^19 + 68T^18 + 90T^17 - 18T^16 - 94T^15 + 303T^14 + 465T^13 + 991T^12 + 1278T^11 + 1209T^10 + 1669T^9 + 3231T^8 + 5364T^7 + 7256T^6 + 7125T^5 + 4470T^4 + 1579T^3 + 261T^2 + 12*T + 1
$3$	$ T^{24} - 3 T^{23} + \cdots + 1 $ T^24 - 3T^23 + 3T^22 + T^21 - 9T^20 + 30T^19 - 13T^18 - 126T^17 + 315T^16 - 392T^15 + 285T^14 + 237T^13 + 19T^12 - 1494T^11 + 1344T^10 + 626T^9 - 450T^8 - 126T^7 - 88T^6 + 183T^5 + 924T^4 + 95T^3 + 189T^2 - 21T + 1
$5$	$ T^{24} $ T^24
$7$	$ T^{24} + 12 T^{22} + \cdots + 1 $ T^24 + 12T^22 + 2T^21 + 90T^20 + 21T^19 + 425T^18 + 135T^17 + 1476T^16 + 524T^15 + 3618T^14 + 1419T^13 + 6625T^12 + 2556T^11 + 8322T^10 + 3139T^9 + 7560T^8 + 2664T^7 + 4193T^6 + 1377T^5 + 1563T^4 + 370T^3 + 180T^2 - 12T + 1
$11$	$ (T^{2} - T + 1)^{12} $ (T^2 - T + 1)^12
$13$	$ (T^{6} - T^{3} + 1)^{4} $ (T^6 - T^3 + 1)^4
$17$	$ T^{24} $ T^24
$19$	$ T^{24} + T^{21} + \cdots + 1 $ T^24 + T^21 - T^15 - T^12 - T^9 + T^3 + 1
$23$	$ T^{24} - 3 T^{23} + \cdots + 1 $ T^24 - 3T^23 + 3T^22 + T^21 - 9T^20 + 30T^19 - 13T^18 - 126T^17 + 315T^16 - 392T^15 + 285T^14 + 237T^13 + 19T^12 - 1494T^11 + 1344T^10 + 626T^9 - 450T^8 - 126T^7 - 88T^6 + 183T^5 + 924T^4 + 95T^3 + 189T^2 - 21T + 1
$29$	$ T^{24} $ T^24
$31$	$ T^{24} $ T^24
$37$	$ T^{24} $ T^24
$41$	$ T^{24} + 3 T^{23} + \cdots + 1 $ T^24 + 3T^23 + 3T^22 - T^21 + 15T^19 + 68T^18 + 90T^17 - 18T^16 - 94T^15 + 303T^14 + 465T^13 + 991T^12 + 1278T^11 + 1209T^10 + 1669T^9 + 3231T^8 + 5364T^7 + 7256T^6 + 7125T^5 + 4470T^4 + 1579T^3 + 261T^2 + 12*T + 1
$43$	$ T^{24} $ T^24
$47$	$ T^{24} $ T^24
$53$	$ T^{24} - 3 T^{23} + \cdots + 1 $ T^24 - 3T^23 + 3T^22 + T^21 - 9T^20 + 30T^19 - 13T^18 - 126T^17 + 315T^16 - 392T^15 + 285T^14 + 237T^13 + 19T^12 - 1494T^11 + 1344T^10 + 626T^9 - 450T^8 - 126T^7 - 88T^6 + 183T^5 + 924T^4 + 95T^3 + 189T^2 - 21T + 1
$59$	$ T^{24} $ T^24
$61$	$ T^{24} $ T^24
$67$	$ T^{24} $ T^24
$71$	$ T^{24} $ T^24
$73$	$ (T^{12} + 4 T^{9} + 17 T^{6} + \cdots + 1)^{2} $ (T^12 + 4T^9 + 17T^6 - 4*T^3 + 1)^2
$79$	$ T^{24} $ T^24
$83$	$ T^{24} + 12 T^{22} + \cdots + 1 $ T^24 + 12T^22 + 2T^21 + 90T^20 + 21T^19 + 425T^18 + 135T^17 + 1476T^16 + 524T^15 + 3618T^14 + 1419T^13 + 6625T^12 + 2556T^11 + 8322T^10 + 3139T^9 + 7560T^8 + 2664T^7 + 4193T^6 + 1377T^5 + 1563T^4 + 370T^3 + 180T^2 - 12T + 1
$89$	$ T^{24} $ T^24
$97$	$ T^{24} $ T^24
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Level:	\( N \)	\(=\)	\( 2717 = 11 \cdot 13 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2717.db (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\( q + (\zeta_{90}^{39} + \zeta_{90}^{11}) q^{2} + (\zeta_{90}^{14} + \zeta_{90}^{6}) q^{3} + ( - \zeta_{90}^{33} + \cdots - \zeta_{90}^{5}) q^{4}+ \cdots + (\zeta_{90}^{28} + \cdots + \zeta_{90}^{12}) q^{9}+O(q^{10}) \) q + (z^39 + z^11) * q^2 + (z^14 + z^6) * q^3 + (-z^33 + z^22 - z^5) * q^4 + (z^25 + z^17 - z^8 - 1) * q^6 + (-z^38 + z^37) * q^7 + (-z^44 + z^33 + z^27 - z^16) * q^8 + (z^28 + z^20 + z^12) * q^9	\( q + (\zeta_{90}^{39} + \zeta_{90}^{11}) q^{2} + (\zeta_{90}^{14} + \zeta_{90}^{6}) q^{3} + ( - \zeta_{90}^{33} + \cdots - \zeta_{90}^{5}) q^{4}+ \cdots + ( - \zeta_{90}^{42} + \cdots + \zeta_{90}^{5}) q^{99}+O(q^{100}) \) q + (z^39 + z^11) * q^2 + (z^14 + z^6) * q^3 + (-z^33 + z^22 - z^5) * q^4 + (z^25 + z^17 - z^8 - 1) * q^6 + (-z^38 + z^37) * q^7 + (-z^44 + z^33 + z^27 - z^16) * q^8 + (z^28 + z^20 + z^12) * q^9 - z^30 * q^11 + (-z^39 + z^36 + z^28 - z^19 - z^11 + z^2) * q^12 + z^35 * q^13 + (z^32 - z^31 + z^4 - z^3) * q^14 + (z^44 + z^38 - z^27 - z^21 + z^10) * q^16 + (z^39 + z^31 + z^23 - z^22 - z^14 - z^6) * q^18 - z^26 * q^19 + (-z^44 + z^43 + z^7 - z^6) * q^21 + (-z^41 + z^24) * q^22 + (-z^7 - z^3) * q^23 + (z^41 + z^39 + z^33 - z^30 - z^22 + z^13 + z^5 - z^2) * q^24 - z^35 * q^25 + (-z^29 - z) * q^26 + (z^42 + z^34 + z^26 + z^18) * q^27 + (z^43 - z^42 - z^26 + z^25 + z^15 - z^14) * q^28 + (-z^38 - z^32 + z^21 + z^15 - z^10 + z^4) * q^32 + (-z^44 - z^36) * q^33 + (z^42 + z^34 - z^33 - z^25 - z^17 + z^16 + z^8 - z^5 + 1) * q^36 + (-z^37 + z^20) * q^38 + (z^41 - z^4) * q^39 + (z^17 + z^3) * q^41 + (z^38 - z^37 + z^18 - z^17 + z^10 - z^9 - z + 1) * q^42 + (z^35 - z^18 + z^7) * q^44 + (-z^42 - z^18 - z^14 + z) * q^46 + (z^44 - z^41 - z^35 - z^33 - z^27 + z^24 + z^16 - z^13 - z^7 - z^5) * q^48 + (-z^31 + z^30 - z^29) * q^49 + (z^29 + z) * q^50 + (-z^40 + z^23 - z^12) * q^52 + (-z^39 + z^16) * q^53 + (z^37 - z^36 + z^29 - z^28 - z^20 - z^12 - z^8 - 1) * q^54 + (-z^37 + z^36 + z^26 - z^25 + z^20 - z^19 - z^9 + z^8) * q^56 + (-z^40 - z^32) * q^57 + (z^21 - z^20 + z^13 - z^12 + z^5 - z^4) * q^63 + (-z^43 + z^32 + z^26 - z^21 - z^15 - z^9 - z^4) * q^64 + (z^38 + z^30 + z^10 + z^2) * q^66 + (-z^21 - z^17 - z^13 - z^9) * q^69 + (-z^44 + z^39 - z^36 - z^28 + z^27 + z^19 - z^16 + z^11 - z^10 - z^8 - z^2 - 1) * q^72 + (z^19 + z) * q^73 + (-z^41 + z^4) * q^75 + (z^31 - z^14 + z^3) * q^76 + (-z^23 + z^22) * q^77 + (-z^43 - z^35 - z^15 - z^7) * q^78 + (z^40 + z^32 + z^24 - z^11 + z^3) * q^81 + (z^42 + z^28 + z^14 - z^11) * q^82 + (-z^28 - z^2) * q^83 + (-z^40 + z^39 - z^32 + z^31 + z^29 - z^28 + z^21 - z^20 - z^12 + z^11 - z^4 + z^3) * q^84 + (-z^29 + z^18 + z^12 - z) * q^88 + (z^28 - z^27) * q^91 + (z^40 + z^36 - z^29 - z^25 + z^12 + z^8) * q^92 + (-z^44 - z^38 + z^35 + z^29 + z^27 - z^24 + z^21 - z^18 - z^16 - z^10 + z^7 + z) * q^96 + (-z^42 + z^41 - z^40 + z^25 - z^24 + z^23) * q^98 + (-z^42 + z^13 + z^5) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 24 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 24 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 24 * q^6 + 3 * q^8 + 3 * q^9	\( 24 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 24 q^{6} + 3 q^{8} + 3 q^{9} + 12 q^{11} - 3 q^{12} + 3 q^{14} - 3 q^{16} - 6 q^{18} - 3 q^{21} + 3 q^{22} + 3 q^{23} + 6 q^{24} - 3 q^{27} + 9 q^{28} + 9 q^{32} + 6 q^{33} + 30 q^{36} - 3 q^{41} + 12 q^{42} + 6 q^{44} + 3 q^{46} - 12 q^{49} - 3 q^{52} + 3 q^{53} - 21 q^{54} - 12 q^{56} - 6 q^{63} - 15 q^{64} - 12 q^{66} - 3 q^{69} - 15 q^{72} - 3 q^{76} - 12 q^{78} + 6 q^{81} + 3 q^{82} - 12 q^{84} - 3 q^{88} - 6 q^{91} - 3 q^{92} + 6 q^{96} - 6 q^{98} - 3 q^{99}+O(q^{100}) \) 24 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 24 * q^6 + 3 * q^8 + 3 * q^9 + 12 * q^11 - 3 * q^12 + 3 * q^14 - 3 * q^16 - 6 * q^18 - 3 * q^21 + 3 * q^22 + 3 * q^23 + 6 * q^24 - 3 * q^27 + 9 * q^28 + 9 * q^32 + 6 * q^33 + 30 * q^36 - 3 * q^41 + 12 * q^42 + 6 * q^44 + 3 * q^46 - 12 * q^49 - 3 * q^52 + 3 * q^53 - 21 * q^54 - 12 * q^56 - 6 * q^63 - 15 * q^64 - 12 * q^66 - 3 * q^69 - 15 * q^72 - 3 * q^76 - 12 * q^78 + 6 * q^81 + 3 * q^82 - 12 * q^84 - 3 * q^88 - 6 * q^91 - 3 * q^92 + 6 * q^96 - 6 * q^98 - 3 * q^99

Introduction

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Modular forms

Varieties

Fields

Representations

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	2717.1.db.c

Level	$2717$
Weight	$1$
Character orbit	2717.db
Analytic conductor	$1.356$
Analytic rank	$0$
Dimension	$24$
Projective image	$D_{45}$
CM discriminant	-143
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.35595963932\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(4\) over \(\Q(\zeta_{18})\)
Coefficient field:	\(\Q(\zeta_{45})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{24} - x^{21} + x^{15} - x^{12} + x^{9} - x^{3} + 1 \) x^24 - x^21 + x^15 - x^12 + x^9 - x^3 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{45}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{45} - \cdots)\)

\(n\)	\(210\)	\(287\)	\(2224\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{90}^{5}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 142.4
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{24} + 3 T^{23} + \cdots + 1 \) T^24 + 3T^23 + 3T^22 - T^21 + 15T^19 + 68T^18 + 90T^17 - 18T^16 - 94T^15 + 303T^14 + 465T^13 + 991T^12 + 1278T^11 + 1209T^10 + 1669T^9 + 3231T^8 + 5364T^7 + 7256T^6 + 7125T^5 + 4470T^4 + 1579T^3 + 261T^2 + 12*T + 1
$3$	\( T^{24} - 3 T^{23} + \cdots + 1 \) T^24 - 3T^23 + 3T^22 + T^21 - 9T^20 + 30T^19 - 13T^18 - 126T^17 + 315T^16 - 392T^15 + 285T^14 + 237T^13 + 19T^12 - 1494T^11 + 1344T^10 + 626T^9 - 450T^8 - 126T^7 - 88T^6 + 183T^5 + 924T^4 + 95T^3 + 189T^2 - 21T + 1
$5$	\( T^{24} \) T^24
$7$	\( T^{24} + 12 T^{22} + \cdots + 1 \) T^24 + 12T^22 + 2T^21 + 90T^20 + 21T^19 + 425T^18 + 135T^17 + 1476T^16 + 524T^15 + 3618T^14 + 1419T^13 + 6625T^12 + 2556T^11 + 8322T^10 + 3139T^9 + 7560T^8 + 2664T^7 + 4193T^6 + 1377T^5 + 1563T^4 + 370T^3 + 180T^2 - 12T + 1
$11$	\( (T^{2} - T + 1)^{12} \) (T^2 - T + 1)^12
$13$	\( (T^{6} - T^{3} + 1)^{4} \) (T^6 - T^3 + 1)^4
$17$	\( T^{24} \) T^24
$19$	\( T^{24} + T^{21} + \cdots + 1 \) T^24 + T^21 - T^15 - T^12 - T^9 + T^3 + 1
$23$	\( T^{24} - 3 T^{23} + \cdots + 1 \) T^24 - 3T^23 + 3T^22 + T^21 - 9T^20 + 30T^19 - 13T^18 - 126T^17 + 315T^16 - 392T^15 + 285T^14 + 237T^13 + 19T^12 - 1494T^11 + 1344T^10 + 626T^9 - 450T^8 - 126T^7 - 88T^6 + 183T^5 + 924T^4 + 95T^3 + 189T^2 - 21T + 1
$29$	\( T^{24} \) T^24
$31$	\( T^{24} \) T^24
$37$	\( T^{24} \) T^24
$41$	\( T^{24} + 3 T^{23} + \cdots + 1 \) T^24 + 3T^23 + 3T^22 - T^21 + 15T^19 + 68T^18 + 90T^17 - 18T^16 - 94T^15 + 303T^14 + 465T^13 + 991T^12 + 1278T^11 + 1209T^10 + 1669T^9 + 3231T^8 + 5364T^7 + 7256T^6 + 7125T^5 + 4470T^4 + 1579T^3 + 261T^2 + 12*T + 1
$43$	\( T^{24} \) T^24
$47$	\( T^{24} \) T^24
$53$	\( T^{24} - 3 T^{23} + \cdots + 1 \) T^24 - 3T^23 + 3T^22 + T^21 - 9T^20 + 30T^19 - 13T^18 - 126T^17 + 315T^16 - 392T^15 + 285T^14 + 237T^13 + 19T^12 - 1494T^11 + 1344T^10 + 626T^9 - 450T^8 - 126T^7 - 88T^6 + 183T^5 + 924T^4 + 95T^3 + 189T^2 - 21T + 1
$59$	\( T^{24} \) T^24
$61$	\( T^{24} \) T^24
$67$	\( T^{24} \) T^24
$71$	\( T^{24} \) T^24
$73$	\( (T^{12} + 4 T^{9} + 17 T^{6} + \cdots + 1)^{2} \) (T^12 + 4T^9 + 17T^6 - 4*T^3 + 1)^2
$79$	\( T^{24} \) T^24
$83$	\( T^{24} + 12 T^{22} + \cdots + 1 \) T^24 + 12T^22 + 2T^21 + 90T^20 + 21T^19 + 425T^18 + 135T^17 + 1476T^16 + 524T^15 + 3618T^14 + 1419T^13 + 6625T^12 + 2556T^11 + 8322T^10 + 3139T^9 + 7560T^8 + 2664T^7 + 4193T^6 + 1377T^5 + 1563T^4 + 370T^3 + 180T^2 - 12T + 1
$89$	\( T^{24} \) T^24
$97$	\( T^{24} \) T^24
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