LMFDB - Newform orbit 1380.1.bn.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1380,1,Mod(359,1380)] 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("1380.359"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1380, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([11, 11, 11, 21])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1380.bn (of order $22$, degree $10$, minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,0,2,0,-2,0,0,2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

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

Self dual:	no
Analytic conductor:	$0.688709717434$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$2$ over $\Q(\zeta_{22})$
Coefficient field:	$\Q(\zeta_{44})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{22}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{22} + \cdots)$

$q$-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_{44}^{7} q^{2} + \zeta_{44}^{3} q^{3} + \zeta_{44}^{14} q^{4} + \zeta_{44}^{5} q^{5} - \zeta_{44}^{10} q^{6} - \zeta_{44}^{21} q^{8} + \zeta_{44}^{6} q^{9} - \zeta_{44}^{12} q^{10} + \zeta_{44}^{17} q^{12} + \cdots + \zeta_{44}^{21} q^{98} +O(q^{100}) $ q - z^7 * q^2 + z^3 * q^3 + z^14 * q^4 + z^5 * q^5 - z^10 * q^6 - z^21 * q^8 + z^6 * q^9 - z^12 * q^10 + z^17 * q^12 + z^8 * q^15 - z^6 * q^16 + (-z^15 - z^11) * q^17 - z^13 * q^18 + (z^16 - z^2) * q^19 + z^19 * q^20 + z * q^23 + z^2 * q^24 + z^10 * q^25 + z^9 * q^27 - z^15 * q^30 + (z^12 - z^4) * q^31 + z^13 * q^32 + (z^18 - 1) * q^34 + z^20 * q^36 + (z^9 + z) * q^38 + z^4 * q^40 + z^11 * q^45 - z^8 * q^46 + (-z^21 - z) * q^47 - z^9 * q^48 - z^14 * q^49 - z^17 * q^50 + (-z^18 - z^14) * q^51 + (-z^19 + z^17) * q^53 - z^16 * q^54 + (z^19 - z^5) * q^57 - q^60 + (z^20 + z^18) * q^61 + (-z^19 + z^11) * q^62 - z^20 * q^64 + (z^7 + z^3) * q^68 + z^4 * q^69 + z^5 * q^72 + z^13 * q^75 + (-z^16 - z^8) * q^76 + (-z^20 + z^10) * q^79 - z^11 * q^80 + z^12 * q^81 + (-z^9 - z^3) * q^83 + (-z^20 - z^16) * q^85 - z^18 * q^90 + z^15 * q^92 + (z^15 - z^7) * q^93 + (z^8 - z^6) * q^94 + (z^21 - z^7) * q^95 + z^16 * q^96 + z^21 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2 q^{4} - 2 q^{6} + 2 q^{9} + 2 q^{10} - 2 q^{15} - 2 q^{16} - 4 q^{19} + 2 q^{24} + 2 q^{25} - 18 q^{34} - 2 q^{36} - 2 q^{40} + 2 q^{46} - 2 q^{49} - 4 q^{51} + 2 q^{54} - 20 q^{60} + 2 q^{64}+ \cdots - 2 q^{96}+O(q^{100}) $ 20 * q + 2 * q^4 - 2 * q^6 + 2 * q^9 + 2 * q^10 - 2 * q^15 - 2 * q^16 - 4 * q^19 + 2 * q^24 + 2 * q^25 - 18 * q^34 - 2 * q^36 - 2 * q^40 + 2 * q^46 - 2 * q^49 - 4 * q^51 + 2 * q^54 - 20 * q^60 + 2 * q^64 - 2 * q^69 + 4 * q^76 + 4 * q^79 - 2 * q^81 + 4 * q^85 - 2 * q^90 - 4 * q^94 - 2 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times$.

$n$	$277$	$461$	$691$	$1201$
$\chi(n)$	$-1$	$-1$	$-1$	$\zeta_{44}^{10}$

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} $

359.1

−0.281733	+	0.959493i
0.281733	−	0.959493i
−0.281733	−	0.959493i
0.281733	+	0.959493i
−0.909632	−	0.415415i
0.909632	+	0.415415i
0.755750	+	0.654861i
−0.755750	−	0.654861i
0.989821	−	0.142315i
−0.989821	+	0.142315i
0.989821	+	0.142315i
−0.989821	−	0.142315i
0.540641	−	0.841254i
−0.540641	+	0.841254i
0.755750	−	0.654861i
−0.755750	+	0.654861i
0.540641	+	0.841254i
−0.540641	−	0.841254i
−0.909632	+	0.415415i
0.909632	−	0.415415i

−0.909632

−

0.415415i

0.755750

−

0.654861i

0.654861

0.755750i

−0.989821

0.142315i

−0.959493

0.281733i

−0.281733

−

0.959493i

0.142315

−

0.989821i

0.959493

0.281733i

359.2

0.909632

0.415415i

−0.755750

0.654861i

0.654861

0.755750i

0.989821

−

0.142315i

−0.959493

0.281733i

0.281733

0.959493i

0.142315

−

0.989821i

0.959493

0.281733i

419.1

−0.909632

0.415415i

0.755750

0.654861i

0.654861

−

0.755750i

−0.989821

−

0.142315i

−0.959493

−

0.281733i

−0.281733

0.959493i

0.142315

0.989821i

0.959493

−

0.281733i

419.2

0.909632

−

0.415415i

−0.755750

−

0.654861i

0.654861

−

0.755750i

0.989821

0.142315i

−0.959493

−

0.281733i

0.281733

−

0.959493i

0.142315

0.989821i

0.959493

−

0.281733i

479.1

−0.989821

0.142315i

−0.281733

−

0.959493i

0.959493

−

0.281733i

0.540641

−

0.841254i

0.415415

0.909632i

−0.909632

0.415415i

−0.841254

0.540641i

−0.415415

0.909632i

479.2

0.989821

−

0.142315i

0.281733

0.959493i

0.959493

−

0.281733i

−0.540641

0.841254i

0.415415

0.909632i

0.909632

−

0.415415i

−0.841254

0.540641i

−0.415415

0.909632i

539.1

−0.281733

0.959493i

−0.540641

0.841254i

−0.841254

−

0.540641i

−0.909632

−

0.415415i

−0.654861

−

0.755750i

0.755750

−

0.654861i

−0.415415

−

0.909632i

0.654861

−

0.755750i

539.2

0.281733

−

0.959493i

0.540641

−

0.841254i

−0.841254

−

0.540641i

0.909632

0.415415i

−0.654861

−

0.755750i

−0.755750

0.654861i

−0.415415

−

0.909632i

0.654861

−

0.755750i

659.1

−0.540641

0.841254i

0.909632

−

0.415415i

−0.415415

−

0.909632i

0.755750

−

0.654861i

−0.142315

0.989821i

0.989821

0.142315i

0.654861

−

0.755750i

0.142315

0.989821i

659.2

0.540641

−

0.841254i

−0.909632

0.415415i

−0.415415

−

0.909632i

−0.755750

0.654861i

−0.142315

0.989821i

−0.989821

−

0.142315i

0.654861

−

0.755750i

0.142315

0.989821i

779.1

−0.540641

−

0.841254i

0.909632

0.415415i

−0.415415

0.909632i

0.755750

0.654861i

−0.142315

−

0.989821i

0.989821

−

0.142315i

0.654861

0.755750i

0.142315

−

0.989821i

779.2

0.540641

0.841254i

−0.909632

−

0.415415i

−0.415415

0.909632i

−0.755750

−

0.654861i

−0.142315

−

0.989821i

−0.989821

0.142315i

0.654861

0.755750i

0.142315

−

0.989821i

839.1

−0.755750

0.654861i

−0.989821

−

0.142315i

0.142315

−

0.989821i

0.281733

0.959493i

0.841254

−

0.540641i

0.540641

0.841254i

0.959493

0.281733i

−0.841254

−

0.540641i

839.2

0.755750

−

0.654861i

0.989821

0.142315i

0.142315

−

0.989821i

−0.281733

−

0.959493i

0.841254

−

0.540641i

−0.540641

−

0.841254i

0.959493

0.281733i

−0.841254

−

0.540641i

1019.1

−0.281733

−

0.959493i

−0.540641

−

0.841254i

−0.841254

0.540641i

−0.909632

0.415415i

−0.654861

0.755750i

0.755750

0.654861i

−0.415415

0.909632i

0.654861

0.755750i

1019.2

0.281733

0.959493i

0.540641

0.841254i

−0.841254

0.540641i

0.909632

−

0.415415i

−0.654861

0.755750i

−0.755750

−

0.654861i

−0.415415

0.909632i

0.654861

0.755750i

1079.1

−0.755750

−

0.654861i

−0.989821

0.142315i

0.142315

0.989821i

0.281733

−

0.959493i

0.841254

0.540641i

0.540641

−

0.841254i

0.959493

−

0.281733i

−0.841254

0.540641i

1079.2

0.755750

0.654861i

0.989821

−

0.142315i

0.142315

0.989821i

−0.281733

0.959493i

0.841254

0.540641i

−0.540641

0.841254i

0.959493

−

0.281733i

−0.841254

0.540641i

1259.1

−0.989821

−

0.142315i

−0.281733

0.959493i

0.959493

0.281733i

0.540641

0.841254i

0.415415

−

0.909632i

−0.909632

−

0.415415i

−0.841254

−

0.540641i

−0.415415

−

0.909632i

1259.2

0.989821

0.142315i

0.281733

−

0.959493i

0.959493

0.281733i

−0.540641

−

0.841254i

0.415415

−

0.909632i

0.909632

0.415415i

−0.841254

−

0.540641i

−0.415415

−

0.909632i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15}) $
3.b	odd	2	1	inner
5.b	even	2	1	inner
92.h	even	22	1	inner
276.j	odd	22	1	inner
460.o	even	22	1	inner
1380.bn	odd	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1380.1.bn.c	yes	20
3.b	odd	2	1	inner	1380.1.bn.c	yes	20
4.b	odd	2	1		1380.1.bn.b	✓	20
5.b	even	2	1	inner	1380.1.bn.c	yes	20
12.b	even	2	1		1380.1.bn.b	✓	20
15.d	odd	2	1	CM	1380.1.bn.c	yes	20
20.d	odd	2	1		1380.1.bn.b	✓	20
23.d	odd	22	1		1380.1.bn.b	✓	20
60.h	even	2	1		1380.1.bn.b	✓	20
69.g	even	22	1		1380.1.bn.b	✓	20
92.h	even	22	1	inner	1380.1.bn.c	yes	20
115.i	odd	22	1		1380.1.bn.b	✓	20
276.j	odd	22	1	inner	1380.1.bn.c	yes	20
345.n	even	22	1		1380.1.bn.b	✓	20
460.o	even	22	1	inner	1380.1.bn.c	yes	20
1380.bn	odd	22	1	inner	1380.1.bn.c	yes	20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1380.1.bn.b	✓	20	4.b	odd	2	1
1380.1.bn.b	✓	20	12.b	even	2	1
1380.1.bn.b	✓	20	20.d	odd	2	1
1380.1.bn.b	✓	20	23.d	odd	22	1
1380.1.bn.b	✓	20	60.h	even	2	1
1380.1.bn.b	✓	20	69.g	even	22	1
1380.1.bn.b	✓	20	115.i	odd	22	1
1380.1.bn.b	✓	20	345.n	even	22	1
1380.1.bn.c	yes	20	1.a	even	1	1	trivial
1380.1.bn.c	yes	20	3.b	odd	2	1	inner
1380.1.bn.c	yes	20	5.b	even	2	1	inner
1380.1.bn.c	yes	20	15.d	odd	2	1	CM
1380.1.bn.c	yes	20	92.h	even	22	1	inner
1380.1.bn.c	yes	20	276.j	odd	22	1	inner
1380.1.bn.c	yes	20	460.o	even	22	1	inner
1380.1.bn.c	yes	20	1380.bn	odd	22	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(1380, [\chi])$:

$ T_{7} $

T7

$ T_{19}^{10} + 2T_{19}^{9} + 4T_{19}^{8} + 8T_{19}^{7} + 5T_{19}^{6} - T_{19}^{5} - 2T_{19}^{4} - 4T_{19}^{3} + 14T_{19}^{2} - 5T_{19} + 1 $

T19^10 + 2*T19^9 + 4*T19^8 + 8*T19^7 + 5*T19^6 - T19^5 - 2*T19^4 - 4*T19^3 + 14*T19^2 - 5*T19 + 1

$ T_{29} $

T29

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{20} - T^{18} + \cdots + 1 $ T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$3$	$ T^{20} - T^{18} + \cdots + 1 $ T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$5$	$ T^{20} - T^{18} + \cdots + 1 $ T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$7$	$ T^{20} $ T^20
$11$	$ T^{20} $ T^20
$13$	$ T^{20} $ T^20
$17$	$ T^{20} + 7 T^{18} + \cdots + 1 $ T^20 + 7T^18 + 27T^16 + 57T^14 + 102T^12 + 54T^10 + 224T^8 - 236T^6 + 185T^4 - 25*T^2 + 1
$19$	$ (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
$23$	$ T^{20} - T^{18} + \cdots + 1 $ T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$29$	$ T^{20} $ T^20
$31$	$ (T^{10} - 22 T^{5} + \cdots + 11)^{2} $ (T^10 - 22T^5 + 33T^3 + 11T^2 - 11T + 11)^2
$37$	$ T^{20} $ T^20
$41$	$ T^{20} $ T^20
$43$	$ T^{20} $ T^20
$47$	$ (T^{10} + 9 T^{8} + 28 T^{6} + \cdots + 1)^{2} $ (T^10 + 9T^8 + 28T^6 + 35T^4 + 15T^2 + 1)^2
$53$	$ T^{20} - 4 T^{18} + \cdots + 1 $ T^20 - 4T^18 + 16T^16 - 9T^14 + 36T^12 + 120T^10 + 125T^8 - 335T^6 + 130T^4 + 8*T^2 + 1
$59$	$ T^{20} $ T^20
$61$	$ (T^{10} + 11 T^{4} + \cdots + 11)^{2} $ (T^10 + 11T^4 - 55T^3 + 77T^2 - 44T + 11)^2
$67$	$ T^{20} $ T^20
$71$	$ T^{20} $ T^20
$73$	$ T^{20} $ T^20
$79$	$ (T^{10} - 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} $ (T^10 - 2T^9 + 4T^8 - 8T^7 + 16T^6 - 32T^5 + 53T^4 - 51T^3 + 25T^2 - 6*T + 1)^2
$83$	$ T^{20} + 22 T^{12} + \cdots + 121 $ T^20 + 22T^12 + 462T^10 + 1452T^8 + 1573T^6 + 847T^4 - 121T^2 + 121
$89$	$ T^{20} $ T^20
$97$	$ T^{20} $ T^20
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1380.bn (of order \(22\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{44}^{7} q^{2} + \zeta_{44}^{3} q^{3} + \zeta_{44}^{14} q^{4} + \zeta_{44}^{5} q^{5} - \zeta_{44}^{10} q^{6} - \zeta_{44}^{21} q^{8} + \zeta_{44}^{6} q^{9} - \zeta_{44}^{12} q^{10} + \zeta_{44}^{17} q^{12} + \cdots + \zeta_{44}^{21} q^{98} +O(q^{100}) \) q - z^7 * q^2 + z^3 * q^3 + z^14 * q^4 + z^5 * q^5 - z^10 * q^6 - z^21 * q^8 + z^6 * q^9 - z^12 * q^10 + z^17 * q^12 + z^8 * q^15 - z^6 * q^16 + (-z^15 - z^11) * q^17 - z^13 * q^18 + (z^16 - z^2) * q^19 + z^19 * q^20 + z * q^23 + z^2 * q^24 + z^10 * q^25 + z^9 * q^27 - z^15 * q^30 + (z^12 - z^4) * q^31 + z^13 * q^32 + (z^18 - 1) * q^34 + z^20 * q^36 + (z^9 + z) * q^38 + z^4 * q^40 + z^11 * q^45 - z^8 * q^46 + (-z^21 - z) * q^47 - z^9 * q^48 - z^14 * q^49 - z^17 * q^50 + (-z^18 - z^14) * q^51 + (-z^19 + z^17) * q^53 - z^16 * q^54 + (z^19 - z^5) * q^57 - q^60 + (z^20 + z^18) * q^61 + (-z^19 + z^11) * q^62 - z^20 * q^64 + (z^7 + z^3) * q^68 + z^4 * q^69 + z^5 * q^72 + z^13 * q^75 + (-z^16 - z^8) * q^76 + (-z^20 + z^10) * q^79 - z^11 * q^80 + z^12 * q^81 + (-z^9 - z^3) * q^83 + (-z^20 - z^16) * q^85 - z^18 * q^90 + z^15 * q^92 + (z^15 - z^7) * q^93 + (z^8 - z^6) * q^94 + (z^21 - z^7) * q^95 + z^16 * q^96 + z^21 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{4} - 2 q^{6} + 2 q^{9} + 2 q^{10} - 2 q^{15} - 2 q^{16} - 4 q^{19} + 2 q^{24} + 2 q^{25} - 18 q^{34} - 2 q^{36} - 2 q^{40} + 2 q^{46} - 2 q^{49} - 4 q^{51} + 2 q^{54} - 20 q^{60} + 2 q^{64}+ \cdots - 2 q^{96}+O(q^{100}) \) 20 * q + 2 * q^4 - 2 * q^6 + 2 * q^9 + 2 * q^10 - 2 * q^15 - 2 * q^16 - 4 * q^19 + 2 * q^24 + 2 * q^25 - 18 * q^34 - 2 * q^36 - 2 * q^40 + 2 * q^46 - 2 * q^49 - 4 * q^51 + 2 * q^54 - 20 * q^60 + 2 * q^64 - 2 * q^69 + 4 * q^76 + 4 * q^79 - 2 * q^81 + 4 * q^85 - 2 * q^90 - 4 * q^94 - 2 * q^96

Introduction

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Properties

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Newspace parameters

Newform invariants

$q$-expansion

Character values

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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	1380.1.bn.c

Level	$1380$
Weight	$1$
Character orbit	1380.bn
Analytic conductor	$0.689$
Analytic rank	$0$
Dimension	$20$
Projective image	$D_{22}$
CM discriminant	-15
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(0.688709717434\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(2\) over \(\Q(\zeta_{22})\)
Coefficient field:	\(\Q(\zeta_{44})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{22}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{22} + \cdots)\)

\(n\)	\(277\)	\(461\)	\(691\)	\(1201\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(\zeta_{44}^{10}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 359.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{20} - T^{18} + \cdots + 1 \) T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$3$	\( T^{20} - T^{18} + \cdots + 1 \) T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$5$	\( T^{20} - T^{18} + \cdots + 1 \) T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$7$	\( T^{20} \) T^20
$11$	\( T^{20} \) T^20
$13$	\( T^{20} \) T^20
$17$	\( T^{20} + 7 T^{18} + \cdots + 1 \) T^20 + 7T^18 + 27T^16 + 57T^14 + 102T^12 + 54T^10 + 224T^8 - 236T^6 + 185T^4 - 25*T^2 + 1
$19$	\( (T^{10} + 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} \) (T^10 + 2T^9 + 4T^8 + 8T^7 + 5T^6 - T^5 - 2T^4 - 4T^3 + 14T^2 - 5T + 1)^2
$23$	\( T^{20} - T^{18} + \cdots + 1 \) T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$29$	\( T^{20} \) T^20
$31$	\( (T^{10} - 22 T^{5} + \cdots + 11)^{2} \) (T^10 - 22T^5 + 33T^3 + 11T^2 - 11T + 11)^2
$37$	\( T^{20} \) T^20
$41$	\( T^{20} \) T^20
$43$	\( T^{20} \) T^20
$47$	\( (T^{10} + 9 T^{8} + 28 T^{6} + \cdots + 1)^{2} \) (T^10 + 9T^8 + 28T^6 + 35T^4 + 15T^2 + 1)^2
$53$	\( T^{20} - 4 T^{18} + \cdots + 1 \) T^20 - 4T^18 + 16T^16 - 9T^14 + 36T^12 + 120T^10 + 125T^8 - 335T^6 + 130T^4 + 8*T^2 + 1
$59$	\( T^{20} \) T^20
$61$	\( (T^{10} + 11 T^{4} + \cdots + 11)^{2} \) (T^10 + 11T^4 - 55T^3 + 77T^2 - 44T + 11)^2
$67$	\( T^{20} \) T^20
$71$	\( T^{20} \) T^20
$73$	\( T^{20} \) T^20
$79$	\( (T^{10} - 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} \) (T^10 - 2T^9 + 4T^8 - 8T^7 + 16T^6 - 32T^5 + 53T^4 - 51T^3 + 25T^2 - 6*T + 1)^2
$83$	\( T^{20} + 22 T^{12} + \cdots + 121 \) T^20 + 22T^12 + 462T^10 + 1452T^8 + 1573T^6 + 847T^4 - 121T^2 + 121
$89$	\( T^{20} \) T^20
$97$	\( T^{20} \) T^20
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