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Properties

Label	65.2.k.b

Level	$65$
Weight	$2$
Character orbit	65.k
Analytic conductor	$0.519$
Analytic rank	$0$
Dimension	$8$
Inner twists	$2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [65,2,Mod(8,65)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(65, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("65.8"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$65 = 5 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	65.k (of order $4$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$0.519027613138$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.619810816.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1$ x^8 - 2x^5 + 14x^4 - 8x^3 + 2x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$ -expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

Coefficients of the $q$ -expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_{5} q^{2} + ( - \beta_{5} + \beta_{4} + \beta_{2} - 1) q^{3} + ( - \beta_{7} - \beta_{5} + \beta_{4} + 1) q^{4} + (\beta_{7} + \beta_{6}) q^{5} + ( - \beta_{6} - \beta_{4} + \beta_{2} - 1) q^{6}+ \cdots + ( - \beta_{7} - \beta_1) q^{99}+O(q^{100})$ q + b5 * q^2 + (-b5 + b4 + b2 - 1) * q^3 + (-b7 - b5 + b4 + 1) * q^4 + (b7 + b6) * q^5 + (-b6 - b4 + b2 - 1) * q^6 + (2b6 - 2b5 + 2b4 - 2b3 + b1) * q^7 + (b7 + b5 + b3) * q^8 + (b5 - b4 + b3 - b2 + b1) * q^9 + (-b7 - b5 - b4 - b3 - 2b2 - b1 - 1) q^10 + (-b6 + b5 + b3 + b2 + 1) * q^11 + (b7 - b6 + b5 - 2b4 + b1) q^12 + (b5 - 2b4 - b3 - 2b2 - b1) * q^13 + (-2b6 + 3b5 - 3b4 + 3b3 - 2b2 - 3b1) * q^14 + (-b7 - b5 + 3b4 + b3) q^15 + (-2b4 - 2b3 - 1) * q^16 + (-b7 - b6 - b4 + 2b2 - b1 - 2) q^17 + (-b6 - 2b2 - b1) q^18 + (-b3 + 2b2 + 2) q^19 + (b7 + 2b6 + b4 + b3 - b2 + b1 - 3) q^20 + (-b7 - b6 + b4 - b2 + b1 - 1) * q^21 + (-b7 - b6 + b4 - b3 + 2b2 + b1 + 2) q^22 + (-b7 - b6 + b4 + b3 + 2b2 + b1 + 2) q^23 + (-b5 + b4) * q^24 + (b7 - 2b6 + b3 + 2b2 - b1 + 1) * q^25 + (b7 + 4b6 - 2b5 + 2b4 - b2 + b1) q^26 + (2b7 + b6 + b5 - 2b4 + b2 - 2b1 + 1) q^27 + (4b6 - 2b5 + 2b4 - 2b3 + 3b1) q^28 + (2b6 - b5 + b4 - b3 - 2b2 + b1) * q^29 + (-b7 + 2b5 + b2 + 3) q^30 + (b6 + 2b5 - b4 - b2 + 1) q^31 + (-3b5 - 2) q^32 + (b7 - b5 - b4 - 2) * q^33 + (2b7 - 2b5 + 2b4 + b2 + 2b1 - 1) * q^34 + (b7 - b6 - b5 + 2b4 - b3 - b2 - 3) q^35 + (4b6 - 3b5 + 3b4 - 3b3 + 5b2) q^36 + (b5 - b4 + b3 - 6b2 - b1) q^37 + (-2b6 + 2b5 + b3 + b2 + 1) * q^38 + (-2b7 - b5 + 3b4 + 4b2 - b1) q^39 + (-b6 - b5 + b3 - b2 - b1 + 2) * q^40 + (-2b7 - b6 - b5 - 2b2 - 2b1 + 2) q^41 + (-b6 + b5 + 2b3 + 3b2 + 3) * q^42 + (-b7 + 3b6 - 4b5 + b4 - 3b3 + b1) q^43 + (-2b6 + 2b5 + b3 + 2b2 + 2) q^44 + (2b7 - 3b6 + 6b5 - 6b4 + 3b3 - 3b2 + 1) * q^45 + (-4b6 + 4b5 + b3 + 2b2 + 2) q^46 + (2b2 + 3b1) * q^47 + (-2b7 + 2b6 - b5 + 3b4 + b2 - 2b1 - 1) * q^48 + (4b5 - 2b4 - 2b3 - 1) q^49 + (-b7 - b5 - 3b3 + 6b2 + 3b1 - 2) q^50 + (3b5 - 3b4 + 3b3 - 4b2) * q^51 + (-b7 - b6 + 2b3 - 5b2 - 3b1 - 3) q^52 + (-b7 - 2b6 + b5 - 3b4 + 3b2 - b1 - 3) q^53 + (-b7 + 3b6 - 4b5 + b4 - 4b3 - 3b2 + b1 - 3) * q^54 + (2b6 + b3 - 3b2 - b1 + 1) * q^55 + (-2b6 - b5 + b4 - b3 - 4b2 - b1) * q^56 + (-b7 - 3b5 + 3b4 + 2b3 - 6) q^57 + (2b5 - 2b4 + 2b3 - 4b2 - 3b1) q^58 + (-4b6 + b5 - 5b4 - 2b2 + 2) q^59 + (b7 - b6 + 4b5 - 3b4 - b3 - b2 + 7) * q^60 + (3b7 - 2b5 - 2b4 + b3) q^61 + (-b7 + b6 - b5 + 2b4 - 4b2 - b1 + 4) * q^62 + (5b7 + 5b5 - 3b4 + 2b3 + 2) * q^63 + (3b7 + b5 + b4 + 4b3 - 7) * q^64 + (-b7 - 2b5 - 2b4 - 5b3 + 5b2 + b1) * q^65 + (b7 - 2b5 - 2b4 - b3 - 6) * q^66 + (-b7 + 2b5 - b3) q^67 + (-3b6 + b5 - 4b4) * q^68 + (3b7 - 4b4 - b3 - 2) * q^69 + (-2b7 + b6 - 3b5 - 2b4 + 7b2 + b1 + 1) * q^70 + (b7 + 2b6 - 4b5 + 6b4 + 2b2 + b1 - 2) * q^71 + (-3b6 + 3b5 - 3b4 + 3b3 - 2b2 - 2b1) * q^72 + (-3b7 - 2b5 + 2b4 - b3) q^73 + (8b6 - 2b5 + 2b4 - 2b3 - 2b2 + b1) q^74 + (-b7 + 2b6 - 4b5 + 4b4 + 2b3 - 5b2 + b1 - 5) q^75 + (-2b7 - b6 - b5 + 2b4 + b3 + b2 + 2b1 + 1) q^76 + (b6 - b5 + 2b4 - 3b2 + 3) * q^77 + (-2b6 + 2b5 + 2b4 + b3 + b2 + b1 + 5) q^78 + (6b6 - 5b5 + 5b4 - 5b3 - 2b2 + 2b1) * q^79 + (-b7 - b6 + 2b5 - 2b4 - 4b3 + 4b2 + 2) * q^80 + (-b7 + 2b5 + 4b4 + 3b3 - 1) q^81 + (3b7 + 6b6 + 3b5 + 3b4 + b2 + 3b1 - 1) q^82 + (-4b6 - 4b2 + 3b1) q^83 + (b7 - b6 + 2b5 - b4 - 2b3 + 3b2 - b1 + 3) q^84 + (-3b7 + 2b5 + b3 + 3b2 - b1 - 1) q^85 + (4b7 - 2b6 + 6b5 - 4b4 + 5b3 - 6b2 - 4b1 - 6) q^86 + (-b6 + b5 - 2b3 + 3b2 + 3) * q^87 + (b3 + b2 + 1) * q^88 + (-b7 - 2b6 - 5b5 + 3b4 + 5b2 - b1 - 5) * q^89 + (-2b7 + 3b6 - 7b5 + 4b4 - 5b3 + 2b2 + 3b1 + 1) q^90 + (-2b7 - b6 - 2b5 - 3b4 + b3 + 5b2 - 3b1 + 1) q^91 + (-2b7 - 2b5 + 2b4 - 3b3 + 7b2 + 2b1 + 7) * q^92 + (-2b6 + 6b2 - b1) * q^93 + (-8b6 + 3b5 - 3b4 + 3b3 - 3b1) q^94 + (b7 + 3b6 + b4 - b3 - 2b1) * q^95 + (3b6 + 2b5 + b4 - 5b2 + 5) q^96 + (2b7 + 3b5 + 3b4 + 5b3 + 2) * q^97 + (-2b7 - 5b5 + 4b4 + 2b3 + 10) * q^98 + (-b7 - b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 4 q^{2} - 6 q^{3} + 8 q^{4} + 2 q^{5} - 6 q^{6} - 6 q^{10} + 6 q^{11} + 2 q^{12} - 2 q^{13} - 2 q^{15} - 8 q^{16} - 16 q^{17} + 14 q^{19} - 22 q^{20} - 12 q^{21} + 10 q^{22} + 14 q^{23} + 2 q^{24}+ \cdots - 2 q^{99}+O(q^{100})$ 8 * q - 4 * q^2 - 6 * q^3 + 8 * q^4 + 2 * q^5 - 6 * q^6 - 6 * q^10 + 6 * q^11 + 2 * q^12 - 2 * q^13 - 2 * q^15 - 8 * q^16 - 16 * q^17 + 14 * q^19 - 22 * q^20 - 12 * q^21 + 10 * q^22 + 14 * q^23 + 2 * q^24 + 12 * q^25 + 6 * q^26 + 12 * q^27 + 14 * q^30 + 2 * q^31 - 4 * q^32 - 8 * q^33 - 24 * q^35 + 2 * q^38 - 6 * q^39 + 22 * q^40 + 16 * q^41 + 24 * q^42 + 6 * q^43 + 10 * q^44 + 6 * q^45 + 2 * q^46 - 14 * q^48 - 24 * q^49 - 20 * q^50 - 22 * q^52 - 24 * q^53 - 20 * q^54 + 10 * q^55 - 40 * q^57 + 22 * q^59 + 46 * q^60 + 20 * q^61 + 30 * q^62 + 16 * q^63 - 48 * q^64 - 36 * q^66 - 12 * q^67 + 4 * q^68 - 4 * q^69 + 20 * q^70 - 10 * q^71 - 4 * q^73 - 30 * q^75 + 6 * q^76 + 24 * q^77 + 30 * q^78 + 2 * q^80 - 20 * q^81 - 20 * q^82 + 16 * q^84 - 20 * q^85 - 46 * q^86 + 16 * q^87 + 10 * q^88 - 28 * q^89 + 14 * q^90 + 20 * q^91 + 50 * q^92 - 2 * q^95 + 30 * q^96 + 12 * q^97 + 92 * q^98 - 2 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1$ x^8 - 2*x^5 + 14*x^4 - 8*x^3 + 2*x^2 + 2*x + 1 :

$\beta_{1}$	$=$	$( 64\nu^{7} + 16\nu^{6} + 4\nu^{5} - 127\nu^{4} + 944\nu^{3} - 276\nu^{2} + 378\nu + 63 ) / 319$ (64v^7 + 16v^6 + 4v^5 - 127v^4 + 944v^3 - 276v^2 + 378*v + 63) / 319
$\beta_{2}$	$=$	$( -63\nu^{7} + 64\nu^{6} + 16\nu^{5} + 130\nu^{4} - 1009\nu^{3} + 1448\nu^{2} - 402\nu - 67 ) / 319$ (-63v^7 + 64v^6 + 16v^5 + 130v^4 - 1009v^3 + 1448v^2 - 402*v - 67) / 319
$\beta_{3}$	$=$	$( -67\nu^{7} + 63\nu^{6} - 64\nu^{5} + 118\nu^{4} - 1068\nu^{3} + 1545\nu^{2} - 1263\nu + 268 ) / 319$ (-67v^7 + 63v^6 - 64v^5 + 118v^4 - 1068v^3 + 1545v^2 - 1263*v + 268) / 319
$\beta_{4}$	$=$	$( 83\nu^{7} - 59\nu^{6} + 65\nu^{5} - 70\nu^{4} + 1304\nu^{3} - 1614\nu^{2} + 1198\nu + 306 ) / 319$ (83v^7 - 59v^6 + 65v^5 - 70v^4 + 1304v^3 - 1614v^2 + 1198*v + 306) / 319
$\beta_{5}$	$=$	$( -172\nu^{7} - 43\nu^{6} + 69\nu^{5} + 441\nu^{4} - 2218\nu^{3} + 662\nu^{2} + 619\nu - 269 ) / 319$ (-172v^7 - 43v^6 + 69v^5 + 441v^4 - 2218v^3 + 662v^2 + 619*v - 269) / 319
$\beta_{6}$	$=$	$( -196\nu^{7} - 49\nu^{6} - 92\nu^{5} + 369\nu^{4} - 2572\nu^{3} + 1244\nu^{2} - 1038\nu - 173 ) / 319$ (-196v^7 - 49v^6 - 92v^5 + 369v^4 - 2572v^3 + 1244v^2 - 1038*v - 173) / 319
$\beta_{7}$	$=$	$\nu^{7} - 2\nu^{4} + 14\nu^{3} - 8\nu^{2} + \nu + 2$ v^7 - 2v^4 + 14v^3 - 8*v^2 + v + 2

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$( -\beta_{7} - \beta_{5} + \beta_{4} + \beta_1 ) / 2$ (-b7 - b5 + b4 + b1) / 2
$\nu^{2}$	$=$	$\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2\beta_{2}$ b6 - b5 + b4 - b3 + 2*b2
$\nu^{3}$	$=$	$( 3\beta_{7} + 5\beta_{5} - 5\beta_{4} - 2\beta_{2} + 3\beta _1 + 2 ) / 2$ (3b7 + 5b5 - 5b4 - 2b2 + 3*b1 + 2) / 2
$\nu^{4}$	$=$	$-\beta_{7} - \beta_{5} + 5\beta_{4} + 4\beta_{3} - 7$ -b7 - b5 + 5b4 + 4b3 - 7
$\nu^{5}$	$=$	$( 11\beta_{7} + 2\beta_{6} + 9\beta_{5} - 11\beta_{4} - 12\beta_{3} + 12\beta_{2} - 11\beta _1 + 12 ) / 2$ (11b7 + 2b6 + 9b5 - 11b4 - 12b3 + 12b2 - 11*b1 + 12) / 2
$\nu^{6}$	$=$	$-15\beta_{6} + 16\beta_{5} - 16\beta_{4} + 16\beta_{3} - 28\beta_{2} + 7\beta_1$ -15b6 + 16b5 - 16b4 + 16b3 - 28b2 + 7b1
$\nu^{7}$	$=$	$( -43\beta_{7} + 16\beta_{6} - 89\beta_{5} + 105\beta_{4} + 60\beta_{2} - 43\beta _1 - 60 ) / 2$ (-43b7 + 16b6 - 89b5 + 105b4 + 60b2 - 43b1 - 60) / 2

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/65\mathbb{Z}\right)^\times$ .

$n$	$27$	$41$
$\chi(n)$	$\beta_{2}$	$-\beta_{2}$

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)

\iota_m(\nu)

a_{2}

a_{3}

a_{4}

a_{5}

a_{6}

a_{7}

a_{8}

a_{9}

a_{10}

1.18254	−	1.18254i
−0.252709	+	0.252709i
−1.49094	+	1.49094i
0.561103	−	0.561103i
1.18254	+	1.18254i
−0.252709	−	0.252709i
−1.49094	−	1.49094i
0.561103	+	0.561103i

−2.31627

−0.240275

−

0.240275i

3.36509

−1.60536

−

1.55654i

0.556540

+

0.556540i

−

3.95872i

−3.16190

−

2.88454i

3.71844

+

3.60536i

−1.57942

0.725850

+

0.725850i

0.494582

2.23127

+

0.146426i

−1.14643

−

1.14643i

4.24997i

2.37769

−

1.94628i

−3.52412

−

0.231269i

−0.134632

−2.15558

−

2.15558i

−1.98187

1.82630

−

1.29021i

0.290209

+

0.290209i

−

1.90970i

0.536087

6.29303i

−0.245878

+

0.173703i

2.03032

−1.33000

−

1.33000i

2.12221

−1.45220

+

1.70032i

−2.70032

−

2.70032i

1.61845i

0.248119

0.537789i

−2.94844

+

3.45220i

−2.31627

−0.240275

+

0.240275i

3.36509

−1.60536

+

1.55654i

0.556540

−

0.556540i

3.95872i

−3.16190

2.88454i

3.71844

−

3.60536i

−1.57942

0.725850

−

0.725850i

0.494582

2.23127

−

0.146426i

−1.14643

+

1.14643i

−

4.24997i

2.37769

1.94628i

−3.52412

+

0.231269i

−0.134632

−2.15558

+

2.15558i

−1.98187

1.82630

+

1.29021i

0.290209

−

0.290209i

1.90970i

0.536087

−

6.29303i

−0.245878

−

0.173703i

2.03032

−1.33000

+

1.33000i

2.12221

−1.45220

−

1.70032i

−2.70032

+

2.70032i

−

1.61845i

0.248119

−

0.537789i

−2.94844

−

3.45220i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
65.k	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	65.2.k.b	yes	8
3.b	odd	2	1		585.2.w.e		8
4.b	odd	2	1		1040.2.bg.n		8
5.b	even	2	1		325.2.k.b		8
5.c	odd	4	1		65.2.f.b	✓	8
5.c	odd	4	1		325.2.f.b		8
13.b	even	2	1		845.2.k.b		8
13.c	even	3	2		845.2.o.d		16
13.d	odd	4	1		65.2.f.b	✓	8
13.d	odd	4	1		845.2.f.b		8
13.e	even	6	2		845.2.o.c		16
13.f	odd	12	2		845.2.t.c		16
13.f	odd	12	2		845.2.t.d		16
15.e	even	4	1		585.2.n.e		8
20.e	even	4	1		1040.2.cd.n		8
39.f	even	4	1		585.2.n.e		8
52.f	even	4	1		1040.2.cd.n		8
65.f	even	4	1		325.2.k.b		8
65.f	even	4	1		845.2.k.b		8
65.g	odd	4	1		325.2.f.b		8
65.h	odd	4	1		845.2.f.b		8
65.k	even	4	1	inner	65.2.k.b	yes	8
65.o	even	12	2		845.2.o.d		16
65.q	odd	12	2		845.2.t.c		16
65.r	odd	12	2		845.2.t.d		16
65.t	even	12	2		845.2.o.c		16
195.j	odd	4	1		585.2.w.e		8
260.s	odd	4	1		1040.2.bg.n		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
65.2.f.b	✓	8	5.c	odd	4	1
65.2.f.b	✓	8	13.d	odd	4	1
65.2.k.b	yes	8	1.a	even	1	1	trivial
65.2.k.b	yes	8	65.k	even	4	1	inner
325.2.f.b		8	5.c	odd	4	1
325.2.f.b		8	65.g	odd	4	1
325.2.k.b		8	5.b	even	2	1
325.2.k.b		8	65.f	even	4	1
585.2.n.e		8	15.e	even	4	1
585.2.n.e		8	39.f	even	4	1
585.2.w.e		8	3.b	odd	2	1
585.2.w.e		8	195.j	odd	4	1
845.2.f.b		8	13.d	odd	4	1
845.2.f.b		8	65.h	odd	4	1
845.2.k.b		8	13.b	even	2	1
845.2.k.b		8	65.f	even	4	1
845.2.o.c		16	13.e	even	6	2
845.2.o.c		16	65.t	even	12	2
845.2.o.d		16	13.c	even	3	2
845.2.o.d		16	65.o	even	12	2
845.2.t.c		16	13.f	odd	12	2
845.2.t.c		16	65.q	odd	12	2
845.2.t.d		16	13.f	odd	12	2
845.2.t.d		16	65.r	odd	12	2
1040.2.bg.n		8	4.b	odd	2	1
1040.2.bg.n		8	260.s	odd	4	1
1040.2.cd.n		8	20.e	even	4	1
1040.2.cd.n		8	52.f	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{2}^{4} + 2T_{2}^{3} - 4T_{2}^{2} - 8T_{2} - 1$ T2^4 + 2*T2^3 - 4*T2^2 - 8*T2 - 1 acting on $S_{2}^{\mathrm{new}}(65, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{4} + 2 T^{3} - 4 T^{2} + \cdots - 1)^{2}$ (T^4 + 2T^3 - 4T^2 - 8*T - 1)^2
$3$	$T^{8} + 6 T^{7} + \cdots + 4$ T^8 + 6T^7 + 18T^6 + 20T^5 + 8T^4 - 4T^3 + 32T^2 + 16*T + 4
$5$	$T^{8} - 2 T^{7} + \cdots + 625$ T^8 - 2T^7 - 4T^6 - 6T^5 + 62T^4 - 30T^3 - 100T^2 - 250*T + 625
$7$	$T^{8} + 40 T^{6} + \cdots + 2704$ T^8 + 40T^6 + 504T^4 + 2096*T^2 + 2704
$11$	$T^{8} - 6 T^{7} + \cdots + 4$ T^8 - 6T^7 + 18T^6 - 8T^5 - 4T^4 + 12T^3 + 32T^2 + 16*T + 4
$13$	$T^{8} + 2 T^{7} + \cdots + 28561$ T^8 + 2T^7 - 26T^5 - 330T^4 - 338T^3 + 4394*T + 28561
$17$	$T^{8} + 16 T^{7} + \cdots + 13456$ T^8 + 16T^7 + 128T^6 + 536T^5 + 1256T^4 + 1088T^3 + 288T^2 + 2784*T + 13456
$19$	$T^{8} - 14 T^{7} + \cdots + 100$ T^8 - 14T^7 + 98T^6 - 396T^5 + 1004T^4 - 1524T^3 + 1352T^2 - 520*T + 100
$23$	$T^{8} - 14 T^{7} + \cdots + 40804$ T^8 - 14T^7 + 98T^6 - 288T^5 + 504T^4 - 1348T^3 + 10952T^2 - 29896*T + 40804
$29$	$T^{8} + 44 T^{6} + \cdots + 10000$ T^8 + 44T^6 + 680T^4 + 4384*T^2 + 10000
$31$	$T^{8} - 2 T^{7} + \cdots + 16900$ T^8 - 2T^7 + 2T^6 + 32T^5 + 316T^4 - 124T^3 + 128T^2 + 2080*T + 16900
$37$	$T^{8} + 140 T^{6} + \cdots + 336400$ T^8 + 140T^6 + 6456T^4 + 105184*T^2 + 336400
$41$	$T^{8} - 16 T^{7} + \cdots + 13456$ T^8 - 16T^7 + 128T^6 - 88T^5 - 216T^4 + 1248T^3 + 11552T^2 + 17632*T + 13456
$43$	$T^{8} - 6 T^{7} + \cdots + 8836$ T^8 - 6T^7 + 18T^6 + 344T^5 + 3176T^4 + 332T^3 + 8T^2 + 376*T + 8836
$47$	$T^{8} + 160 T^{6} + \cdots + 26896$ T^8 + 160T^6 + 7800T^4 + 116752*T^2 + 26896
$53$	$T^{8} + 24 T^{7} + \cdots + 19600$ T^8 + 24T^7 + 288T^6 + 1664T^5 + 4904T^4 + 1184T^3 + 512T^2 + 4480*T + 19600
$59$	$T^{8} - 22 T^{7} + \cdots + 119716$ T^8 - 22T^7 + 242T^6 - 572T^5 - 116T^4 - 18788T^3 + 605000T^2 + 380600*T + 119716
$61$	$(T^{4} - 10 T^{3} + \cdots - 3628)^{2}$ (T^4 - 10T^3 - 120T^2 + 1512*T - 3628)^2
$67$	$(T^{4} + 6 T^{3} + \cdots - 148)^{2}$ (T^4 + 6T^3 - 32T^2 - 192*T - 148)^2
$71$	$T^{8} + 10 T^{7} + \cdots + 1223236$ T^8 + 10T^7 + 50T^6 + 108T^5 + 12212T^4 + 121860T^3 + 613832T^2 + 1225448*T + 1223236
$73$	$(T^{4} + 2 T^{3} + \cdots + 740)^{2}$ (T^4 + 2T^3 - 56T^2 - 56*T + 740)^2
$79$	$T^{8} + 292 T^{6} + \cdots + 13719616$ T^8 + 292T^6 + 29552T^4 + 1178624*T^2 + 13719616
$83$	$T^{8} + 416 T^{6} + \cdots + 54346384$ T^8 + 416T^6 + 57896T^4 + 3083920*T^2 + 54346384
$89$	$T^{8} + 28 T^{7} + \cdots + 1795600$ T^8 + 28T^7 + 392T^6 + 1528T^5 - 2616T^4 - 47952T^3 + 850208T^2 - 1747360*T + 1795600
$97$	$(T^{4} - 6 T^{3} + \cdots + 3704)^{2}$ (T^4 - 6T^3 - 128T^2 + 480*T + 3704)^2
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