# Properties

 Label 429.2.e.d Level $429$ Weight $2$ Character orbit 429.e Analytic conductor $3.426$ Analytic rank $0$ Dimension $20$ CM discriminant -143 Inner twists $8$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 429.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.42558224671$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ Defining polynomial: $$x^{20} - 53 x^{10} + 1024$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{12}\cdot 3^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + \beta_{1} q^{3} + ( -2 + \beta_{8} ) q^{4} + \beta_{7} q^{6} + \beta_{14} q^{7} + ( -2 \beta_{4} - \beta_{10} ) q^{8} + \beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{2} + \beta_{1} q^{3} + ( -2 + \beta_{8} ) q^{4} + \beta_{7} q^{6} + \beta_{14} q^{7} + ( -2 \beta_{4} - \beta_{10} ) q^{8} + \beta_{2} q^{9} -\beta_{5} q^{11} + ( -2 \beta_{1} - \beta_{12} + \beta_{15} - \beta_{16} ) q^{12} + ( \beta_{7} + \beta_{11} ) q^{13} + ( \beta_{1} - \beta_{2} - \beta_{12} - \beta_{16} - \beta_{18} ) q^{14} + ( 4 + \beta_{1} - 2 \beta_{8} - \beta_{12} + \beta_{15} ) q^{16} + ( \beta_{3} + \beta_{7} + \beta_{13} + \beta_{14} ) q^{18} + ( -\beta_{3} + \beta_{5} - \beta_{7} + \beta_{9} - \beta_{11} - \beta_{19} ) q^{19} + ( \beta_{4} - \beta_{10} - \beta_{13} + \beta_{19} ) q^{21} + ( -2 \beta_{1} - \beta_{6} - \beta_{12} + \beta_{15} ) q^{22} + ( -\beta_{2} + \beta_{8} - \beta_{15} + \beta_{16} + \beta_{18} ) q^{23} + ( -\beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{14} + \beta_{19} ) q^{24} -5 q^{25} + ( -2 \beta_{1} + \beta_{6} - \beta_{12} - \beta_{16} ) q^{26} + ( -\beta_{8} + \beta_{18} ) q^{27} + ( -\beta_{3} - \beta_{4} - \beta_{7} - \beta_{9} - \beta_{11} - 2 \beta_{14} - \beta_{19} ) q^{28} + ( 4 \beta_{4} + \beta_{5} + 2 \beta_{10} ) q^{32} + ( \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{9} + \beta_{11} ) q^{33} + ( 1 - 2 \beta_{2} - \beta_{6} - \beta_{12} + \beta_{15} - \beta_{16} + \beta_{17} ) q^{36} + ( -1 + 2 \beta_{2} - \beta_{8} - 2 \beta_{17} ) q^{38} + ( \beta_{3} - \beta_{4} + \beta_{9} ) q^{39} + ( 2 \beta_{5} + 2 \beta_{7} - 2 \beta_{11} + 2 \beta_{13} + \beta_{14} ) q^{41} + ( -2 + \beta_{2} + \beta_{6} - \beta_{8} - \beta_{12} + \beta_{15} - \beta_{16} + \beta_{17} - 2 \beta_{18} ) q^{42} + ( -2 \beta_{7} + \beta_{10} + 2 \beta_{11} - 2 \beta_{13} - \beta_{14} ) q^{44} + ( -\beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{7} + 3 \beta_{9} - \beta_{10} - 3 \beta_{11} - 2 \beta_{13} - \beta_{19} ) q^{46} + ( 4 + 4 \beta_{1} + \beta_{2} + 2 \beta_{8} + 2 \beta_{12} - 2 \beta_{15} + 2 \beta_{16} + \beta_{17} + \beta_{18} ) q^{48} + ( 7 + \beta_{1} + \beta_{6} + 2 \beta_{12} - 2 \beta_{15} ) q^{49} -5 \beta_{4} q^{50} + ( -\beta_{5} - 3 \beta_{7} + \beta_{10} - \beta_{11} + 2 \beta_{13} - \beta_{14} ) q^{52} + ( \beta_{1} - \beta_{6} + 2 \beta_{12} + 2 \beta_{16} ) q^{53} + ( 3 \beta_{4} - \beta_{7} + \beta_{9} + \beta_{10} - \beta_{13} + \beta_{14} + \beta_{19} ) q^{54} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{6} + \beta_{12} - \beta_{15} + 2 \beta_{16} - 2 \beta_{17} + 2 \beta_{18} ) q^{56} + ( -2 \beta_{5} - \beta_{7} - \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} ) q^{57} + ( -3 \beta_{4} + 3 \beta_{7} - 2 \beta_{9} + 3 \beta_{11} + \beta_{19} ) q^{63} + ( -8 + 2 \beta_{1} + \beta_{6} + 4 \beta_{8} + \beta_{12} - \beta_{15} ) q^{64} + ( -5 - 2 \beta_{2} + 2 \beta_{8} + \beta_{17} + \beta_{18} ) q^{66} + ( \beta_{6} + \beta_{8} + 2 \beta_{12} - \beta_{18} ) q^{69} + ( -2 \beta_{3} + 3 \beta_{5} - 2 \beta_{7} + \beta_{9} - 2 \beta_{13} - 2 \beta_{14} - 2 \beta_{19} ) q^{72} + ( -\beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{7} - 3 \beta_{9} + \beta_{11} - \beta_{19} ) q^{73} -5 \beta_{1} q^{75} + ( 4 \beta_{3} + \beta_{4} - 4 \beta_{5} + 2 \beta_{7} - 2 \beta_{9} + \beta_{10} + 4 \beta_{11} + 2 \beta_{13} + 2 \beta_{14} + 4 \beta_{19} ) q^{76} + ( -\beta_{2} - \beta_{8} + \beta_{15} - \beta_{16} - 3 \beta_{18} ) q^{77} + ( 7 - 2 \beta_{2} - 2 \beta_{8} + \beta_{15} - \beta_{16} + \beta_{17} - \beta_{18} ) q^{78} + ( -\beta_{6} + 2 \beta_{12} - \beta_{15} + \beta_{16} ) q^{81} + ( -5 \beta_{1} - 3 \beta_{2} - 2 \beta_{6} - \beta_{12} + 2 \beta_{15} - \beta_{16} + \beta_{18} ) q^{82} + ( -2 \beta_{5} - 2 \beta_{7} - 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{13} - \beta_{14} ) q^{83} + ( -\beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} - \beta_{9} + \beta_{10} - 4 \beta_{11} + 3 \beta_{13} - \beta_{14} - 2 \beta_{19} ) q^{84} + ( 4 \beta_{1} + 3 \beta_{2} + 2 \beta_{6} + 2 \beta_{8} + 2 \beta_{12} - 3 \beta_{15} + \beta_{16} - \beta_{18} ) q^{88} + ( 3 \beta_{2} - \beta_{8} - \beta_{15} + \beta_{16} - \beta_{18} ) q^{91} + ( -1 - 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{6} - 2 \beta_{8} - \beta_{12} + \beta_{15} - 2 \beta_{16} - 2 \beta_{17} - 2 \beta_{18} ) q^{92} + ( -\beta_{3} + \beta_{4} + 3 \beta_{5} + 3 \beta_{7} + 3 \beta_{9} - 2 \beta_{10} - 3 \beta_{11} + 2 \beta_{14} - 2 \beta_{19} ) q^{96} + ( 7 \beta_{4} - 3 \beta_{5} + 2 \beta_{7} - 3 \beta_{10} - 2 \beta_{11} + 2 \beta_{13} + \beta_{14} ) q^{98} + ( \beta_{5} + \beta_{7} + \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q - 40q^{4} + O(q^{10})$$ $$20q - 40q^{4} + 80q^{16} - 100q^{25} + 10q^{36} - 50q^{42} + 70q^{48} + 140q^{49} - 160q^{64} - 110q^{66} + 130q^{78} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} - 53 x^{10} + 1024$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{16} + 11 \nu^{6}$$$$)/192$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{12} - 25 \nu^{2}$$$$)/12$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{15} + 16 \nu^{11} - 11 \nu^{5} - 400 \nu$$$$)/96$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{19} + 53 \nu^{9} - 512 \nu$$$$)/512$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{15} + 21 \nu^{5}$$$$)/32$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{14} + 37 \nu^{4}$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{17} + 2 \nu^{15} - 11 \nu^{7} + 22 \nu^{5}$$$$)/192$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{18} + 53 \nu^{8} + 256 \nu^{2}$$$$)/256$$ $$\beta_{9}$$ $$=$$ $$($$$$-3 \nu^{19} + 16 \nu^{15} + 159 \nu^{9} + 176 \nu^{5} + 3072 \nu$$$$)/1536$$ $$\beta_{10}$$ $$=$$ $$($$$$\nu^{17} - 53 \nu^{7} + 128 \nu^{3}$$$$)/128$$ $$\beta_{11}$$ $$=$$ $$($$$$\nu^{17} - 4 \nu^{15} + 11 \nu^{7} + 148 \nu^{5}$$$$)/192$$ $$\beta_{12}$$ $$=$$ $$($$$$\nu^{18} - 4 \nu^{14} + 11 \nu^{8} - 44 \nu^{4}$$$$)/192$$ $$\beta_{13}$$ $$=$$ $$($$$$-3 \nu^{17} - 16 \nu^{13} + 159 \nu^{7} + 592 \nu^{3}$$$$)/384$$ $$\beta_{14}$$ $$=$$ $$($$$$5 \nu^{17} - 16 \nu^{13} - 137 \nu^{7} + 208 \nu^{3}$$$$)/384$$ $$\beta_{15}$$ $$=$$ $$($$$$\nu^{18} - 4 \nu^{16} - 4 \nu^{14} + 11 \nu^{8} + 148 \nu^{6} + 148 \nu^{4}$$$$)/192$$ $$\beta_{16}$$ $$=$$ $$($$$$-\nu^{18} - 4 \nu^{16} - 4 \nu^{14} - 11 \nu^{8} + 148 \nu^{6} + 148 \nu^{4}$$$$)/192$$ $$\beta_{17}$$ $$=$$ $$($$$$-\nu^{18} + 64 \nu^{10} - 11 \nu^{8} - 1792$$$$)/192$$ $$\beta_{18}$$ $$=$$ $$($$$$5 \nu^{18} - 137 \nu^{8} + 384 \nu^{2}$$$$)/384$$ $$\beta_{19}$$ $$=$$ $$($$$$-7 \nu^{19} + 115 \nu^{9} + 768 \nu$$$$)/768$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{11} + 2 \beta_{9} - \beta_{7} + \beta_{5} - 2 \beta_{4}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{18} + \beta_{16} - \beta_{15} + 4 \beta_{8}$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{14} + 2 \beta_{13} + \beta_{11} + 4 \beta_{10} - \beta_{7} - \beta_{5}$$$$)/6$$ $$\nu^{4}$$ $$=$$ $$($$$$-3 \beta_{16} + 3 \beta_{15} - 6 \beta_{12} + 2 \beta_{6}$$$$)/6$$ $$\nu^{5}$$ $$=$$ $$($$$$3 \beta_{11} + 3 \beta_{7} - \beta_{5}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$3 \beta_{16} + 3 \beta_{15} - 2 \beta_{6} + 24 \beta_{1}$$$$)/6$$ $$\nu^{7}$$ $$=$$ $$($$$$-4 \beta_{14} + 4 \beta_{13} + 11 \beta_{11} - 4 \beta_{10} - 11 \beta_{7} - 11 \beta_{5}$$$$)/6$$ $$\nu^{8}$$ $$=$$ $$($$$$-8 \beta_{18} - 13 \beta_{16} + 13 \beta_{15} + 8 \beta_{8}$$$$)/6$$ $$\nu^{9}$$ $$=$$ $$($$$$-18 \beta_{19} - 17 \beta_{11} + 34 \beta_{9} - 17 \beta_{7} + 17 \beta_{5} + 50 \beta_{4}$$$$)/6$$ $$\nu^{10}$$ $$=$$ $$($$$$6 \beta_{17} - 3 \beta_{16} + 3 \beta_{15} + 56$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-7 \beta_{11} + 50 \beta_{9} - 7 \beta_{7} + 7 \beta_{5} - 50 \beta_{4} + 36 \beta_{3}$$$$)/6$$ $$\nu^{12}$$ $$=$$ $$($$$$50 \beta_{18} + 25 \beta_{16} - 25 \beta_{15} + 100 \beta_{8} + 72 \beta_{2}$$$$)/6$$ $$\nu^{13}$$ $$=$$ $$($$$$-122 \beta_{14} - 22 \beta_{13} + 61 \beta_{11} + 100 \beta_{10} - 61 \beta_{7} - 61 \beta_{5}$$$$)/6$$ $$\nu^{14}$$ $$=$$ $$($$$$-111 \beta_{16} + 111 \beta_{15} - 222 \beta_{12} - 22 \beta_{6}$$$$)/6$$ $$\nu^{15}$$ $$=$$ $$($$$$63 \beta_{11} + 63 \beta_{7} - 85 \beta_{5}$$$$)/2$$ $$\nu^{16}$$ $$=$$ $$($$$$-33 \beta_{16} - 33 \beta_{15} + 22 \beta_{6} + 888 \beta_{1}$$$$)/6$$ $$\nu^{17}$$ $$=$$ $$($$$$44 \beta_{14} - 44 \beta_{13} + 455 \beta_{11} + 44 \beta_{10} - 455 \beta_{7} - 455 \beta_{5}$$$$)/6$$ $$\nu^{18}$$ $$=$$ $$($$$$88 \beta_{18} - 433 \beta_{16} + 433 \beta_{15} - 88 \beta_{8}$$$$)/6$$ $$\nu^{19}$$ $$=$$ $$($$$$-954 \beta_{19} - 389 \beta_{11} + 778 \beta_{9} - 389 \beta_{7} + 389 \beta_{5} + 602 \beta_{4}$$$$)/6$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/429\mathbb{Z}\right)^\times$$.

 $$n$$ $$67$$ $$79$$ $$287$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
428.1
 −0.356257 + 1.36861i 0.356257 + 1.36861i −0.516228 + 1.31663i 0.516228 + 1.31663i 1.09266 + 0.897823i −1.09266 + 0.897823i −1.19153 + 0.761743i 1.19153 + 0.761743i −1.41171 + 0.0841020i 1.41171 + 0.0841020i 1.41171 − 0.0841020i −1.41171 − 0.0841020i 1.19153 − 0.761743i −1.19153 − 0.761743i −1.09266 − 0.897823i 1.09266 − 0.897823i 0.516228 − 1.31663i −0.516228 − 1.31663i 0.356257 − 1.36861i −0.356257 − 1.36861i
2.73721i −0.813662 1.52904i −5.49232 0 −4.18530 + 2.22717i 0.851686 9.55923i −1.67591 + 2.48824i 0
428.2 2.73721i −0.813662 + 1.52904i −5.49232 0 4.18530 + 2.22717i −0.851686 9.55923i −1.67591 2.48824i 0
428.3 2.63326i 1.55701 0.758758i −4.93403 0 −1.99800 4.10001i −4.70372 7.72606i 1.84857 2.36279i 0
428.4 2.63326i 1.55701 + 0.758758i −4.93403 0 1.99800 4.10001i 4.70372 7.72606i 1.84857 + 2.36279i 0
428.5 1.79565i −0.240479 1.71528i −1.22434 0 −3.08003 + 0.431815i 5.23009 1.39281i −2.88434 + 0.824975i 0
428.6 1.79565i −0.240479 + 1.71528i −1.22434 0 3.08003 + 0.431815i −5.23009 1.39281i −2.88434 0.824975i 0
428.7 1.52349i −1.70564 0.301340i −0.321008 0 −0.459088 + 2.59851i 3.75874 2.55792i 2.81839 + 1.02795i 0
428.8 1.52349i −1.70564 + 0.301340i −0.321008 0 0.459088 + 2.59851i −3.75874 2.55792i 2.81839 1.02795i 0
428.9 0.168204i 1.20277 1.24634i 1.97171 0 −0.209639 0.202310i 2.38069 0.668057i −0.106712 2.99810i 0
428.10 0.168204i 1.20277 + 1.24634i 1.97171 0 0.209639 0.202310i −2.38069 0.668057i −0.106712 + 2.99810i 0
428.11 0.168204i 1.20277 1.24634i 1.97171 0 0.209639 + 0.202310i −2.38069 0.668057i −0.106712 2.99810i 0
428.12 0.168204i 1.20277 + 1.24634i 1.97171 0 −0.209639 + 0.202310i 2.38069 0.668057i −0.106712 + 2.99810i 0
428.13 1.52349i −1.70564 0.301340i −0.321008 0 0.459088 2.59851i −3.75874 2.55792i 2.81839 + 1.02795i 0
428.14 1.52349i −1.70564 + 0.301340i −0.321008 0 −0.459088 2.59851i 3.75874 2.55792i 2.81839 1.02795i 0
428.15 1.79565i −0.240479 1.71528i −1.22434 0 3.08003 0.431815i −5.23009 1.39281i −2.88434 + 0.824975i 0
428.16 1.79565i −0.240479 + 1.71528i −1.22434 0 −3.08003 0.431815i 5.23009 1.39281i −2.88434 0.824975i 0
428.17 2.63326i 1.55701 0.758758i −4.93403 0 1.99800 + 4.10001i 4.70372 7.72606i 1.84857 2.36279i 0
428.18 2.63326i 1.55701 + 0.758758i −4.93403 0 −1.99800 + 4.10001i −4.70372 7.72606i 1.84857 + 2.36279i 0
428.19 2.73721i −0.813662 1.52904i −5.49232 0 4.18530 2.22717i −0.851686 9.55923i −1.67591 + 2.48824i 0
428.20 2.73721i −0.813662 + 1.52904i −5.49232 0 −4.18530 2.22717i 0.851686 9.55923i −1.67591 2.48824i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 428.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
143.d odd 2 1 CM by $$\Q(\sqrt{-143})$$
3.b odd 2 1 inner
11.b odd 2 1 inner
13.b even 2 1 inner
33.d even 2 1 inner
39.d odd 2 1 inner
429.e even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.e.d 20
3.b odd 2 1 inner 429.2.e.d 20
11.b odd 2 1 inner 429.2.e.d 20
13.b even 2 1 inner 429.2.e.d 20
33.d even 2 1 inner 429.2.e.d 20
39.d odd 2 1 inner 429.2.e.d 20
143.d odd 2 1 CM 429.2.e.d 20
429.e even 2 1 inner 429.2.e.d 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.e.d 20 1.a even 1 1 trivial
429.2.e.d 20 3.b odd 2 1 inner
429.2.e.d 20 11.b odd 2 1 inner
429.2.e.d 20 13.b even 2 1 inner
429.2.e.d 20 33.d even 2 1 inner
429.2.e.d 20 39.d odd 2 1 inner
429.2.e.d 20 143.d odd 2 1 CM
429.2.e.d 20 429.e even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} + 20 T_{2}^{8} + 140 T_{2}^{6} + 400 T_{2}^{4} + 400 T_{2}^{2} + 11$$ acting on $$S_{2}^{\mathrm{new}}(429, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 11 + 400 T^{2} + 400 T^{4} + 140 T^{6} + 20 T^{8} + T^{10} )^{2}$$
$3$ $$( 243 + 20 T^{5} + T^{10} )^{2}$$
$5$ $$T^{20}$$
$7$ $$( -35152 + 60025 T^{2} - 17150 T^{4} + 1715 T^{6} - 70 T^{8} + T^{10} )^{2}$$
$11$ $$( 11 + T^{2} )^{10}$$
$13$ $$( -13 + T^{2} )^{10}$$
$17$ $$T^{20}$$
$19$ $$( -9884992 + 3258025 T^{2} - 342950 T^{4} + 12635 T^{6} - 190 T^{8} + T^{10} )^{2}$$
$23$ $$( 3391388 + 6996025 T^{2} + 608350 T^{4} + 18515 T^{6} + 230 T^{8} + T^{10} )^{2}$$
$29$ $$T^{20}$$
$31$ $$T^{20}$$
$37$ $$T^{20}$$
$41$ $$( 93104 + 70644025 T^{2} + 3446050 T^{4} + 58835 T^{6} + 410 T^{8} + T^{10} )^{2}$$
$43$ $$T^{20}$$
$47$ $$T^{20}$$
$53$ $$( 1054861808 + 197262025 T^{2} + 7443850 T^{4} + 98315 T^{6} + 530 T^{8} + T^{10} )^{2}$$
$59$ $$T^{20}$$
$61$ $$T^{20}$$
$67$ $$T^{20}$$
$71$ $$T^{20}$$
$73$ $$( -7761264628 + 709956025 T^{2} - 19450850 T^{4} + 186515 T^{6} - 730 T^{8} + T^{10} )^{2}$$
$79$ $$T^{20}$$
$83$ $$( 529494284 + 1186458025 T^{2} + 28589350 T^{4} + 241115 T^{6} + 830 T^{8} + T^{10} )^{2}$$
$89$ $$T^{20}$$
$97$ $$T^{20}$$
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