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 $8$

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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:

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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