Properties

Label 1512.2.c.e
Level 1512
Weight 2
Character orbit 1512.c
Analytic conductor 12.073
Analytic rank 0
Dimension 20
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1512.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.0733807856\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + \beta_{14} q^{5} - q^{7} + \beta_{3} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + \beta_{14} q^{5} - q^{7} + \beta_{3} q^{8} + ( -1 + \beta_{17} ) q^{10} + ( \beta_{3} + \beta_{6} - \beta_{12} - \beta_{14} ) q^{11} + ( -\beta_{2} - \beta_{13} ) q^{13} -\beta_{1} q^{14} + ( -1 + \beta_{4} ) q^{16} + ( \beta_{3} + \beta_{8} + \beta_{9} ) q^{17} + ( -\beta_{10} - \beta_{13} + \beta_{17} + \beta_{18} ) q^{19} + ( \beta_{3} + \beta_{6} + \beta_{8} - \beta_{12} - \beta_{14} ) q^{20} + ( \beta_{4} + \beta_{10} + 2 \beta_{13} - \beta_{17} ) q^{22} + ( \beta_{3} + \beta_{12} ) q^{23} + ( -2 + \beta_{4} - \beta_{15} ) q^{25} + ( -\beta_{3} - \beta_{16} ) q^{26} -\beta_{2} q^{28} + ( -2 \beta_{1} - \beta_{6} + \beta_{14} - \beta_{16} ) q^{29} + ( 2 + \beta_{2} - \beta_{13} ) q^{31} + ( -\beta_{1} + \beta_{6} + \beta_{7} ) q^{32} + ( -\beta_{2} + \beta_{5} - \beta_{18} ) q^{34} -\beta_{14} q^{35} + ( \beta_{4} + \beta_{5} - \beta_{11} + \beta_{15} + \beta_{17} ) q^{37} + ( -\beta_{6} + \beta_{8} + \beta_{9} + \beta_{14} + \beta_{19} ) q^{38} + ( -1 - \beta_{2} + \beta_{10} + \beta_{11} + 2 \beta_{13} - \beta_{17} - \beta_{18} ) q^{40} + ( \beta_{1} + \beta_{3} - \beta_{7} + \beta_{8} + \beta_{9} + \beta_{19} ) q^{41} + ( 1 - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{10} - \beta_{13} ) q^{43} + ( -\beta_{1} - \beta_{3} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{14} + \beta_{16} - \beta_{19} ) q^{44} + ( -1 + \beta_{4} - 2 \beta_{13} ) q^{46} + ( -2 \beta_{1} - \beta_{3} - \beta_{6} - \beta_{8} + \beta_{9} + \beta_{14} + \beta_{16} ) q^{47} + q^{49} + ( -\beta_{1} + \beta_{6} + \beta_{7} + 2 \beta_{12} ) q^{50} + ( 5 - \beta_{4} ) q^{52} + ( -\beta_{1} - 2 \beta_{6} - \beta_{7} + 2 \beta_{14} - \beta_{16} - \beta_{19} ) q^{53} + ( 1 - 2 \beta_{2} + 2 \beta_{10} + \beta_{11} + 3 \beta_{13} - \beta_{15} + \beta_{18} ) q^{55} -\beta_{3} q^{56} + ( 3 - 2 \beta_{2} - \beta_{10} + \beta_{17} ) q^{58} + ( -2 \beta_{1} + \beta_{6} - \beta_{16} ) q^{59} + ( -1 - \beta_{11} - \beta_{13} - \beta_{15} + 2 \beta_{17} + \beta_{18} ) q^{61} + ( 2 \beta_{1} + \beta_{3} - \beta_{16} ) q^{62} + ( 1 - \beta_{2} + \beta_{5} + \beta_{17} + \beta_{18} ) q^{64} + ( -\beta_{3} - \beta_{6} - 2 \beta_{8} + \beta_{12} + \beta_{14} ) q^{65} + ( \beta_{4} + \beta_{15} ) q^{67} + ( \beta_{1} - \beta_{3} + 2 \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{12} - 3 \beta_{14} ) q^{68} + ( 1 - \beta_{17} ) q^{70} + ( -3 \beta_{1} + \beta_{7} + \beta_{16} - \beta_{19} ) q^{71} + ( -3 \beta_{2} - \beta_{4} - \beta_{10} + 2 \beta_{13} + \beta_{15} + \beta_{17} - \beta_{18} ) q^{73} + ( 3 \beta_{6} + \beta_{7} - 2 \beta_{9} - 2 \beta_{12} ) q^{74} + ( 2 - \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{10} + 2 \beta_{15} + \beta_{17} - \beta_{18} ) q^{76} + ( -\beta_{3} - \beta_{6} + \beta_{12} + \beta_{14} ) q^{77} + ( 2 - \beta_{2} + \beta_{10} + \beta_{11} + \beta_{13} - \beta_{15} + \beta_{17} ) q^{79} + ( -\beta_{1} - \beta_{3} + \beta_{7} - \beta_{8} - \beta_{9} - 3 \beta_{14} + \beta_{16} - \beta_{19} ) q^{80} + ( 1 + 2 \beta_{15} - \beta_{17} - 2 \beta_{18} ) q^{82} + ( 3 \beta_{1} + 2 \beta_{3} - \beta_{6} - \beta_{7} - 2 \beta_{12} + \beta_{16} - \beta_{19} ) q^{83} + ( 1 - 3 \beta_{2} + \beta_{5} - 2 \beta_{13} + \beta_{15} - \beta_{17} - \beta_{18} ) q^{85} + ( \beta_{1} - 2 \beta_{3} - \beta_{6} - \beta_{8} - \beta_{9} + 2 \beta_{12} - \beta_{14} + \beta_{19} ) q^{86} + ( -6 - 2 \beta_{15} + 2 \beta_{17} + 2 \beta_{18} ) q^{88} + ( -\beta_{1} - \beta_{7} + \beta_{16} + \beta_{19} ) q^{89} + ( \beta_{2} + \beta_{13} ) q^{91} + ( -\beta_{1} + \beta_{6} + \beta_{7} - 2 \beta_{16} ) q^{92} + ( -1 - \beta_{2} + \beta_{5} - \beta_{10} - 2 \beta_{11} + \beta_{17} + \beta_{18} ) q^{94} + ( -\beta_{1} + 2 \beta_{3} - \beta_{6} - \beta_{7} + 2 \beta_{9} + 2 \beta_{12} + \beta_{14} + \beta_{16} + \beta_{19} ) q^{95} + ( 3 - \beta_{2} + \beta_{4} - \beta_{10} - \beta_{11} + \beta_{13} - \beta_{17} ) q^{97} + \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 2q^{4} - 20q^{7} + O(q^{10}) \) \( 20q - 2q^{4} - 20q^{7} - 12q^{10} - 14q^{16} + 8q^{22} - 28q^{25} + 2q^{28} + 36q^{31} - 6q^{34} - 16q^{40} - 18q^{46} + 20q^{49} + 94q^{52} + 48q^{55} + 66q^{58} + 22q^{64} + 12q^{70} + 12q^{76} + 64q^{79} - 92q^{88} - 24q^{94} + 56q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} + x^{18} + 4 x^{16} + 8 x^{12} + 4 x^{10} + 32 x^{8} + 256 x^{4} + 256 x^{2} + 1024\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\( \nu^{3} \)
\(\beta_{4}\)\(=\)\( \nu^{4} + 1 \)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{18} + 11 \nu^{16} - 24 \nu^{14} - 48 \nu^{12} + 120 \nu^{10} - 100 \nu^{8} + 1168 \nu^{6} - 128 \nu^{4} + 1280 \nu^{2} + 512 \)\()/1792\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{19} - 3 \nu^{17} + 4 \nu^{15} - 6 \nu^{13} + 8 \nu^{11} - 100 \nu^{9} - 120 \nu^{7} + 40 \nu^{5} - 288 \nu^{3} - 832 \nu \)\()/896\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{19} + 3 \nu^{17} - 4 \nu^{15} + 6 \nu^{13} - 8 \nu^{11} + 100 \nu^{9} + 120 \nu^{7} + 856 \nu^{5} + 288 \nu^{3} + 1728 \nu \)\()/896\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{19} - 10 \nu^{17} + 11 \nu^{15} - 20 \nu^{13} - 48 \nu^{11} - 44 \nu^{9} + 188 \nu^{7} - 128 \nu^{5} - 288 \nu^{3} - 3072 \nu \)\()/896\)
\(\beta_{9}\)\(=\)\((\)\( \nu^{19} + 3 \nu^{17} - 4 \nu^{15} + 6 \nu^{13} - 8 \nu^{11} + 100 \nu^{9} - 328 \nu^{7} + 408 \nu^{5} - 608 \nu^{3} + 1728 \nu \)\()/896\)
\(\beta_{10}\)\(=\)\((\)\( -\nu^{18} + 4 \nu^{16} - 3 \nu^{14} + 8 \nu^{12} - 48 \nu^{10} - 44 \nu^{8} + 20 \nu^{6} - 16 \nu^{4} - 288 \nu^{2} + 512 \)\()/448\)
\(\beta_{11}\)\(=\)\((\)\( -5 \nu^{18} + 27 \nu^{16} - 36 \nu^{14} - 16 \nu^{12} - 72 \nu^{10} + 620 \nu^{8} + 352 \nu^{6} + 1600 \nu^{4} - 1664 \nu^{2} + 6144 \)\()/1792\)
\(\beta_{12}\)\(=\)\((\)\( \nu^{19} - 3 \nu^{17} - 16 \nu^{13} + 8 \nu^{11} - 28 \nu^{9} + 16 \nu^{7} - 128 \nu^{5} + 256 \nu^{3} - 768 \nu \)\()/512\)
\(\beta_{13}\)\(=\)\((\)\( \nu^{18} + \nu^{16} + 4 \nu^{14} + 8 \nu^{10} + 4 \nu^{8} + 32 \nu^{6} + 256 \nu^{2} + 256 \)\()/256\)
\(\beta_{14}\)\(=\)\((\)\( 9 \nu^{19} - 15 \nu^{17} + 20 \nu^{15} + 40 \nu^{13} + 40 \nu^{11} - 220 \nu^{9} + 128 \nu^{7} + 32 \nu^{5} + 2816 \nu^{3} - 1024 \nu \)\()/3584\)
\(\beta_{15}\)\(=\)\((\)\( -\nu^{18} + 3 \nu^{16} + 16 \nu^{12} - 8 \nu^{10} + 28 \nu^{8} - 16 \nu^{6} + 128 \nu^{4} - 256 \nu^{2} + 512 \)\()/256\)
\(\beta_{16}\)\(=\)\((\)\( \nu^{19} + \nu^{17} + 4 \nu^{15} + 8 \nu^{11} + 4 \nu^{9} + 32 \nu^{7} + 256 \nu^{3} + 256 \nu \)\()/256\)
\(\beta_{17}\)\(=\)\((\)\( -3 \nu^{18} - 2 \nu^{16} + 5 \nu^{14} - 4 \nu^{12} - 32 \nu^{10} - 20 \nu^{8} + 4 \nu^{6} + 64 \nu^{4} - 416 \nu^{2} - 704 \)\()/448\)
\(\beta_{18}\)\(=\)\((\)\( 13 \nu^{18} - 3 \nu^{16} + 4 \nu^{14} + 64 \nu^{12} + 8 \nu^{10} + 180 \nu^{8} + 608 \nu^{6} - 128 \nu^{4} + 2176 \nu^{2} + 512 \)\()/1792\)
\(\beta_{19}\)\(=\)\((\)\( -17 \nu^{19} - 37 \nu^{17} - 16 \nu^{15} + 24 \nu^{13} + 248 \nu^{11} + 92 \nu^{9} + 592 \nu^{7} - 608 \nu^{5} - 640 \nu^{3} - 9216 \nu \)\()/3584\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{3}\)
\(\nu^{4}\)\(=\)\(\beta_{4} - 1\)
\(\nu^{5}\)\(=\)\(\beta_{7} + \beta_{6} - \beta_{1}\)
\(\nu^{6}\)\(=\)\(\beta_{18} + \beta_{17} + \beta_{5} - \beta_{2} + 1\)
\(\nu^{7}\)\(=\)\(-2 \beta_{9} + \beta_{7} - \beta_{6} - 2 \beta_{3} + \beta_{1}\)
\(\nu^{8}\)\(=\)\(\beta_{18} + \beta_{17} + 2 \beta_{11} - 2 \beta_{10} - \beta_{5} - 2 \beta_{4} + \beta_{2} - 1\)
\(\nu^{9}\)\(=\)\(2 \beta_{19} + 2 \beta_{16} - 2 \beta_{14} + 4 \beta_{9} + 2 \beta_{8} - \beta_{7} - 5 \beta_{6} + 2 \beta_{3} - \beta_{1}\)
\(\nu^{10}\)\(=\)\(-3 \beta_{18} - 3 \beta_{17} + 4 \beta_{15} - 2 \beta_{11} - 6 \beta_{10} + 3 \beta_{5} - 3 \beta_{2} + 1\)
\(\nu^{11}\)\(=\)\(6 \beta_{19} + 6 \beta_{16} - 2 \beta_{14} + 4 \beta_{12} - 6 \beta_{8} + \beta_{7} + 5 \beta_{6} - 6 \beta_{3} - 3 \beta_{1}\)
\(\nu^{12}\)\(=\)\(7 \beta_{18} - \beta_{17} + 12 \beta_{15} - 8 \beta_{13} - 6 \beta_{11} - 2 \beta_{10} + \beta_{5} + 3 \beta_{2} + 3\)
\(\nu^{13}\)\(=\)\(2 \beta_{19} - 6 \beta_{16} + 26 \beta_{14} - 20 \beta_{12} - 2 \beta_{8} - 5 \beta_{7} - 9 \beta_{6} - 6 \beta_{3} - 17 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-3 \beta_{18} + 21 \beta_{17} + 4 \beta_{15} + 40 \beta_{13} - 2 \beta_{11} - 6 \beta_{10} - 5 \beta_{5} - 4 \beta_{4} - 15 \beta_{2} + 5\)
\(\nu^{15}\)\(=\)\(6 \beta_{19} + 46 \beta_{16} - 18 \beta_{14} - 28 \beta_{12} + 16 \beta_{9} + 26 \beta_{8} - 11 \beta_{7} + 25 \beta_{6} + 14 \beta_{3} + 25 \beta_{1}\)
\(\nu^{16}\)\(=\)\(-37 \beta_{18} - 29 \beta_{17} + 12 \beta_{15} + 56 \beta_{13} + 10 \beta_{11} + 30 \beta_{10} + 5 \beta_{5} - 12 \beta_{4} - \beta_{2} - 173\)
\(\nu^{17}\)\(=\)\(-30 \beta_{19} + 26 \beta_{16} - 70 \beta_{14} - 20 \beta_{12} - 40 \beta_{9} - 34 \beta_{8} + 3 \beta_{7} + 23 \beta_{6} + 2 \beta_{3} - 177 \beta_{1}\)
\(\nu^{18}\)\(=\)\(37 \beta_{18} - 67 \beta_{17} - 60 \beta_{15} + 40 \beta_{13} + 6 \beta_{11} + 50 \beta_{10} - 37 \beta_{5} + 36 \beta_{4} - 143 \beta_{2} - 139\)
\(\nu^{19}\)\(=\)\(-50 \beta_{19} - 10 \beta_{16} + 166 \beta_{14} + 100 \beta_{12} + 24 \beta_{9} - 30 \beta_{8} + 5 \beta_{7} - 111 \beta_{6} - 210 \beta_{3} - 183 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
757.1
−1.37874 0.314750i
−1.37874 + 0.314750i
−1.19566 0.755240i
−1.19566 + 0.755240i
−0.885915 1.10234i
−0.885915 + 1.10234i
−0.725842 1.21374i
−0.725842 + 1.21374i
−0.328272 1.37559i
−0.328272 + 1.37559i
0.328272 1.37559i
0.328272 + 1.37559i
0.725842 1.21374i
0.725842 + 1.21374i
0.885915 1.10234i
0.885915 + 1.10234i
1.19566 0.755240i
1.19566 + 0.755240i
1.37874 0.314750i
1.37874 + 0.314750i
−1.37874 0.314750i 0 1.80186 + 0.867919i 0.114591i 0 −1.00000 −2.21113 1.76377i 0 −0.0360676 + 0.157992i
757.2 −1.37874 + 0.314750i 0 1.80186 0.867919i 0.114591i 0 −1.00000 −2.21113 + 1.76377i 0 −0.0360676 0.157992i
757.3 −1.19566 0.755240i 0 0.859226 + 1.80603i 3.16969i 0 −1.00000 0.336637 2.80832i 0 −2.39387 + 3.78988i
757.4 −1.19566 + 0.755240i 0 0.859226 1.80603i 3.16969i 0 −1.00000 0.336637 + 2.80832i 0 −2.39387 3.78988i
757.5 −0.885915 1.10234i 0 −0.430309 + 1.95316i 3.50133i 0 −1.00000 2.53426 1.25599i 0 3.85966 3.10188i
757.6 −0.885915 + 1.10234i 0 −0.430309 1.95316i 3.50133i 0 −1.00000 2.53426 + 1.25599i 0 3.85966 + 3.10188i
757.7 −0.725842 1.21374i 0 −0.946308 + 1.76196i 3.06888i 0 −1.00000 2.82542 0.130337i 0 −3.72481 + 2.22752i
757.8 −0.725842 + 1.21374i 0 −0.946308 1.76196i 3.06888i 0 −1.00000 2.82542 + 0.130337i 0 −3.72481 2.22752i
757.9 −0.328272 1.37559i 0 −1.78447 + 0.903134i 0.512447i 0 −1.00000 1.82813 + 2.15822i 0 −0.704915 + 0.168222i
757.10 −0.328272 + 1.37559i 0 −1.78447 0.903134i 0.512447i 0 −1.00000 1.82813 2.15822i 0 −0.704915 0.168222i
757.11 0.328272 1.37559i 0 −1.78447 0.903134i 0.512447i 0 −1.00000 −1.82813 + 2.15822i 0 −0.704915 0.168222i
757.12 0.328272 + 1.37559i 0 −1.78447 + 0.903134i 0.512447i 0 −1.00000 −1.82813 2.15822i 0 −0.704915 + 0.168222i
757.13 0.725842 1.21374i 0 −0.946308 1.76196i 3.06888i 0 −1.00000 −2.82542 0.130337i 0 −3.72481 2.22752i
757.14 0.725842 + 1.21374i 0 −0.946308 + 1.76196i 3.06888i 0 −1.00000 −2.82542 + 0.130337i 0 −3.72481 + 2.22752i
757.15 0.885915 1.10234i 0 −0.430309 1.95316i 3.50133i 0 −1.00000 −2.53426 1.25599i 0 3.85966 + 3.10188i
757.16 0.885915 + 1.10234i 0 −0.430309 + 1.95316i 3.50133i 0 −1.00000 −2.53426 + 1.25599i 0 3.85966 3.10188i
757.17 1.19566 0.755240i 0 0.859226 1.80603i 3.16969i 0 −1.00000 −0.336637 2.80832i 0 −2.39387 3.78988i
757.18 1.19566 + 0.755240i 0 0.859226 + 1.80603i 3.16969i 0 −1.00000 −0.336637 + 2.80832i 0 −2.39387 + 3.78988i
757.19 1.37874 0.314750i 0 1.80186 0.867919i 0.114591i 0 −1.00000 2.21113 1.76377i 0 −0.0360676 0.157992i
757.20 1.37874 + 0.314750i 0 1.80186 + 0.867919i 0.114591i 0 −1.00000 2.21113 + 1.76377i 0 −0.0360676 + 0.157992i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 757.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1512.2.c.e 20
3.b odd 2 1 inner 1512.2.c.e 20
4.b odd 2 1 6048.2.c.e 20
8.b even 2 1 inner 1512.2.c.e 20
8.d odd 2 1 6048.2.c.e 20
12.b even 2 1 6048.2.c.e 20
24.f even 2 1 6048.2.c.e 20
24.h odd 2 1 inner 1512.2.c.e 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.c.e 20 1.a even 1 1 trivial
1512.2.c.e 20 3.b odd 2 1 inner
1512.2.c.e 20 8.b even 2 1 inner
1512.2.c.e 20 24.h odd 2 1 inner
6048.2.c.e 20 4.b odd 2 1
6048.2.c.e 20 8.d odd 2 1
6048.2.c.e 20 12.b even 2 1
6048.2.c.e 20 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1512, [\chi])\):

\( T_{5}^{10} + 32 T_{5}^{8} + 342 T_{5}^{6} + 1252 T_{5}^{4} + 321 T_{5}^{2} + 4 \)
\( T_{17}^{10} - 131 T_{17}^{8} + 5266 T_{17}^{6} - 69022 T_{17}^{4} + 319757 T_{17}^{2} - 395839 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} + 4 T^{4} + 8 T^{8} + 4 T^{10} + 32 T^{12} + 256 T^{16} + 256 T^{18} + 1024 T^{20} \)
$3$ 1
$5$ \( ( 1 - 18 T^{2} + 187 T^{4} - 1608 T^{6} + 10781 T^{8} - 57906 T^{10} + 269525 T^{12} - 1005000 T^{14} + 2921875 T^{16} - 7031250 T^{18} + 9765625 T^{20} )^{2} \)
$7$ \( ( 1 + T )^{20} \)
$11$ \( ( 1 - 34 T^{2} + 589 T^{4} - 8472 T^{6} + 122234 T^{8} - 1528172 T^{10} + 14790314 T^{12} - 124038552 T^{14} + 1043449429 T^{16} - 7288201954 T^{18} + 25937424601 T^{20} )^{2} \)
$13$ \( ( 1 - 83 T^{2} + 3533 T^{4} - 98556 T^{6} + 1975738 T^{8} - 29558418 T^{10} + 333899722 T^{12} - 2814857916 T^{14} + 17053116197 T^{16} - 67705649843 T^{18} + 137858491849 T^{20} )^{2} \)
$17$ \( ( 1 + 39 T^{2} + 455 T^{4} - 2382 T^{6} - 47987 T^{8} + 145305 T^{10} - 13868243 T^{12} - 198947022 T^{14} + 10982593895 T^{16} + 272054540199 T^{18} + 2015993900449 T^{20} )^{2} \)
$19$ \( ( 1 - 70 T^{2} + 3325 T^{4} - 107448 T^{6} + 2823098 T^{8} - 58351460 T^{10} + 1019138378 T^{12} - 14002730808 T^{14} + 156427554325 T^{16} - 1188849412870 T^{18} + 6131066257801 T^{20} )^{2} \)
$23$ \( ( 1 + 139 T^{2} + 10081 T^{4} + 485484 T^{6} + 16965806 T^{8} + 447075730 T^{10} + 8974911374 T^{12} + 135858328044 T^{14} + 1492349797009 T^{16} + 10885226954059 T^{18} + 41426511213649 T^{20} )^{2} \)
$29$ \( ( 1 - 189 T^{2} + 17145 T^{4} - 999612 T^{6} + 42327774 T^{8} - 1383568894 T^{10} + 35597657934 T^{12} - 707006574972 T^{14} + 10198245838545 T^{16} - 94546572049629 T^{18} + 420707233300201 T^{20} )^{2} \)
$31$ \( ( 1 - 9 T + 171 T^{2} - 1092 T^{3} + 11058 T^{4} - 50402 T^{5} + 342798 T^{6} - 1049412 T^{7} + 5094261 T^{8} - 8311689 T^{9} + 28629151 T^{10} )^{4} \)
$37$ \( ( 1 - 22 T^{2} + 1771 T^{4} + 13920 T^{6} + 526445 T^{8} + 112600450 T^{10} + 720703205 T^{12} + 26088321120 T^{14} + 4543901470339 T^{16} - 77274547986262 T^{18} + 4808584372417849 T^{20} )^{2} \)
$41$ \( ( 1 + 190 T^{2} + 16651 T^{4} + 899952 T^{6} + 36306749 T^{8} + 1400716342 T^{10} + 61031645069 T^{12} + 2543049263472 T^{14} + 79093985716891 T^{16} + 1517135793532990 T^{18} + 13422659310152401 T^{20} )^{2} \)
$43$ \( ( 1 - 131 T^{2} + 5615 T^{4} - 93162 T^{6} + 5954173 T^{8} - 455929005 T^{10} + 11009265877 T^{12} - 318502338762 T^{14} + 35494453520135 T^{16} - 1531154236365731 T^{18} + 21611482313284249 T^{20} )^{2} \)
$47$ \( ( 1 + 110 T^{2} + 11187 T^{4} + 742128 T^{6} + 47768349 T^{8} + 2326511142 T^{10} + 105520282941 T^{12} + 3621347901168 T^{14} + 120587081885523 T^{16} + 2619241532793710 T^{18} + 52599132235830049 T^{20} )^{2} \)
$53$ \( ( 1 - 193 T^{2} + 20557 T^{4} - 1818204 T^{6} + 128583722 T^{8} - 7326778022 T^{10} + 361191675098 T^{12} - 14346504116124 T^{14} + 455632771728853 T^{16} - 12016120249392673 T^{18} + 174887470365513049 T^{20} )^{2} \)
$59$ \( ( 1 - 449 T^{2} + 96439 T^{4} - 13053078 T^{6} + 1232082461 T^{8} - 84671750375 T^{10} + 4288879046741 T^{12} - 158168858287158 T^{14} + 4067848483804399 T^{16} - 65926866484340129 T^{18} + 511116753300641401 T^{20} )^{2} \)
$61$ \( ( 1 - 186 T^{2} + 23517 T^{4} - 2152152 T^{6} + 175128378 T^{8} - 11500636220 T^{10} + 651652694538 T^{12} - 29798354399832 T^{14} + 1211604643847637 T^{16} - 35657560217494266 T^{18} + 713342911662882601 T^{20} )^{2} \)
$67$ \( ( 1 - 495 T^{2} + 120093 T^{4} - 18487716 T^{6} + 1986970986 T^{8} - 155229602234 T^{10} + 8919512756154 T^{12} - 372548202129636 T^{14} + 10863418489821717 T^{16} - 201003500390537295 T^{18} + 1822837804551761449 T^{20} )^{2} \)
$71$ \( ( 1 + 343 T^{2} + 52613 T^{4} + 4802116 T^{6} + 309151594 T^{8} + 19321038474 T^{10} + 1558433185354 T^{12} + 122029839916996 T^{14} + 6739740237935573 T^{16} + 221493461217296023 T^{18} + 3255243551009881201 T^{20} )^{2} \)
$73$ \( ( 1 + 149 T^{2} + 432 T^{3} + 14410 T^{4} + 51408 T^{5} + 1051930 T^{6} + 2302128 T^{7} + 57963533 T^{8} + 2073071593 T^{10} )^{4} \)
$79$ \( ( 1 - 16 T + 335 T^{2} - 3202 T^{3} + 40117 T^{4} - 296322 T^{5} + 3169243 T^{6} - 19983682 T^{7} + 165168065 T^{8} - 623201296 T^{9} + 3077056399 T^{10} )^{4} \)
$83$ \( ( 1 - 138 T^{2} + 13627 T^{4} - 1173792 T^{6} + 82503245 T^{8} - 3517947618 T^{10} + 568364854805 T^{12} - 55706197523232 T^{14} + 4455216467899363 T^{16} - 310816328035187658 T^{18} + 15516041187205853449 T^{20} )^{2} \)
$89$ \( ( 1 + 547 T^{2} + 153205 T^{4} + 28413396 T^{6} + 3828024122 T^{8} + 389779473874 T^{10} + 30321779070362 T^{12} + 1782720139460436 T^{14} + 76140018681680005 T^{16} + 2153314076719038307 T^{18} + 31181719929966183601 T^{20} )^{2} \)
$97$ \( ( 1 - 14 T + 341 T^{2} - 3480 T^{3} + 53038 T^{4} - 434964 T^{5} + 5144686 T^{6} - 32743320 T^{7} + 311221493 T^{8} - 1239409934 T^{9} + 8587340257 T^{10} )^{4} \)
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