Properties

Label 930.2.h.c
Level $930$
Weight $2$
Character orbit 930.h
Analytic conductor $7.426$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 2 x^{14} + 10 x^{12} - 42 x^{10} + 82 x^{8} - 378 x^{6} + 810 x^{4} - 1458 x^{2} + 6561\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} -\beta_{5} q^{3} - q^{4} + \beta_{3} q^{5} + \beta_{1} q^{6} + ( -1 - \beta_{4} ) q^{7} -\beta_{3} q^{8} + ( -\beta_{2} - \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{3} q^{2} -\beta_{5} q^{3} - q^{4} + \beta_{3} q^{5} + \beta_{1} q^{6} + ( -1 - \beta_{4} ) q^{7} -\beta_{3} q^{8} + ( -\beta_{2} - \beta_{3} ) q^{9} - q^{10} -\beta_{12} q^{11} + \beta_{5} q^{12} + ( \beta_{1} - 2 \beta_{10} + \beta_{14} ) q^{13} + ( -\beta_{3} - \beta_{7} ) q^{14} + \beta_{1} q^{15} + q^{16} + ( -\beta_{5} - \beta_{6} - \beta_{11} ) q^{17} + ( 1 + \beta_{9} ) q^{18} + ( -1 + \beta_{2} + \beta_{4} + \beta_{15} ) q^{19} -\beta_{3} q^{20} + ( \beta_{5} - \beta_{6} - \beta_{10} - \beta_{12} + \beta_{14} ) q^{21} + \beta_{8} q^{22} + ( \beta_{1} - \beta_{12} - \beta_{14} ) q^{23} -\beta_{1} q^{24} - q^{25} + ( \beta_{5} - 2 \beta_{6} + \beta_{11} ) q^{26} + ( \beta_{5} + \beta_{8} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{14} ) q^{27} + ( 1 + \beta_{4} ) q^{28} + ( -\beta_{6} - 2 \beta_{12} ) q^{29} + \beta_{5} q^{30} + ( \beta_{1} + \beta_{2} - \beta_{4} + \beta_{8} + \beta_{9} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{31} + \beta_{3} q^{32} + ( -1 - \beta_{4} + \beta_{9} - \beta_{13} + \beta_{15} ) q^{33} + ( \beta_{1} + \beta_{10} + \beta_{14} ) q^{34} + ( -\beta_{3} - \beta_{7} ) q^{35} + ( \beta_{2} + \beta_{3} ) q^{36} + ( \beta_{5} - \beta_{11} ) q^{37} + ( -\beta_{3} + \beta_{7} - \beta_{9} + \beta_{13} ) q^{38} + ( -1 + \beta_{3} + 2 \beta_{4} + 2 \beta_{7} + \beta_{9} + 2 \beta_{15} ) q^{39} + q^{40} + ( -2 \beta_{3} + 2 \beta_{7} + \beta_{9} - \beta_{13} ) q^{41} + ( -\beta_{1} - \beta_{6} + \beta_{8} + \beta_{10} + \beta_{11} ) q^{42} + ( 2 \beta_{1} - 3 \beta_{5} + \beta_{8} + \beta_{10} + 3 \beta_{11} + 2 \beta_{14} ) q^{43} + \beta_{12} q^{44} + ( 1 + \beta_{9} ) q^{45} + ( \beta_{5} + \beta_{8} - \beta_{11} ) q^{46} + ( \beta_{2} - 2 \beta_{9} + 2 \beta_{13} - \beta_{15} ) q^{47} -\beta_{5} q^{48} + ( -\beta_{2} + 3 \beta_{4} - \beta_{15} ) q^{49} -\beta_{3} q^{50} + ( 4 - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{7} + \beta_{13} ) q^{51} + ( -\beta_{1} + 2 \beta_{10} - \beta_{14} ) q^{52} + ( -\beta_{5} + \beta_{6} - \beta_{11} - 3 \beta_{12} ) q^{53} + ( -\beta_{1} - \beta_{6} - \beta_{8} - \beta_{11} + \beta_{12} - \beta_{14} ) q^{54} + \beta_{8} q^{55} + ( \beta_{3} + \beta_{7} ) q^{56} + ( \beta_{6} - \beta_{8} + 2 \beta_{10} + 2 \beta_{11} ) q^{57} + ( 2 \beta_{8} + \beta_{10} ) q^{58} + ( \beta_{2} - 4 \beta_{3} - 2 \beta_{7} + \beta_{9} - \beta_{13} - \beta_{15} ) q^{59} -\beta_{1} q^{60} + ( 3 \beta_{1} - \beta_{5} + \beta_{10} + \beta_{11} + 3 \beta_{14} ) q^{61} + ( \beta_{2} + \beta_{5} - \beta_{7} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{15} ) q^{62} + ( -1 + \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{7} + \beta_{9} + 2 \beta_{15} ) q^{63} - q^{64} + ( \beta_{5} - 2 \beta_{6} + \beta_{11} ) q^{65} + ( \beta_{2} - \beta_{3} - \beta_{7} + \beta_{13} + \beta_{15} ) q^{66} + ( 8 - 2 \beta_{2} - 2 \beta_{15} ) q^{67} + ( \beta_{5} + \beta_{6} + \beta_{11} ) q^{68} + ( -3 \beta_{3} - \beta_{4} + 2 \beta_{9} - \beta_{13} + \beta_{15} ) q^{69} + ( 1 + \beta_{4} ) q^{70} + ( -\beta_{3} + 3 \beta_{7} - \beta_{9} + \beta_{13} ) q^{71} + ( -1 - \beta_{9} ) q^{72} + ( 3 \beta_{5} - \beta_{8} - 4 \beta_{10} - 3 \beta_{11} ) q^{73} + ( -\beta_{1} + \beta_{14} ) q^{74} + \beta_{5} q^{75} + ( 1 - \beta_{2} - \beta_{4} - \beta_{15} ) q^{76} + ( -2 \beta_{5} + \beta_{6} - 2 \beta_{11} + 3 \beta_{12} ) q^{77} + ( -1 + \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{7} + 2 \beta_{13} ) q^{78} + ( \beta_{1} + \beta_{5} + \beta_{8} - 3 \beta_{10} - \beta_{11} + \beta_{14} ) q^{79} + \beta_{3} q^{80} + ( -3 + 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - \beta_{9} + 2 \beta_{13} + \beta_{15} ) q^{81} + ( 2 + \beta_{2} - 2 \beta_{4} + \beta_{15} ) q^{82} + ( -\beta_{5} - \beta_{11} ) q^{83} + ( -\beta_{5} + \beta_{6} + \beta_{10} + \beta_{12} - \beta_{14} ) q^{84} + ( \beta_{1} + \beta_{10} + \beta_{14} ) q^{85} + ( 3 \beta_{1} + 2 \beta_{5} + \beta_{6} + 2 \beta_{11} + \beta_{12} - 3 \beta_{14} ) q^{86} + ( -1 - \beta_{3} - 3 \beta_{4} + \beta_{7} + 2 \beta_{9} - \beta_{13} + 2 \beta_{15} ) q^{87} -\beta_{8} q^{88} + ( -\beta_{1} - 3 \beta_{5} - \beta_{6} - 3 \beta_{11} - 3 \beta_{12} + \beta_{14} ) q^{89} + ( \beta_{2} + \beta_{3} ) q^{90} + ( -3 \beta_{1} - 3 \beta_{5} + 2 \beta_{8} + 2 \beta_{10} + 3 \beta_{11} - 3 \beta_{14} ) q^{91} + ( -\beta_{1} + \beta_{12} + \beta_{14} ) q^{92} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{93} + ( -2 \beta_{2} - \beta_{9} - \beta_{13} - 2 \beta_{15} ) q^{94} + ( -\beta_{3} + \beta_{7} - \beta_{9} + \beta_{13} ) q^{95} + \beta_{1} q^{96} + ( -2 + 2 \beta_{2} + 4 \beta_{4} - \beta_{9} - \beta_{13} + 2 \beta_{15} ) q^{97} + ( 3 \beta_{7} + \beta_{9} - \beta_{13} ) q^{98} + ( \beta_{5} - 2 \beta_{6} - \beta_{8} - \beta_{10} + 2 \beta_{11} + 3 \beta_{14} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16q^{4} - 20q^{7} - 4q^{9} + O(q^{10}) \) \( 16q - 16q^{4} - 20q^{7} - 4q^{9} - 16q^{10} + 16q^{16} + 12q^{18} - 4q^{19} - 16q^{25} + 20q^{28} - 4q^{31} - 16q^{33} + 4q^{36} - 4q^{39} + 16q^{40} + 12q^{45} + 4q^{49} + 52q^{51} - 12q^{63} - 16q^{64} + 4q^{66} + 112q^{67} - 4q^{69} + 20q^{70} - 12q^{72} + 4q^{76} - 28q^{78} - 32q^{81} + 32q^{82} - 24q^{87} + 4q^{90} + 16q^{93} - 8q^{94} + 8q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 2 x^{14} + 10 x^{12} - 42 x^{10} + 82 x^{8} - 378 x^{6} + 810 x^{4} - 1458 x^{2} + 6561\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -23 \nu^{14} + 325 \nu^{12} - 2003 \nu^{10} - 375 \nu^{8} - 23567 \nu^{6} + 61461 \nu^{4} + 452061 \nu^{2} + 449793 \)\()/571536\)
\(\beta_{3}\)\(=\)\((\)\( 23 \nu^{14} - 325 \nu^{12} + 2003 \nu^{10} + 375 \nu^{8} + 23567 \nu^{6} - 61461 \nu^{4} + 119475 \nu^{2} - 449793 \)\()/571536\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{14} - 52 \nu^{12} + 179 \nu^{10} - 660 \nu^{8} + 2267 \nu^{6} - 1620 \nu^{4} + 10935 \nu^{2} - 23328 \)\()/20412\)
\(\beta_{5}\)\(=\)\((\)\( 23 \nu^{15} - 325 \nu^{13} + 2003 \nu^{11} + 375 \nu^{9} + 23567 \nu^{7} - 61461 \nu^{5} + 119475 \nu^{3} - 449793 \nu \)\()/571536\)
\(\beta_{6}\)\(=\)\((\)\( 11 \nu^{15} - 265 \nu^{13} + 569 \nu^{11} - 1623 \nu^{9} + 4763 \nu^{7} - 2781 \nu^{5} + 68661 \nu^{3} - 84807 \nu \)\()/142884\)
\(\beta_{7}\)\(=\)\((\)\( -40 \nu^{14} + 431 \nu^{12} - 454 \nu^{10} + 249 \nu^{8} + 1580 \nu^{6} - 2997 \nu^{4} + 13446 \nu^{2} - 51759 \)\()/142884\)
\(\beta_{8}\)\(=\)\((\)\( 49 \nu^{15} + 361 \nu^{13} - 347 \nu^{11} + 4557 \nu^{9} - 12263 \nu^{7} + 26649 \nu^{5} - 46251 \nu^{3} + 79461 \nu \)\()/244944\)
\(\beta_{9}\)\(=\)\((\)\( 31 \nu^{14} - 197 \nu^{12} - 149 \nu^{10} - 2409 \nu^{8} + 5863 \nu^{6} - 11205 \nu^{4} + 46251 \nu^{2} - 46737 \)\()/63504\)
\(\beta_{10}\)\(=\)\((\)\( 35 \nu^{15} - 169 \nu^{13} - 181 \nu^{11} - 273 \nu^{9} + 467 \nu^{7} - 7497 \nu^{5} + 12555 \nu^{3} + 40095 \nu \)\()/122472\)
\(\beta_{11}\)\(=\)\((\)\( -617 \nu^{15} + 1027 \nu^{13} - 3245 \nu^{11} + 7887 \nu^{9} - 53969 \nu^{7} + 21123 \nu^{5} + 53379 \nu^{3} - 175689 \nu \)\()/1714608\)
\(\beta_{12}\)\(=\)\((\)\( 107 \nu^{15} - 199 \nu^{13} + 1499 \nu^{11} - 2103 \nu^{9} - 3061 \nu^{7} - 20343 \nu^{5} - 87669 \nu^{3} - 10935 \nu \)\()/285768\)
\(\beta_{13}\)\(=\)\((\)\( 617 \nu^{14} - 1027 \nu^{12} + 3245 \nu^{10} - 7887 \nu^{8} + 53969 \nu^{6} - 21123 \nu^{4} - 53379 \nu^{2} - 395847 \)\()/571536\)
\(\beta_{14}\)\(=\)\((\)\( \nu^{15} - 2 \nu^{13} + 10 \nu^{11} - 42 \nu^{9} + 82 \nu^{7} - 378 \nu^{5} + 810 \nu^{3} - 1458 \nu \)\()/2187\)
\(\beta_{15}\)\(=\)\((\)\( -761 \nu^{14} + 1243 \nu^{12} - 5837 \nu^{10} + 33303 \nu^{8} - 40721 \nu^{6} + 234891 \nu^{4} - 515565 \nu^{2} + 693279 \)\()/571536\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{14} - \beta_{12} + \beta_{11} + \beta_{8} + \beta_{6} + \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{15} + 2 \beta_{13} - \beta_{9} + 2 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 3\)
\(\nu^{5}\)\(=\)\(-2 \beta_{14} - \beta_{12} - 5 \beta_{11} - 3 \beta_{10} + \beta_{8} + 4 \beta_{6} - 4 \beta_{5} - 2 \beta_{1}\)
\(\nu^{6}\)\(=\)\(4 \beta_{15} + 5 \beta_{13} + 3 \beta_{9} + 6 \beta_{7} + 2 \beta_{4} + 5 \beta_{3} - \beta_{2} + 10\)
\(\nu^{7}\)\(=\)\(-4 \beta_{14} - 8 \beta_{12} - 21 \beta_{11} - 5 \beta_{10} - 6 \beta_{8} - 5 \beta_{6} + 7 \beta_{5} + \beta_{1}\)
\(\nu^{8}\)\(=\)\(9 \beta_{15} + 7 \beta_{13} - \beta_{9} - 8 \beta_{7} - 16 \beta_{4} + 56 \beta_{3} + 9 \beta_{2} + 9\)
\(\nu^{9}\)\(=\)\(-27 \beta_{14} + 5 \beta_{11} + 23 \beta_{10} + 26 \beta_{8} + \beta_{6} + 57 \beta_{5} + 12 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-25 \beta_{15} + 3 \beta_{13} - 83 \beta_{9} - 22 \beta_{7} + 50 \beta_{4} + 19 \beta_{3} + 12 \beta_{2} + 26\)
\(\nu^{11}\)\(=\)\(-18 \beta_{14} + 93 \beta_{12} + 36 \beta_{11} - 111 \beta_{10} + 45 \beta_{8} + 84 \beta_{6} + 65 \beta_{5} + 118 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-54 \beta_{15} + 63 \beta_{13} - 110 \beta_{9} + 288 \beta_{7} + 18 \beta_{4} - 92 \beta_{3} + 25 \beta_{2} + 61\)
\(\nu^{13}\)\(=\)\(365 \beta_{14} - 203 \beta_{12} - 72 \beta_{11} - 416 \beta_{10} + 117 \beta_{8} - 245 \beta_{6} - 61 \beta_{5} + 133 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-159 \beta_{15} + 736 \beta_{13} - 56 \beta_{9} - 32 \beta_{7} - 544 \beta_{4} - 25 \beta_{3} + 336 \beta_{2} - 256\)
\(\nu^{15}\)\(=\)\(725 \beta_{14} - 248 \beta_{12} - 1272 \beta_{11} + 520 \beta_{10} + 936 \beta_{8} - 176 \beta_{6} - 464 \beta_{5} - 600 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).

\(n\) \(187\) \(311\) \(871\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
371.1
−0.687009 + 1.58997i
−1.06195 + 1.36831i
1.39665 + 1.02439i
−1.71746 + 0.224352i
1.71746 0.224352i
−1.39665 1.02439i
1.06195 1.36831i
0.687009 1.58997i
−0.687009 1.58997i
−1.06195 1.36831i
1.39665 1.02439i
−1.71746 0.224352i
1.71746 + 0.224352i
−1.39665 + 1.02439i
1.06195 + 1.36831i
0.687009 + 1.58997i
1.00000i −1.58997 0.687009i −1.00000 1.00000i −0.687009 + 1.58997i −4.71902 1.00000i 2.05604 + 2.18465i −1.00000
371.2 1.00000i −1.36831 1.06195i −1.00000 1.00000i −1.06195 + 1.36831i 1.28192 1.00000i 0.744540 + 2.90614i −1.00000
371.3 1.00000i −1.02439 + 1.39665i −1.00000 1.00000i 1.39665 + 1.02439i 0.608912 1.00000i −0.901245 2.86143i −1.00000
371.4 1.00000i −0.224352 1.71746i −1.00000 1.00000i −1.71746 + 0.224352i −2.17181 1.00000i −2.89933 + 0.770630i −1.00000
371.5 1.00000i 0.224352 + 1.71746i −1.00000 1.00000i 1.71746 0.224352i −2.17181 1.00000i −2.89933 + 0.770630i −1.00000
371.6 1.00000i 1.02439 1.39665i −1.00000 1.00000i −1.39665 1.02439i 0.608912 1.00000i −0.901245 2.86143i −1.00000
371.7 1.00000i 1.36831 + 1.06195i −1.00000 1.00000i 1.06195 1.36831i 1.28192 1.00000i 0.744540 + 2.90614i −1.00000
371.8 1.00000i 1.58997 + 0.687009i −1.00000 1.00000i 0.687009 1.58997i −4.71902 1.00000i 2.05604 + 2.18465i −1.00000
371.9 1.00000i −1.58997 + 0.687009i −1.00000 1.00000i −0.687009 1.58997i −4.71902 1.00000i 2.05604 2.18465i −1.00000
371.10 1.00000i −1.36831 + 1.06195i −1.00000 1.00000i −1.06195 1.36831i 1.28192 1.00000i 0.744540 2.90614i −1.00000
371.11 1.00000i −1.02439 1.39665i −1.00000 1.00000i 1.39665 1.02439i 0.608912 1.00000i −0.901245 + 2.86143i −1.00000
371.12 1.00000i −0.224352 + 1.71746i −1.00000 1.00000i −1.71746 0.224352i −2.17181 1.00000i −2.89933 0.770630i −1.00000
371.13 1.00000i 0.224352 1.71746i −1.00000 1.00000i 1.71746 + 0.224352i −2.17181 1.00000i −2.89933 0.770630i −1.00000
371.14 1.00000i 1.02439 + 1.39665i −1.00000 1.00000i −1.39665 + 1.02439i 0.608912 1.00000i −0.901245 + 2.86143i −1.00000
371.15 1.00000i 1.36831 1.06195i −1.00000 1.00000i 1.06195 + 1.36831i 1.28192 1.00000i 0.744540 2.90614i −1.00000
371.16 1.00000i 1.58997 0.687009i −1.00000 1.00000i 0.687009 + 1.58997i −4.71902 1.00000i 2.05604 2.18465i −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 371.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.b odd 2 1 inner
93.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.h.c 16
3.b odd 2 1 inner 930.2.h.c 16
31.b odd 2 1 inner 930.2.h.c 16
93.c even 2 1 inner 930.2.h.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.h.c 16 1.a even 1 1 trivial
930.2.h.c 16 3.b odd 2 1 inner
930.2.h.c 16 31.b odd 2 1 inner
930.2.h.c 16 93.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 5 T_{7}^{3} - 2 T_{7}^{2} - 14 T_{7} + 8 \) acting on \(S_{2}^{\mathrm{new}}(930, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{8} \)
$3$ \( 6561 + 1458 T^{2} + 810 T^{4} + 378 T^{6} + 82 T^{8} + 42 T^{10} + 10 T^{12} + 2 T^{14} + T^{16} \)
$5$ \( ( 1 + T^{2} )^{8} \)
$7$ \( ( 8 - 14 T - 2 T^{2} + 5 T^{3} + T^{4} )^{4} \)
$11$ \( ( 64 - 454 T^{2} + 258 T^{4} - 31 T^{6} + T^{8} )^{2} \)
$13$ \( ( 200704 + 38556 T^{2} + 2746 T^{4} + 86 T^{6} + T^{8} )^{2} \)
$17$ \( ( 256 - 576 T^{2} + 328 T^{4} - 50 T^{6} + T^{8} )^{2} \)
$19$ \( ( -32 - 64 T - 30 T^{2} + T^{3} + T^{4} )^{4} \)
$23$ \( ( 4096 - 3584 T^{2} + 810 T^{4} - 61 T^{6} + T^{8} )^{2} \)
$29$ \( ( 123904 - 54512 T^{2} + 4730 T^{4} - 136 T^{6} + T^{8} )^{2} \)
$31$ \( ( 923521 + 59582 T - 46128 T^{2} + 1054 T^{3} + 2334 T^{4} + 34 T^{5} - 48 T^{6} + 2 T^{7} + T^{8} )^{2} \)
$37$ \( ( 784 + 756 T^{2} + 226 T^{4} + 26 T^{6} + T^{8} )^{2} \)
$41$ \( ( 3444736 + 357184 T^{2} + 13060 T^{4} + 196 T^{6} + T^{8} )^{2} \)
$43$ \( ( 988036 + 223874 T^{2} + 14788 T^{4} + 245 T^{6} + T^{8} )^{2} \)
$47$ \( ( 50176 + 76944 T^{2} + 7060 T^{4} + 176 T^{6} + T^{8} )^{2} \)
$53$ \( ( 9604 - 632786 T^{2} + 23580 T^{4} - 277 T^{6} + T^{8} )^{2} \)
$59$ \( ( 14258176 + 1218816 T^{2} + 33344 T^{4} + 324 T^{6} + T^{8} )^{2} \)
$61$ \( ( 4528384 + 511056 T^{2} + 17098 T^{4} + 224 T^{6} + T^{8} )^{2} \)
$67$ \( ( -2816 + 288 T + 184 T^{2} - 28 T^{3} + T^{4} )^{4} \)
$71$ \( ( 3564544 + 410128 T^{2} + 14856 T^{4} + 209 T^{6} + T^{8} )^{2} \)
$73$ \( ( 62980096 + 4034560 T^{2} + 70746 T^{4} + 461 T^{6} + T^{8} )^{2} \)
$79$ \( ( 1149184 + 210528 T^{2} + 11752 T^{4} + 223 T^{6} + T^{8} )^{2} \)
$83$ \( ( 64 - 348 T^{2} + 154 T^{4} - 22 T^{6} + T^{8} )^{2} \)
$89$ \( ( 19219456 - 1465728 T^{2} + 36712 T^{4} - 335 T^{6} + T^{8} )^{2} \)
$97$ \( ( -1744 + 1388 T - 274 T^{2} - 2 T^{3} + T^{4} )^{4} \)
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