Properties

Label 1078.2.c.c
Level $1078$
Weight $2$
Character orbit 1078.c
Analytic conductor $8.608$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1078.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.60787333789\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 32 x^{14} + 512 x^{12} - 2272 x^{10} - 1087 x^{8} + 72448 x^{6} + 819200 x^{4} + 1310720 x^{2} + 1048576\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 32 x^{14} + 512 x^{12} - 2272 x^{10} - 1087 x^{8} + 72448 x^{6} + 819200 x^{4} + 1310720 x^{2} + 1048576\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -32173 \nu^{14} + 668960 \nu^{12} - 6919680 \nu^{10} - 46669984 \nu^{8} - 133787245 \nu^{6} + 474538112 \nu^{4} + 1011889152 \nu^{2} - 481919500288 \)\()/ 341451177984 \)
\(\beta_{2}\)\(=\)\((\)\(21649515 \nu^{14} - 500615872 \nu^{12} + 4404178432 \nu^{10} + 73242356320 \nu^{8} - 905783138901 \nu^{6} + 4403763700640 \nu^{4} + 35129391631360 \nu^{2} + 115772308922368\)\()/ 28937987334144 \)
\(\beta_{3}\)\(=\)\((\)\( 240905 \nu^{14} - 7913160 \nu^{12} + 129966848 \nu^{10} - 662732768 \nu^{8} + 491616713 \nu^{6} + 13417758552 \nu^{4} + 194978259968 \nu^{2} + 167526400000 \)\()/ 149937758208 \)
\(\beta_{4}\)\(=\)\((\)\( 42729 \nu^{14} - 1400096 \nu^{12} + 23098880 \nu^{10} - 119264224 \nu^{8} + 119589033 \nu^{6} + 2634257152 \nu^{4} + 35772670976 \nu^{2} + 30704009216 \)\()/ 19295207424 \)
\(\beta_{5}\)\(=\)\((\)\(116266153 \nu^{15} - 3358638720 \nu^{13} + 46096726528 \nu^{11} - 28654574560 \nu^{9} - 1703395919255 \nu^{7} + 10664594701728 \nu^{5} + 114707998022656 \nu^{3} + 298669982056448 \nu\)\()/ 308671864897536 \)
\(\beta_{6}\)\(=\)\((\)\(-26548369 \nu^{14} + 933827904 \nu^{12} - 16598608384 \nu^{10} + 111831131872 \nu^{8} - 267579354193 \nu^{6} - 2009136336864 \nu^{4} - 11825933968384 \nu^{2} + 8620523487232\)\()/ 9645995778048 \)
\(\beta_{7}\)\(=\)\((\)\(1040915 \nu^{14} - 33066624 \nu^{12} + 519454208 \nu^{10} - 2062231712 \nu^{8} - 5141852845 \nu^{6} + 101561457888 \nu^{4} + 743042069504 \nu^{2} + 849802264576\)\()/ 256088383488 \)
\(\beta_{8}\)\(=\)\((\)\(-135220701 \nu^{14} + 4561306816 \nu^{12} - 76278327808 \nu^{10} + 410096159072 \nu^{8} - 107080965789 \nu^{6} - 12255806975456 \nu^{4} - 84320719473664 \nu^{2} - 49543323222016\)\()/ 28937987334144 \)
\(\beta_{9}\)\(=\)\((\)\(-263077499 \nu^{15} + 7568932160 \nu^{13} - 104813186560 \nu^{11} + 66556609440 \nu^{9} + 3864561461317 \nu^{7} - 27621977981728 \nu^{5} - 279805449960448 \nu^{3} - 723250826477568 \nu\)\()/ 308671864897536 \)
\(\beta_{10}\)\(=\)\((\)\(15339039 \nu^{15} - 510774904 \nu^{13} + 8514025600 \nu^{11} - 45705450272 \nu^{9} + 42003795807 \nu^{7} + 997736067176 \nu^{5} + 13433955225472 \nu^{3} - 13167578624000 \nu\)\()/ 14468993667072 \)
\(\beta_{11}\)\(=\)\((\)\(133618959 \nu^{15} - 4312976672 \nu^{13} + 70457976320 \nu^{11} - 349822477600 \nu^{9} + 381384242511 \nu^{7} + 7821020117440 \nu^{5} + 107128611381248 \nu^{3} + 289533980278784 \nu\)\()/ 115751949336576 \)
\(\beta_{12}\)\(=\)\((\)\(-10897203 \nu^{15} + 359824640 \nu^{13} - 5954700800 \nu^{11} + 30984053920 \nu^{9} - 17555552883 \nu^{7} - 851122418272 \nu^{5} - 7459200822272 \nu^{3} - 4233948004352 \nu\)\()/ 8194828271616 \)
\(\beta_{13}\)\(=\)\((\)\(77172743 \nu^{15} - 2548789688 \nu^{13} + 42111652864 \nu^{11} - 219318324768 \nu^{9} + 178889617223 \nu^{7} + 4585099534888 \nu^{5} + 64498714624768 \nu^{3} + 34935425286144 \nu\)\()/ 28937987334144 \)
\(\beta_{14}\)\(=\)\((\)\(-319597619 \nu^{15} + 10421936192 \nu^{13} - 170792382976 \nu^{11} + 861452174496 \nu^{9} - 760912344947 \nu^{7} - 18179529719584 \nu^{5} - 257651828076544 \nu^{3} - 303112411742208 \nu\)\()/ 115751949336576 \)
\(\beta_{15}\)\(=\)\((\)\(-25933907 \nu^{15} + 863194304 \nu^{13} - 14336292352 \nu^{11} + 75544371360 \nu^{9} - 37801257875 \nu^{7} - 2043398985376 \nu^{5} - 17995089961984 \nu^{3} - 10214744358912 \nu\)\()/ 8194828271616 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{14} + \beta_{13} + \beta_{11} - \beta_{10}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{8} + \beta_{7} + \beta_{6} + 8 \beta_{4} - 8 \beta_{3} - \beta_{2} + 8 \beta_{1} + 9\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(4 \beta_{15} + 17 \beta_{14} + 9 \beta_{13} - 4 \beta_{12} + 17 \beta_{11} + 9 \beta_{10} + 4 \beta_{9} + 12 \beta_{5}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-17 \beta_{8} + 17 \beta_{7} + 33 \beta_{6} + 136 \beta_{4} - 208 \beta_{3} - 33 \beta_{2} + 33\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-68 \beta_{15} + 121 \beta_{14} - 121 \beta_{13} + 132 \beta_{12} + 273 \beta_{11} + 273 \beta_{10} + 132 \beta_{9} + 332 \beta_{5}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-737 \beta_{8} - 321 \beta_{7} + 737 \beta_{6} + 1576 \beta_{4} - 2184 \beta_{3} - 321 \beta_{2} - 1576 \beta_{1} - 1863\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-2948 \beta_{15} - 1897 \beta_{14} - 4497 \beta_{13} + 7180 \beta_{12} + 1897 \beta_{11} + 4497 \beta_{10} + 1284 \beta_{9} + 2948 \beta_{5}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-14625 \beta_{8} - 14625 \beta_{7} + 6129 \beta_{6} + 6129 \beta_{2} - 35976 \beta_{1} - 57281\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-58500 \beta_{15} - 76177 \beta_{14} - 76177 \beta_{13} + 141516 \beta_{12} - 31705 \beta_{11} + 31705 \beta_{10} - 24516 \beta_{9} - 58500 \beta_{5}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(-114721 \beta_{8} - 276193 \beta_{7} - 114721 \beta_{6} - 431528 \beta_{4} + 609416 \beta_{3} + 276193 \beta_{2} - 431528 \beta_{1} - 885609\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-458884 \beta_{15} - 1317137 \beta_{14} - 546249 \beta_{13} + 1104772 \beta_{12} - 1317137 \beta_{11} - 546249 \beta_{10} - 1104772 \beta_{9} - 2668428 \beta_{5}\)\()/2\)
\(\nu^{12}\)\(=\)\((\)\(2109905 \beta_{8} - 2109905 \beta_{7} - 5090337 \beta_{6} - 10537096 \beta_{4} + 14907088 \beta_{3} + 5090337 \beta_{2} - 5090337\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(8439620 \beta_{15} - 9563449 \beta_{14} + 9563449 \beta_{13} - 20361348 \beta_{12} - 23080977 \beta_{11} - 23080977 \beta_{10} - 20361348 \beta_{9} - 49162316 \beta_{5}\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(92604641 \beta_{8} + 38364417 \beta_{7} - 92604641 \beta_{6} - 130577704 \beta_{4} + 184647816 \beta_{3} + 38364417 \beta_{2} + 130577704 \beta_{1} + 146283399\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(370418564 \beta_{15} + 168942121 \beta_{14} + 407830161 \beta_{13} - 894294796 \beta_{12} - 168942121 \beta_{11} - 407830161 \beta_{10} - 153457668 \beta_{9} - 370418564 \beta_{5}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(981\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1077.1
2.98338 1.23576i
−3.90726 + 1.61844i
−0.807586 1.94969i
0.424903 + 1.02581i
−0.424903 1.02581i
0.807586 + 1.94969i
3.90726 1.61844i
−2.98338 + 1.23576i
−2.98338 1.23576i
3.90726 + 1.61844i
0.807586 1.94969i
−0.424903 + 1.02581i
0.424903 1.02581i
−0.807586 + 1.94969i
−3.90726 1.61844i
2.98338 + 1.23576i
1.00000i 1.84776i −1.00000 3.23688i −1.84776 0 1.00000i −0.414214 −3.23688
1077.2 1.00000i 1.84776i −1.00000 2.47151i −1.84776 0 1.00000i −0.414214 2.47151
1077.3 1.00000i 0.765367i −1.00000 2.05161i −0.765367 0 1.00000i 2.41421 −2.05161
1077.4 1.00000i 0.765367i −1.00000 3.89937i −0.765367 0 1.00000i 2.41421 3.89937
1077.5 1.00000i 0.765367i −1.00000 3.89937i 0.765367 0 1.00000i 2.41421 −3.89937
1077.6 1.00000i 0.765367i −1.00000 2.05161i 0.765367 0 1.00000i 2.41421 2.05161
1077.7 1.00000i 1.84776i −1.00000 2.47151i 1.84776 0 1.00000i −0.414214 −2.47151
1077.8 1.00000i 1.84776i −1.00000 3.23688i 1.84776 0 1.00000i −0.414214 3.23688
1077.9 1.00000i 1.84776i −1.00000 3.23688i 1.84776 0 1.00000i −0.414214 3.23688
1077.10 1.00000i 1.84776i −1.00000 2.47151i 1.84776 0 1.00000i −0.414214 −2.47151
1077.11 1.00000i 0.765367i −1.00000 2.05161i 0.765367 0 1.00000i 2.41421 2.05161
1077.12 1.00000i 0.765367i −1.00000 3.89937i 0.765367 0 1.00000i 2.41421 −3.89937
1077.13 1.00000i 0.765367i −1.00000 3.89937i −0.765367 0 1.00000i 2.41421 3.89937
1077.14 1.00000i 0.765367i −1.00000 2.05161i −0.765367 0 1.00000i 2.41421 −2.05161
1077.15 1.00000i 1.84776i −1.00000 2.47151i −1.84776 0 1.00000i −0.414214 2.47151
1077.16 1.00000i 1.84776i −1.00000 3.23688i −1.84776 0 1.00000i −0.414214 −3.23688
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1077.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1078.2.c.c 16
7.b odd 2 1 inner 1078.2.c.c 16
7.c even 3 2 1078.2.i.d 32
7.d odd 6 2 1078.2.i.d 32
11.b odd 2 1 inner 1078.2.c.c 16
77.b even 2 1 inner 1078.2.c.c 16
77.h odd 6 2 1078.2.i.d 32
77.i even 6 2 1078.2.i.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1078.2.c.c 16 1.a even 1 1 trivial
1078.2.c.c 16 7.b odd 2 1 inner
1078.2.c.c 16 11.b odd 2 1 inner
1078.2.c.c 16 77.b even 2 1 inner
1078.2.i.d 32 7.c even 3 2
1078.2.i.d 32 7.d odd 6 2
1078.2.i.d 32 77.h odd 6 2
1078.2.i.d 32 77.i even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 4 T_{3}^{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(1078, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{8} \)
$3$ \( ( 2 + 4 T^{2} + T^{4} )^{4} \)
$5$ \( ( 4096 + 2304 T^{2} + 450 T^{4} + 36 T^{6} + T^{8} )^{2} \)
$7$ \( T^{16} \)
$11$ \( ( 14641 - 10648 T + 4114 T^{2} - 1408 T^{3} + 450 T^{4} - 128 T^{5} + 34 T^{6} - 8 T^{7} + T^{8} )^{2} \)
$13$ \( ( 20736 - 12672 T^{2} + 2216 T^{4} - 88 T^{6} + T^{8} )^{2} \)
$17$ \( ( 1016064 - 152640 T^{2} + 7490 T^{4} - 148 T^{6} + T^{8} )^{2} \)
$19$ \( ( 1296 - 3312 T^{2} + 2186 T^{4} - 92 T^{6} + T^{8} )^{2} \)
$23$ \( ( -144 - 192 T - 38 T^{2} + 4 T^{3} + T^{4} )^{4} \)
$29$ \( ( 331776 + 207360 T^{2} + 10768 T^{4} + 184 T^{6} + T^{8} )^{2} \)
$31$ \( ( 322624 + 85536 T^{2} + 5634 T^{4} + 132 T^{6} + T^{8} )^{2} \)
$37$ \( ( -2592 - 576 T + 68 T^{2} + 20 T^{3} + T^{4} )^{4} \)
$41$ \( ( 5184 - 4896 T^{2} + 1058 T^{4} - 68 T^{6} + T^{8} )^{2} \)
$43$ \( ( 419904 + 85536 T^{2} + 5620 T^{4} + 132 T^{6} + T^{8} )^{2} \)
$47$ \( ( 64 + 1568 T^{2} + 674 T^{4} + 52 T^{6} + T^{8} )^{2} \)
$53$ \( ( -56 + 136 T - 10 T^{2} - 8 T^{3} + T^{4} )^{4} \)
$59$ \( ( 21316 + 25968 T^{2} + 3444 T^{4} + 120 T^{6} + T^{8} )^{2} \)
$61$ \( ( 107495424 - 4396032 T^{2} + 65672 T^{4} - 424 T^{6} + T^{8} )^{2} \)
$67$ \( ( -376 + 296 T - 54 T^{2} - 4 T^{3} + T^{4} )^{4} \)
$71$ \( ( -1568 - 912 T - 62 T^{2} + 12 T^{3} + T^{4} )^{4} \)
$73$ \( ( 5184 - 1670688 T^{2} + 42242 T^{4} - 356 T^{6} + T^{8} )^{2} \)
$79$ \( ( 72 + T^{2} )^{8} \)
$83$ \( ( 12194064 - 1101744 T^{2} + 31658 T^{4} - 316 T^{6} + T^{8} )^{2} \)
$89$ \( ( 454276 + 193232 T^{2} + 12500 T^{4} + 232 T^{6} + T^{8} )^{2} \)
$97$ \( ( 1250 + 100 T^{2} + T^{4} )^{4} \)
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