# Properties

 Label 2394.2.e.c Level $2394$ Weight $2$ Character orbit 2394.e Analytic conductor $19.116$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2394.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.1161862439$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 44 x^{14} + 708 x^{12} + 5378 x^{10} + 20592 x^{8} + 38856 x^{6} + 33265 x^{4} + 10216 x^{2} + 900$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 266) 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 44 x^{14} + 708 x^{12} + 5378 x^{10} + 20592 x^{8} + 38856 x^{6} + 33265 x^{4} + 10216 x^{2} + 900$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$4684 \nu^{14} - 62369 \nu^{12} - 7419046 \nu^{10} - 122870856 \nu^{8} - 791532402 \nu^{6} - 2218314285 \nu^{4} - 2439918896 \nu^{2} - 581108283$$$$)/44810181$$ $$\beta_{2}$$ $$=$$ $$($$$$40712 \nu^{14} + 1832436 \nu^{12} + 30245083 \nu^{10} + 233190572 \nu^{8} + 869437060 \nu^{6} + 1433713403 \nu^{4} + 816806418 \nu^{2} + 61394868$$$$)/29873454$$ $$\beta_{3}$$ $$=$$ $$($$$$-245873 \nu^{15} + 1038150 \nu^{14} - 10399862 \nu^{13} + 44598900 \nu^{12} - 157844884 \nu^{11} + 687143520 \nu^{10} - 1110556014 \nu^{9} + 4815739140 \nu^{8} - 3920343936 \nu^{7} + 15801979620 \nu^{6} - 7152308028 \nu^{5} + 22206093720 \nu^{4} - 8408943245 \nu^{3} + 13957453470 \nu^{2} - 9073010958 \nu + 5245163640$$$$)/ 1792407240$$ $$\beta_{4}$$ $$=$$ $$($$$$-11500 \nu^{14} - 479398 \nu^{12} - 7072472 \nu^{10} - 46978521 \nu^{8} - 146700948 \nu^{6} - 199375920 \nu^{4} - 99300229 \nu^{2} - 18087156$$$$)/5271786$$ $$\beta_{5}$$ $$=$$ $$($$$$245873 \nu^{15} + 1038150 \nu^{14} + 10399862 \nu^{13} + 44598900 \nu^{12} + 157844884 \nu^{11} + 687143520 \nu^{10} + 1110556014 \nu^{9} + 4815739140 \nu^{8} + 3920343936 \nu^{7} + 15801979620 \nu^{6} + 7152308028 \nu^{5} + 22206093720 \nu^{4} + 8408943245 \nu^{3} + 13957453470 \nu^{2} + 9073010958 \nu + 5245163640$$$$)/ 1792407240$$ $$\beta_{6}$$ $$=$$ $$($$$$-11500 \nu^{14} - 479398 \nu^{12} - 7072472 \nu^{10} - 46978521 \nu^{8} - 146700948 \nu^{6} - 199375920 \nu^{4} - 96664336 \nu^{2} - 2271798$$$$)/2635893$$ $$\beta_{7}$$ $$=$$ $$($$$$-216968 \nu^{15} - 9869032 \nu^{13} - 167295244 \nu^{11} - 1372581589 \nu^{9} - 5836366106 \nu^{7} - 12372693628 \nu^{5} - 10730971855 \nu^{3} - 1736652438 \nu$$$$)/ 149367270$$ $$\beta_{8}$$ $$=$$ $$($$$$-2118938 \nu^{15} + 1093745 \nu^{14} - 94420697 \nu^{13} + 47832545 \nu^{12} - 1546793269 \nu^{11} + 769839325 \nu^{10} - 12017106969 \nu^{9} + 5979077205 \nu^{8} - 47154831381 \nu^{7} + 24482065395 \nu^{6} - 90984872283 \nu^{5} + 51735928575 \nu^{4} - 77856115445 \nu^{3} + 46813521590 \nu^{2} - 19662214278 \nu + 9715121460$$$$)/ 896203620$$ $$\beta_{9}$$ $$=$$ $$($$$$-2118938 \nu^{15} - 1093745 \nu^{14} - 94420697 \nu^{13} - 47832545 \nu^{12} - 1546793269 \nu^{11} - 769839325 \nu^{10} - 12017106969 \nu^{9} - 5979077205 \nu^{8} - 47154831381 \nu^{7} - 24482065395 \nu^{6} - 90984872283 \nu^{5} - 51735928575 \nu^{4} - 77856115445 \nu^{3} - 46813521590 \nu^{2} - 19662214278 \nu - 9715121460$$$$)/ 896203620$$ $$\beta_{10}$$ $$=$$ $$($$$$142393 \nu^{15} + 6150292 \nu^{13} + 96020264 \nu^{11} + 695064834 \nu^{9} + 2462371446 \nu^{7} + 4065812928 \nu^{5} + 2742943945 \nu^{3} + 514402458 \nu$$$$)/52717860$$ $$\beta_{11}$$ $$=$$ $$($$$$142393 \nu^{15} + 6150292 \nu^{13} + 96020264 \nu^{11} + 695064834 \nu^{9} + 2462371446 \nu^{7} + 4065812928 \nu^{5} + 2742943945 \nu^{3} + 461684598 \nu$$$$)/52717860$$ $$\beta_{12}$$ $$=$$ $$($$$$-4980447 \nu^{15} + 6972320 \nu^{14} - 221025498 \nu^{13} + 301307180 \nu^{12} - 3595438836 \nu^{11} + 4694740000 \nu^{10} - 27586528446 \nu^{9} + 33637197720 \nu^{8} - 105681300324 \nu^{7} + 115445192280 \nu^{6} - 193633680612 \nu^{5} + 176246688600 \nu^{4} - 147209600175 \nu^{3} + 103289088800 \nu^{2} - 27280359702 \nu + 14857203060$$$$)/ 1792407240$$ $$\beta_{13}$$ $$=$$ $$($$$$-4980447 \nu^{15} - 6972320 \nu^{14} - 221025498 \nu^{13} - 301307180 \nu^{12} - 3595438836 \nu^{11} - 4694740000 \nu^{10} - 27586528446 \nu^{9} - 33637197720 \nu^{8} - 105681300324 \nu^{7} - 115445192280 \nu^{6} - 193633680612 \nu^{5} - 176246688600 \nu^{4} - 147209600175 \nu^{3} - 103289088800 \nu^{2} - 27280359702 \nu - 14857203060$$$$)/ 1792407240$$ $$\beta_{14}$$ $$=$$ $$($$$$3212683 \nu^{15} + 142253242 \nu^{13} + 2316632594 \nu^{11} + 17996184174 \nu^{9} + 71636896776 \nu^{7} + 142720800858 \nu^{5} + 124669637035 \nu^{3} + 29377335738 \nu$$$$)/ 896203620$$ $$\beta_{15}$$ $$=$$ $$($$$$-75762 \nu^{15} - 3301198 \nu^{13} - 52226056 \nu^{11} - 385079021 \nu^{9} - 1397079384 \nu^{7} - 2378248752 \nu^{5} - 1663128915 \nu^{3} - 268858162 \nu$$$$)/8786310$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$-\beta_{11} + \beta_{10}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} - 2 \beta_{4} - 6$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{15} - 2 \beta_{13} - 2 \beta_{12} + 16 \beta_{11} - 10 \beta_{10} + \beta_{7}$$ $$\nu^{4}$$ $$=$$ $$3 \beta_{13} - 3 \beta_{12} - 18 \beta_{6} + 8 \beta_{5} + 32 \beta_{4} + 8 \beta_{3} + 4 \beta_{2} - \beta_{1} + 76$$ $$\nu^{5}$$ $$=$$ $$-70 \beta_{15} - 5 \beta_{14} + 47 \beta_{13} + 47 \beta_{12} - 276 \beta_{11} + 137 \beta_{10} - \beta_{9} - \beta_{8} - 29 \beta_{7} + 15 \beta_{5} - 15 \beta_{3}$$ $$\nu^{6}$$ $$=$$ $$-91 \beta_{13} + 91 \beta_{12} - 6 \beta_{9} + 6 \beta_{8} + 330 \beta_{6} - 202 \beta_{5} - 518 \beta_{4} - 202 \beta_{3} - 134 \beta_{2} + 36 \beta_{1} - 1207$$ $$\nu^{7}$$ $$=$$ $$1386 \beta_{15} + 182 \beta_{14} - 953 \beta_{13} - 953 \beta_{12} + 4873 \beta_{11} - 2167 \beta_{10} + 42 \beta_{9} + 42 \beta_{8} + 682 \beta_{7} - 427 \beta_{5} + 427 \beta_{3}$$ $$\nu^{8}$$ $$=$$ $$2062 \beta_{13} - 2062 \beta_{12} + 224 \beta_{9} - 224 \beta_{8} - 6141 \beta_{6} + 4152 \beta_{5} + 8776 \beta_{4} + 4152 \beta_{3} + 3104 \beta_{2} - 948 \beta_{1} + 20884$$ $$\nu^{9}$$ $$=$$ $$-26325 \beta_{15} - 4500 \beta_{14} + 18496 \beta_{13} + 18496 \beta_{12} - 87580 \beta_{11} + 36870 \beta_{10} - 1172 \beta_{9} - 1172 \beta_{8} - 14317 \beta_{7} + 9318 \beta_{5} - 9318 \beta_{3}$$ $$\nu^{10}$$ $$=$$ $$-42131 \beta_{13} + 42131 \beta_{12} - 5672 \beta_{9} + 5672 \beta_{8} + 114504 \beta_{6} - 80464 \beta_{5} - 153964 \beta_{4} - 80464 \beta_{3} - 63778 \beta_{2} + 21161 \beta_{1} - 374548$$ $$\nu^{11}$$ $$=$$ $$493174 \beta_{15} + 96283 \beta_{14} - 351603 \beta_{13} - 351603 \beta_{12} + 1593780 \beta_{11} - 652097 \beta_{10} + 26833 \beta_{9} + 26833 \beta_{8} + 283705 \beta_{7} - 186373 \beta_{5} + 186373 \beta_{3}$$ $$\nu^{12}$$ $$=$$ $$821681 \beta_{13} - 821681 \beta_{12} + 123116 \beta_{9} - 123116 \beta_{8} - 2132182 \beta_{6} + 1524324 \beta_{5} + 2763196 \beta_{4} + 1524324 \beta_{3} + 1245908 \beta_{2} - 433654 \beta_{1} + 6828087$$ $$\nu^{13}$$ $$=$$ $$-9189934 \beta_{15} - 1925794 \beta_{14} + 6610369 \beta_{13} + 6610369 \beta_{12} - 29233179 \beta_{11} + 11778631 \beta_{10} - 556770 \beta_{9} - 556770 \beta_{8} - 5455106 \beta_{7} + 3591913 \beta_{5} - 3591913 \beta_{3}$$ $$\nu^{14}$$ $$=$$ $$-15657388 \beta_{13} + 15657388 \beta_{12} - 2482564 \beta_{9} + 2482564 \beta_{8} + 39644409 \beta_{6} - 28582150 \beta_{5} - 50282058 \beta_{4} - 28582150 \beta_{3} - 23754660 \beta_{2} + 8494440 \beta_{1} - 125478022$$ $$\nu^{15}$$ $$=$$ $$170845427 \beta_{15} + 37214228 \beta_{14} - 123528356 \beta_{13} - 123528356 \beta_{12} + 538726232 \beta_{11} - 215239682 \beta_{10} + 10977004 \beta_{9} + 10977004 \beta_{8} + 103208285 \beta_{7} - 67994198 \beta_{5} + 67994198 \beta_{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times$$.

 $$n$$ $$533$$ $$1009$$ $$1711$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1063.1
 1.23100i − 1.41533i − 4.30643i − 2.38481i 0.384810i 2.30643i − 0.584675i − 3.23100i 3.23100i 0.584675i − 2.30643i − 0.384810i 2.38481i 4.30643i 1.41533i − 1.23100i
1.00000i 0 −1.00000 3.34318i 0 2.62955 0.292339i 1.00000i 0 −3.34318
1063.2 1.00000i 0 −1.00000 2.74394i 0 −2.24817 + 1.39489i 1.00000i 0 −2.74394
1063.3 1.00000i 0 −1.00000 2.68862i 0 −0.592978 2.57844i 1.00000i 0 −2.68862
1063.4 1.00000i 0 −1.00000 1.03210i 0 1.71160 + 2.01753i 1.00000i 0 −1.03210
1063.5 1.00000i 0 −1.00000 1.03210i 0 1.71160 2.01753i 1.00000i 0 1.03210
1063.6 1.00000i 0 −1.00000 2.68862i 0 −0.592978 + 2.57844i 1.00000i 0 2.68862
1063.7 1.00000i 0 −1.00000 2.74394i 0 −2.24817 1.39489i 1.00000i 0 2.74394
1063.8 1.00000i 0 −1.00000 3.34318i 0 2.62955 + 0.292339i 1.00000i 0 3.34318
1063.9 1.00000i 0 −1.00000 3.34318i 0 2.62955 0.292339i 1.00000i 0 3.34318
1063.10 1.00000i 0 −1.00000 2.74394i 0 −2.24817 + 1.39489i 1.00000i 0 2.74394
1063.11 1.00000i 0 −1.00000 2.68862i 0 −0.592978 2.57844i 1.00000i 0 2.68862
1063.12 1.00000i 0 −1.00000 1.03210i 0 1.71160 + 2.01753i 1.00000i 0 1.03210
1063.13 1.00000i 0 −1.00000 1.03210i 0 1.71160 2.01753i 1.00000i 0 −1.03210
1063.14 1.00000i 0 −1.00000 2.68862i 0 −0.592978 + 2.57844i 1.00000i 0 −2.68862
1063.15 1.00000i 0 −1.00000 2.74394i 0 −2.24817 1.39489i 1.00000i 0 −2.74394
1063.16 1.00000i 0 −1.00000 3.34318i 0 2.62955 + 0.292339i 1.00000i 0 −3.34318
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1063.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
19.b odd 2 1 inner
133.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2394.2.e.c 16
3.b odd 2 1 266.2.d.a 16
7.b odd 2 1 inner 2394.2.e.c 16
12.b even 2 1 2128.2.m.f 16
19.b odd 2 1 inner 2394.2.e.c 16
21.c even 2 1 266.2.d.a 16
57.d even 2 1 266.2.d.a 16
84.h odd 2 1 2128.2.m.f 16
133.c even 2 1 inner 2394.2.e.c 16
228.b odd 2 1 2128.2.m.f 16
399.h odd 2 1 266.2.d.a 16
1596.p even 2 1 2128.2.m.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
266.2.d.a 16 3.b odd 2 1
266.2.d.a 16 21.c even 2 1
266.2.d.a 16 57.d even 2 1
266.2.d.a 16 399.h odd 2 1
2128.2.m.f 16 12.b even 2 1
2128.2.m.f 16 84.h odd 2 1
2128.2.m.f 16 228.b odd 2 1
2128.2.m.f 16 1596.p even 2 1
2394.2.e.c 16 1.a even 1 1 trivial
2394.2.e.c 16 7.b odd 2 1 inner
2394.2.e.c 16 19.b odd 2 1 inner
2394.2.e.c 16 133.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2394, [\chi])$$:

 $$T_{5}^{8} + 27 T_{5}^{6} + 247 T_{5}^{4} + 842 T_{5}^{2} + 648$$ $$T_{13}^{8} - 93 T_{13}^{6} + 3214 T_{13}^{4} - 48920 T_{13}^{2} + 276768$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{8}$$
$3$ $$T^{16}$$
$5$ $$( 648 + 842 T^{2} + 247 T^{4} + 27 T^{6} + T^{8} )^{2}$$
$7$ $$( 2401 - 1029 T + 98 T^{2} - 49 T^{3} + 26 T^{4} - 7 T^{5} + 2 T^{6} - 3 T^{7} + T^{8} )^{2}$$
$11$ $$( -48 + 70 T - 23 T^{2} - 3 T^{3} + T^{4} )^{4}$$
$13$ $$( 276768 - 48920 T^{2} + 3214 T^{4} - 93 T^{6} + T^{8} )^{2}$$
$17$ $$( 1152 + 3632 T^{2} + 784 T^{4} + 51 T^{6} + T^{8} )^{2}$$
$19$ $$16983563041 + 282275286 T^{2} + 83405440 T^{4} + 1401402 T^{6} + 227102 T^{8} + 3882 T^{10} + 640 T^{12} + 6 T^{14} + T^{16}$$
$23$ $$( 96 - 52 T - 20 T^{2} + 5 T^{3} + T^{4} )^{4}$$
$29$ $$( 5062500 + 500625 T^{2} + 16300 T^{4} + 216 T^{6} + T^{8} )^{2}$$
$31$ $$( 4608 - 7616 T^{2} + 1408 T^{4} - 72 T^{6} + T^{8} )^{2}$$
$37$ $$( 11664 + 28440 T^{2} + 3721 T^{4} + 139 T^{6} + T^{8} )^{2}$$
$41$ $$( 41472 - 59544 T^{2} + 7013 T^{4} - 185 T^{6} + T^{8} )^{2}$$
$43$ $$( 128 - 154 T - 57 T^{2} + T^{3} + T^{4} )^{4}$$
$47$ $$( 5971968 + 588060 T^{2} + 19593 T^{4} + 253 T^{6} + T^{8} )^{2}$$
$53$ $$( 26244 + 18585 T^{2} + 3676 T^{4} + 176 T^{6} + T^{8} )^{2}$$
$59$ $$( 36450 - 13383 T^{2} + 1604 T^{4} - 70 T^{6} + T^{8} )^{2}$$
$61$ $$( 2880000 + 867878 T^{2} + 29307 T^{4} + 311 T^{6} + T^{8} )^{2}$$
$67$ $$( 82944 + 164880 T^{2} + 16396 T^{4} + 269 T^{6} + T^{8} )^{2}$$
$71$ $$( 2073600 + 584784 T^{2} + 26353 T^{4} + 299 T^{6} + T^{8} )^{2}$$
$73$ $$( 10616832 + 1529856 T^{2} + 46688 T^{4} + 395 T^{6} + T^{8} )^{2}$$
$79$ $$( 5308416 + 576576 T^{2} + 19921 T^{4} + 247 T^{6} + T^{8} )^{2}$$
$83$ $$( 93312 + 46656 T^{2} + 6336 T^{4} + 238 T^{6} + T^{8} )^{2}$$
$89$ $$( 6480000 - 872892 T^{2} + 30929 T^{4} - 365 T^{6} + T^{8} )^{2}$$
$97$ $$( 7558272 - 1089288 T^{2} + 38493 T^{4} - 361 T^{6} + T^{8} )^{2}$$