# Properties

 Label 1386.2.g.b Level $1386$ Weight $2$ Character orbit 1386.g Analytic conductor $11.067$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1386.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.0672657201$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 12 x^{14} + 72 x^{12} + 1296 x^{10} + 8245 x^{8} + 18780 x^{6} + 20808 x^{4} - 3672 x^{2} + 324$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{9}\cdot 3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{2} - q^{4} + \beta_{1} q^{5} -\beta_{10} q^{7} -\beta_{5} q^{8} +O(q^{10})$$ $$q + \beta_{5} q^{2} - q^{4} + \beta_{1} q^{5} -\beta_{10} q^{7} -\beta_{5} q^{8} + \beta_{2} q^{10} + \beta_{5} q^{11} + ( 1 + \beta_{14} + \beta_{15} ) q^{13} -\beta_{8} q^{14} + q^{16} + ( -\beta_{1} - \beta_{7} - \beta_{8} ) q^{17} + \beta_{6} q^{19} -\beta_{1} q^{20} - q^{22} + ( 3 + \beta_{4} - \beta_{9} - \beta_{10} ) q^{25} + ( \beta_{11} - \beta_{12} ) q^{26} + \beta_{10} q^{28} + ( -\beta_{7} + \beta_{8} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{29} + ( -\beta_{6} - \beta_{9} + \beta_{10} ) q^{31} + \beta_{5} q^{32} + ( -\beta_{2} - \beta_{9} + \beta_{10} ) q^{34} + ( -\beta_{3} - \beta_{5} - 2 \beta_{7} + \beta_{11} ) q^{35} + ( -1 - \beta_{14} + \beta_{15} ) q^{37} + \beta_{3} q^{38} -\beta_{2} q^{40} + ( -\beta_{3} + \beta_{7} + \beta_{8} + \beta_{11} - \beta_{12} ) q^{41} + ( -5 - \beta_{9} - \beta_{10} - \beta_{14} + \beta_{15} ) q^{43} -\beta_{5} q^{44} + ( 2 \beta_{1} + \beta_{3} + \beta_{7} + \beta_{8} ) q^{47} + ( 2 - 2 \beta_{2} - \beta_{9} + \beta_{10} + \beta_{15} ) q^{49} + ( 3 \beta_{5} + \beta_{7} - \beta_{8} + \beta_{13} ) q^{50} + ( -1 - \beta_{14} - \beta_{15} ) q^{52} + ( 6 \beta_{5} - \beta_{7} + \beta_{8} + \beta_{11} + \beta_{12} ) q^{53} + \beta_{2} q^{55} + \beta_{8} q^{56} + ( 1 - \beta_{4} - \beta_{9} - \beta_{10} + \beta_{14} - \beta_{15} ) q^{58} + ( -2 \beta_{1} + 2 \beta_{11} - 2 \beta_{12} ) q^{59} + ( 1 - 2 \beta_{9} + 2 \beta_{10} + \beta_{14} + \beta_{15} ) q^{61} + ( -\beta_{3} + \beta_{7} + \beta_{8} ) q^{62} - q^{64} + ( -2 \beta_{7} + 2 \beta_{8} + 2 \beta_{11} + 2 \beta_{12} ) q^{65} + ( 1 + \beta_{4} + \beta_{9} + \beta_{10} - \beta_{14} + \beta_{15} ) q^{67} + ( \beta_{1} + \beta_{7} + \beta_{8} ) q^{68} + ( 1 + \beta_{6} - 2 \beta_{9} - \beta_{15} ) q^{70} + 2 \beta_{13} q^{71} + ( 1 + 2 \beta_{2} - 3 \beta_{6} - \beta_{9} + \beta_{10} + \beta_{14} + \beta_{15} ) q^{73} + ( \beta_{11} + \beta_{12} ) q^{74} -\beta_{6} q^{76} -\beta_{8} q^{77} + ( 6 + \beta_{4} - 2 \beta_{9} - 2 \beta_{10} ) q^{79} + \beta_{1} q^{80} + ( -1 + \beta_{6} + \beta_{9} - \beta_{10} - \beta_{14} - \beta_{15} ) q^{82} + ( -\beta_{1} + 2 \beta_{3} - \beta_{7} - \beta_{8} + \beta_{11} - \beta_{12} ) q^{83} + ( -5 - \beta_{4} - \beta_{9} - \beta_{10} + \beta_{14} - \beta_{15} ) q^{85} + ( -4 \beta_{5} + \beta_{7} - \beta_{8} + \beta_{11} + \beta_{12} ) q^{86} + q^{88} + ( 2 \beta_{3} - 3 \beta_{7} - 3 \beta_{8} - \beta_{11} + \beta_{12} ) q^{89} + ( -3 - 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{9} - \beta_{14} - \beta_{15} ) q^{91} + ( 2 \beta_{2} - \beta_{6} + \beta_{9} - \beta_{10} ) q^{94} + ( 4 \beta_{5} + \beta_{7} - \beta_{8} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{95} + ( 2 \beta_{2} - 2 \beta_{6} ) q^{97} + ( 2 \beta_{1} + 2 \beta_{5} + \beta_{7} + \beta_{8} + \beta_{11} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 16q^{4} + O(q^{10})$$ $$16q - 16q^{4} + 16q^{16} - 16q^{22} + 48q^{25} - 16q^{37} - 80q^{43} + 24q^{49} + 16q^{58} - 16q^{64} + 16q^{67} + 24q^{70} + 96q^{79} - 80q^{85} + 16q^{88} - 32q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 12 x^{14} + 72 x^{12} + 1296 x^{10} + 8245 x^{8} + 18780 x^{6} + 20808 x^{4} - 3672 x^{2} + 324$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$1381965647 \nu^{15} - 16053899943 \nu^{13} + 91281494448 \nu^{11} + 1854613004958 \nu^{9} + 11898941379431 \nu^{7} + 28364190325341 \nu^{5} + 24819818006916 \nu^{3} - 7903986405426 \nu$$$$)/ 8313550996212$$ $$\beta_{2}$$ $$=$$ $$($$$$3221232346 \nu^{15} - 39189954813 \nu^{13} + 239020561026 \nu^{11} + 4127085846450 \nu^{9} + 25935569845462 \nu^{7} + 56767773870939 \nu^{5} + 61881879334122 \nu^{3} - 2619108000162 \nu$$$$)/ 8313550996212$$ $$\beta_{3}$$ $$=$$ $$($$$$-3221232346 \nu^{15} + 39189954813 \nu^{13} - 239020561026 \nu^{11} - 4127085846450 \nu^{9} - 25935569845462 \nu^{7} - 56767773870939 \nu^{5} - 61881879334122 \nu^{3} + 19246209992586 \nu$$$$)/ 8313550996212$$ $$\beta_{4}$$ $$=$$ $$($$$$133007 \nu^{14} - 1743582 \nu^{12} + 11322414 \nu^{10} + 162109944 \nu^{8} + 903956675 \nu^{6} + 1249135962 \nu^{4} - 217432278 \nu^{2} - 2021921136$$$$)/ 481024764$$ $$\beta_{5}$$ $$=$$ $$($$$$-12219569 \nu^{14} + 143946111 \nu^{12} - 844183530 \nu^{10} - 16075743516 \nu^{8} - 103907059937 \nu^{6} - 248202840507 \nu^{4} - 280109314314 \nu^{2} + 22892859036$$$$)/ 26476277058$$ $$\beta_{6}$$ $$=$$ $$($$$$2097307895 \nu^{15} - 24781419255 \nu^{13} + 146288587464 \nu^{11} + 2746984641030 \nu^{9} + 17784897420335 \nu^{7} + 42502397837685 \nu^{5} + 50362943794092 \nu^{3} - 2157576356394 \nu$$$$)/ 2771183665404$$ $$\beta_{7}$$ $$=$$ $$($$$$19480188302 \nu^{15} + 13882041737 \nu^{14} - 234062963571 \nu^{13} - 161840906400 \nu^{12} + 1400972065902 \nu^{11} + 934218818370 \nu^{10} + 25294834981458 \nu^{9} + 18439559986428 \nu^{8} + 159769990661966 \nu^{7} + 119968078560641 \nu^{6} + 356840743850133 \nu^{5} + 288480282523008 \nu^{4} + 370209846741390 \nu^{3} + 324339270312042 \nu^{2} - 115709358795318 \nu - 26508114581148$$$$)/ 16627101992424$$ $$\beta_{8}$$ $$=$$ $$($$$$19480188302 \nu^{15} - 13882041737 \nu^{14} - 234062963571 \nu^{13} + 161840906400 \nu^{12} + 1400972065902 \nu^{11} - 934218818370 \nu^{10} + 25294834981458 \nu^{9} - 18439559986428 \nu^{8} + 159769990661966 \nu^{7} - 119968078560641 \nu^{6} + 356840743850133 \nu^{5} - 288480282523008 \nu^{4} + 370209846741390 \nu^{3} - 324339270312042 \nu^{2} - 115709358795318 \nu + 26508114581148$$$$)/ 16627101992424$$ $$\beta_{9}$$ $$=$$ $$($$$$39894432671 \nu^{15} + 384169521 \nu^{14} - 475048585479 \nu^{13} - 4764275946 \nu^{12} + 2827831403844 \nu^{11} + 25319556342 \nu^{10} + 51984546028254 \nu^{9} + 563556418488 \nu^{8} + 333491777592671 \nu^{7} + 2291272948425 \nu^{6} + 780606940001001 \nu^{5} + 3182590103886 \nu^{4} + 907759525301604 \nu^{3} - 553821784434 \nu^{2} - 38786965485858 \nu + 30719356487784$$$$)/ 16627101992424$$ $$\beta_{10}$$ $$=$$ $$($$$$-39894432671 \nu^{15} + 384169521 \nu^{14} + 475048585479 \nu^{13} - 4764275946 \nu^{12} - 2827831403844 \nu^{11} + 25319556342 \nu^{10} - 51984546028254 \nu^{9} + 563556418488 \nu^{8} - 333491777592671 \nu^{7} + 2291272948425 \nu^{6} - 780606940001001 \nu^{5} + 3182590103886 \nu^{4} - 907759525301604 \nu^{3} - 553821784434 \nu^{2} + 38786965485858 \nu + 30719356487784$$$$)/ 16627101992424$$ $$\beta_{11}$$ $$=$$ $$($$$$27614604307 \nu^{15} + 17650113884 \nu^{14} - 335527493352 \nu^{13} - 212774422656 \nu^{12} + 2043949302306 \nu^{11} + 1289410603416 \nu^{10} + 35408015840484 \nu^{9} + 22721134737168 \nu^{8} + 222837281341987 \nu^{7} + 144651653487260 \nu^{6} + 491627421010416 \nu^{5} + 334165131463704 \nu^{4} + 525737247375474 \nu^{3} + 387312732459288 \nu^{2} - 163820467378116 \nu - 31653237281040$$$$)/ 16627101992424$$ $$\beta_{12}$$ $$=$$ $$($$$$-27614604307 \nu^{15} + 17650113884 \nu^{14} + 335527493352 \nu^{13} - 212774422656 \nu^{12} - 2043949302306 \nu^{11} + 1289410603416 \nu^{10} - 35408015840484 \nu^{9} + 22721134737168 \nu^{8} - 222837281341987 \nu^{7} + 144651653487260 \nu^{6} - 491627421010416 \nu^{5} + 334165131463704 \nu^{4} - 525737247375474 \nu^{3} + 387312732459288 \nu^{2} + 163820467378116 \nu - 31653237281040$$$$)/ 16627101992424$$ $$\beta_{13}$$ $$=$$ $$($$$$66695 \nu^{14} - 802932 \nu^{12} + 4850514 \nu^{10} + 86012388 \nu^{8} + 548291507 \nu^{6} + 1249477428 \nu^{4} + 1468931886 \nu^{2} - 120049236$$$$)/32561172$$ $$\beta_{14}$$ $$=$$ $$($$$$-54470935301 \nu^{15} - 1838672364 \nu^{14} + 645220229322 \nu^{13} + 23712216120 \nu^{12} - 3821988100914 \nu^{11} - 145901660616 \nu^{10} - 71169624928584 \nu^{9} - 2374094531592 \nu^{8} - 460394505565205 \nu^{7} - 12036485070732 \nu^{6} - 1092006367892142 \nu^{5} - 16656226850664 \nu^{4} - 1285878937382370 \nu^{3} + 2899055925144 \nu^{2} + 55039531212000 \nu + 19609438426344$$$$)/ 16627101992424$$ $$\beta_{15}$$ $$=$$ $$($$$$-54470935301 \nu^{15} + 1838672364 \nu^{14} + 645220229322 \nu^{13} - 23712216120 \nu^{12} - 3821988100914 \nu^{11} + 145901660616 \nu^{10} - 71169624928584 \nu^{9} + 2374094531592 \nu^{8} - 460394505565205 \nu^{7} + 12036485070732 \nu^{6} - 1092006367892142 \nu^{5} + 16656226850664 \nu^{4} - 1285878937382370 \nu^{3} - 2899055925144 \nu^{2} + 55039531212000 \nu - 36236540418768$$$$)/ 16627101992424$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - 4 \beta_{5} - \beta_{4} + 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{15} + \beta_{14} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + 13 \beta_{6} + 4 \beta_{3} + 4 \beta_{2} - 5 \beta_{1} + 1$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$8 \beta_{13} - 17 \beta_{12} - 17 \beta_{11} + 28 \beta_{8} - 28 \beta_{7} - 144 \beta_{5}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$14 \beta_{15} + 14 \beta_{14} + 3 \beta_{12} - 3 \beta_{11} + 17 \beta_{10} - 17 \beta_{9} - 11 \beta_{8} - 11 \beta_{7} + 191 \beta_{6} - 74 \beta_{3} + 74 \beta_{2} + 43 \beta_{1} + 14$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-202 \beta_{15} + 202 \beta_{14} + 117 \beta_{13} - 202 \beta_{12} - 202 \beta_{11} + 271 \beta_{10} + 271 \beta_{9} + 271 \beta_{8} - 271 \beta_{7} - 1390 \beta_{5} + 117 \beta_{4} - 1188$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$69 \beta_{15} + 69 \beta_{14} + 202 \beta_{12} - 202 \beta_{11} + 133 \beta_{10} - 133 \beta_{9} - 271 \beta_{8} - 271 \beta_{7} + 1168 \beta_{6} - 2869 \beta_{3} + 533 \beta_{2} + 1168 \beta_{1} + 69$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-4175 \beta_{15} + 4175 \beta_{14} + 6004 \beta_{10} + 6004 \beta_{9} + 2336 \beta_{4} - 26389$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-1173 \beta_{15} - 1173 \beta_{14} + 3002 \beta_{12} - 3002 \beta_{11} - 1829 \beta_{10} + 1829 \beta_{9} - 4175 \beta_{8} - 4175 \beta_{7} - 17858 \beta_{6} - 43319 \beta_{3} - 7603 \beta_{2} + 17858 \beta_{1} - 1173$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-45148 \beta_{15} + 45148 \beta_{14} - 25461 \beta_{13} + 45148 \beta_{12} + 45148 \beta_{11} + 63523 \beta_{10} + 63523 \beta_{9} - 63523 \beta_{8} + 63523 \beta_{7} + 324490 \beta_{5} + 25461 \beta_{4} - 279342$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$-45148 \beta_{15} - 45148 \beta_{14} + 18375 \beta_{12} - 18375 \beta_{11} - 63523 \beta_{10} + 63523 \beta_{9} - 26773 \beta_{8} - 26773 \beta_{7} - 654937 \beta_{6} - 270916 \beta_{3} - 270916 \beta_{2} + 113105 \beta_{1} - 45148$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$($$$$-541832 \beta_{13} + 962603 \beta_{12} + 962603 \beta_{11} - 1363420 \beta_{8} + 1363420 \beta_{7} + 6956172 \beta_{5}$$$$)/2$$ $$\nu^{13}$$ $$=$$ $$($$$$-681710 \beta_{15} - 681710 \beta_{14} - 280893 \beta_{12} + 280893 \beta_{11} - 962603 \beta_{10} + 962603 \beta_{9} + 400817 \beta_{8} + 400817 \beta_{7} - 9905483 \beta_{6} + 4101374 \beta_{3} - 4101374 \beta_{2} - 1702735 \beta_{1} - 681710$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$10306300 \beta_{15} - 10306300 \beta_{14} - 5804109 \beta_{13} + 10306300 \beta_{12} + 10306300 \beta_{11} - 14568643 \beta_{10} - 14568643 \beta_{9} - 14568643 \beta_{8} + 14568643 \beta_{7} + 74358130 \beta_{5} - 5804109 \beta_{4} + 64051830$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$-4262343 \beta_{15} - 4262343 \beta_{14} - 10306300 \beta_{12} + 10306300 \beta_{11} - 6043957 \beta_{10} + 6043957 \beta_{9} + 14568643 \beta_{8} + 14568643 \beta_{7} - 62054008 \beta_{6} + 149828821 \beta_{3} - 25720805 \beta_{2} - 62054008 \beta_{1} - 4262343$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$$.

 $$n$$ $$155$$ $$199$$ $$1135$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
881.1
 −0.808328 + 1.95148i 3.59322 + 1.48836i −0.579826 + 1.39982i 0.314903 + 0.130437i −0.314903 − 0.130437i 0.579826 − 1.39982i −3.59322 − 1.48836i 0.808328 − 1.95148i −0.808328 − 1.95148i 3.59322 − 1.48836i −0.579826 − 1.39982i 0.314903 − 0.130437i −0.314903 + 0.130437i 0.579826 + 1.39982i −3.59322 + 1.48836i 0.808328 + 1.95148i
1.00000i 0 −1.00000 −3.90295 0 1.49520 2.18274i 1.00000i 0 3.90295i
881.2 1.00000i 0 −1.00000 −2.97672 0 2.55176 0.698947i 1.00000i 0 2.97672i
881.3 1.00000i 0 −1.00000 −2.79965 0 −2.20231 + 1.46623i 1.00000i 0 2.79965i
881.4 1.00000i 0 −1.00000 −0.260874 0 −1.84465 + 1.89664i 1.00000i 0 0.260874i
881.5 1.00000i 0 −1.00000 0.260874 0 −1.84465 1.89664i 1.00000i 0 0.260874i
881.6 1.00000i 0 −1.00000 2.79965 0 −2.20231 1.46623i 1.00000i 0 2.79965i
881.7 1.00000i 0 −1.00000 2.97672 0 2.55176 + 0.698947i 1.00000i 0 2.97672i
881.8 1.00000i 0 −1.00000 3.90295 0 1.49520 + 2.18274i 1.00000i 0 3.90295i
881.9 1.00000i 0 −1.00000 −3.90295 0 1.49520 + 2.18274i 1.00000i 0 3.90295i
881.10 1.00000i 0 −1.00000 −2.97672 0 2.55176 + 0.698947i 1.00000i 0 2.97672i
881.11 1.00000i 0 −1.00000 −2.79965 0 −2.20231 1.46623i 1.00000i 0 2.79965i
881.12 1.00000i 0 −1.00000 −0.260874 0 −1.84465 1.89664i 1.00000i 0 0.260874i
881.13 1.00000i 0 −1.00000 0.260874 0 −1.84465 + 1.89664i 1.00000i 0 0.260874i
881.14 1.00000i 0 −1.00000 2.79965 0 −2.20231 + 1.46623i 1.00000i 0 2.79965i
881.15 1.00000i 0 −1.00000 2.97672 0 2.55176 0.698947i 1.00000i 0 2.97672i
881.16 1.00000i 0 −1.00000 3.90295 0 1.49520 2.18274i 1.00000i 0 3.90295i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 881.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.2.g.b 16
3.b odd 2 1 inner 1386.2.g.b 16
7.b odd 2 1 inner 1386.2.g.b 16
21.c even 2 1 inner 1386.2.g.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.g.b 16 1.a even 1 1 trivial
1386.2.g.b 16 3.b odd 2 1 inner
1386.2.g.b 16 7.b odd 2 1 inner
1386.2.g.b 16 21.c even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 32 T_{5}^{6} + 326 T_{5}^{4} - 1080 T_{5}^{2} + 72$$ acting on $$S_{2}^{\mathrm{new}}(1386, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{8}$$
$3$ $$T^{16}$$
$5$ $$( 72 - 1080 T^{2} + 326 T^{4} - 32 T^{6} + T^{8} )^{2}$$
$7$ $$( 2401 - 294 T^{2} - 56 T^{3} + 66 T^{4} - 8 T^{5} - 6 T^{6} + T^{8} )^{2}$$
$11$ $$( 1 + T^{2} )^{8}$$
$13$ $$( 73728 + 27648 T^{2} + 2576 T^{4} + 88 T^{6} + T^{8} )^{2}$$
$17$ $$( 288 - 1488 T^{2} + 674 T^{4} - 52 T^{6} + T^{8} )^{2}$$
$19$ $$( 72 + 264 T^{2} + 230 T^{4} + 56 T^{6} + T^{8} )^{2}$$
$23$ $$T^{16}$$
$29$ $$( 254016 + 167328 T^{2} + 10420 T^{4} + 188 T^{6} + T^{8} )^{2}$$
$31$ $$( 73728 + 31488 T^{2} + 2834 T^{4} + 92 T^{6} + T^{8} )^{2}$$
$37$ $$( 448 - 224 T - 76 T^{2} + 4 T^{3} + T^{4} )^{4}$$
$41$ $$( 288 - 7920 T^{2} + 2018 T^{4} - 116 T^{6} + T^{8} )^{2}$$
$43$ $$( -7792 - 1680 T + 10 T^{2} + 20 T^{3} + T^{4} )^{4}$$
$47$ $$( 225792 - 60480 T^{2} + 4946 T^{4} - 140 T^{6} + T^{8} )^{2}$$
$53$ $$( 11451456 + 946080 T^{2} + 26356 T^{4} + 284 T^{6} + T^{8} )^{2}$$
$59$ $$( 40716288 - 2990592 T^{2} + 57440 T^{4} - 416 T^{6} + T^{8} )^{2}$$
$61$ $$( 73728 + 39936 T^{2} + 5648 T^{4} + 200 T^{6} + T^{8} )^{2}$$
$67$ $$( 144 - 144 T - 86 T^{2} - 4 T^{3} + T^{4} )^{4}$$
$71$ $$( 72 + T^{2} )^{8}$$
$73$ $$( 206613792 + 6952176 T^{2} + 87266 T^{4} + 484 T^{6} + T^{8} )^{2}$$
$79$ $$( -1564 + 1040 T + 68 T^{2} - 24 T^{3} + T^{4} )^{4}$$
$83$ $$( 4608 - 1789632 T^{2} + 55250 T^{4} - 476 T^{6} + T^{8} )^{2}$$
$89$ $$( 12221568 - 2587200 T^{2} + 64868 T^{4} - 500 T^{6} + T^{8} )^{2}$$
$97$ $$( 294912 + 552960 T^{2} + 20864 T^{4} + 256 T^{6} + T^{8} )^{2}$$