# Properties

 Label 7168.2.a.bi Level $7168$ Weight $2$ Character orbit 7168.a Self dual yes Analytic conductor $57.237$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7168 = 2^{10} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7168.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$57.2367681689$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 24x^{10} + 221x^{8} - 968x^{6} + 2008x^{4} - 1640x^{2} + 196$$ x^12 - 24*x^10 + 221*x^8 - 968*x^6 + 2008*x^4 - 1640*x^2 + 196 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: no (minimal twist has level 112) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{5} - q^{7} + (\beta_{9} + \beta_{8} + 1) q^{9}+O(q^{10})$$ q + b1 * q^3 + (-b4 - b2 - b1) * q^5 - q^7 + (b9 + b8 + 1) * q^9 $$q + \beta_1 q^{3} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{5} - q^{7} + (\beta_{9} + \beta_{8} + 1) q^{9} + \beta_{5} q^{11} - \beta_{6} q^{13} + ( - \beta_{8} - \beta_{3} - 2) q^{15} + (\beta_{10} + 1) q^{17} + (\beta_{5} - \beta_{2} + \beta_1) q^{19} - \beta_1 q^{21} + ( - \beta_{10} - \beta_{7} - 2) q^{23} + (\beta_{10} + \beta_{7} + \beta_{3} + 2) q^{25} + (\beta_{11} + \beta_{4} + 2 \beta_1) q^{27} + ( - \beta_{6} - \beta_{5} + \beta_1) q^{29} + ( - \beta_{3} + 1) q^{31} + ( - \beta_{10} + \beta_{8} - \beta_{7} - \beta_{3}) q^{33} + (\beta_{4} + \beta_{2} + \beta_1) q^{35} + (\beta_{4} - 2 \beta_{2}) q^{37} + (\beta_{10} - \beta_{8} + 2 \beta_{3} - 2) q^{39} + (\beta_{9} + \beta_{3} + 2) q^{41} + ( - \beta_{11} + \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{2} - \beta_1) q^{43} + ( - \beta_{11} + \beta_{6} + \beta_{5} - 5 \beta_{4} - \beta_{2} - 4 \beta_1) q^{45} + (\beta_{9} + \beta_{8} + \beta_{7} - \beta_{3} + 2) q^{47} + q^{49} + ( - \beta_{6} - \beta_{5} - \beta_{4} + \beta_1) q^{51} + (\beta_{11} + \beta_{5} - \beta_{2}) q^{53} + ( - \beta_{9} + \beta_{8} + \beta_{7} - 1) q^{55} + ( - \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - \beta_{7} - \beta_{3} + 5) q^{57} + ( - \beta_{11} - \beta_{4} + \beta_1) q^{59} + (\beta_{11} - \beta_{5} - 2 \beta_{4} - \beta_1) q^{61} + ( - \beta_{9} - \beta_{8} - 1) q^{63} + ( - \beta_{9} - \beta_{8} + 2 \beta_{7} - 2 \beta_{3} + 4) q^{65} + ( - \beta_{6} - \beta_{4} + 2 \beta_{2} + \beta_1) q^{67} + (\beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{2} - 3 \beta_1) q^{69} + (\beta_{10} + 2 \beta_{8} + \beta_{3}) q^{71} + ( - \beta_{10} + \beta_{9} - 2 \beta_{8} - \beta_{3} - 1) q^{73} + ( - 2 \beta_{6} - 3 \beta_{5} + 6 \beta_{4} + 3 \beta_{2} + 5 \beta_1) q^{75} - \beta_{5} q^{77} + ( - \beta_{10} + \beta_{8} - \beta_{7} - \beta_{3} + 2) q^{79} + ( - \beta_{10} + 2 \beta_{8} - \beta_{7} - \beta_{3} + 4) q^{81} + ( - \beta_{11} + 2 \beta_{6} + \beta_{5} - 3 \beta_{4} - 3 \beta_{2} - \beta_1) q^{83} + ( - \beta_{11} + \beta_{6} - 2 \beta_{4} - 3 \beta_{2} - \beta_1) q^{85} + (2 \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + 3 \beta_{3} + 2) q^{87} + ( - \beta_{10} - \beta_{8} - \beta_{7} + 3 \beta_{3}) q^{89} + \beta_{6} q^{91} + (\beta_{6} + \beta_{5} - 5 \beta_{4} - 2 \beta_{2} - \beta_1) q^{93} + (\beta_{10} - 2 \beta_{9} - \beta_{8} + 2 \beta_{7} - 2 \beta_{3} + 2) q^{95} + (\beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + 2 \beta_{3} + 4) q^{97} + (\beta_{11} + 2 \beta_{6} - 3 \beta_{4} - \beta_{2}) q^{99}+O(q^{100})$$ q + b1 * q^3 + (-b4 - b2 - b1) * q^5 - q^7 + (b9 + b8 + 1) * q^9 + b5 * q^11 - b6 * q^13 + (-b8 - b3 - 2) * q^15 + (b10 + 1) * q^17 + (b5 - b2 + b1) * q^19 - b1 * q^21 + (-b10 - b7 - 2) * q^23 + (b10 + b7 + b3 + 2) * q^25 + (b11 + b4 + 2*b1) * q^27 + (-b6 - b5 + b1) * q^29 + (-b3 + 1) * q^31 + (-b10 + b8 - b7 - b3) * q^33 + (b4 + b2 + b1) * q^35 + (b4 - 2*b2) * q^37 + (b10 - b8 + 2*b3 - 2) * q^39 + (b9 + b3 + 2) * q^41 + (-b11 + b6 + b5 - 2*b4 - b2 - b1) * q^43 + (-b11 + b6 + b5 - 5*b4 - b2 - 4*b1) * q^45 + (b9 + b8 + b7 - b3 + 2) * q^47 + q^49 + (-b6 - b5 - b4 + b1) * q^51 + (b11 + b5 - b2) * q^53 + (-b9 + b8 + b7 - 1) * q^55 + (-b10 + 2*b9 + 2*b8 - b7 - b3 + 5) * q^57 + (-b11 - b4 + b1) * q^59 + (b11 - b5 - 2*b4 - b1) * q^61 + (-b9 - b8 - 1) * q^63 + (-b9 - b8 + 2*b7 - 2*b3 + 4) * q^65 + (-b6 - b4 + 2*b2 + b1) * q^67 + (b6 + 2*b5 - b4 - b2 - 3*b1) * q^69 + (b10 + 2*b8 + b3) * q^71 + (-b10 + b9 - 2*b8 - b3 - 1) * q^73 + (-2*b6 - 3*b5 + 6*b4 + 3*b2 + 5*b1) * q^75 - b5 * q^77 + (-b10 + b8 - b7 - b3 + 2) * q^79 + (-b10 + 2*b8 - b7 - b3 + 4) * q^81 + (-b11 + 2*b6 + b5 - 3*b4 - 3*b2 - b1) * q^83 + (-b11 + b6 - 2*b4 - 3*b2 - b1) * q^85 + (2*b10 + b9 - b8 + b7 + 3*b3 + 2) * q^87 + (-b10 - b8 - b7 + 3*b3) * q^89 + b6 * q^91 + (b6 + b5 - 5*b4 - 2*b2 - b1) * q^93 + (b10 - 2*b9 - b8 + 2*b7 - 2*b3 + 2) * q^95 + (b10 - b9 + b8 - b7 + 2*b3 + 4) * q^97 + (b11 + 2*b6 - 3*b4 - b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 12 q^{7} + 12 q^{9}+O(q^{10})$$ 12 * q - 12 * q^7 + 12 * q^9 $$12 q - 12 q^{7} + 12 q^{9} - 24 q^{15} + 8 q^{17} - 16 q^{23} + 20 q^{25} + 8 q^{31} - 16 q^{39} + 32 q^{41} + 16 q^{47} + 12 q^{49} - 24 q^{55} + 64 q^{57} - 12 q^{63} + 32 q^{65} - 8 q^{71} + 24 q^{79} + 44 q^{81} + 32 q^{87} + 24 q^{89} + 48 q^{97}+O(q^{100})$$ 12 * q - 12 * q^7 + 12 * q^9 - 24 * q^15 + 8 * q^17 - 16 * q^23 + 20 * q^25 + 8 * q^31 - 16 * q^39 + 32 * q^41 + 16 * q^47 + 12 * q^49 - 24 * q^55 + 64 * q^57 - 12 * q^63 + 32 * q^65 - 8 * q^71 + 24 * q^79 + 44 * q^81 + 32 * q^87 + 24 * q^89 + 48 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 24x^{10} + 221x^{8} - 968x^{6} + 2008x^{4} - 1640x^{2} + 196$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{11} + 17\nu^{9} - 95\nu^{7} + 177\nu^{5} - 6\nu^{3} - 68\nu ) / 42$$ (-v^11 + 17*v^9 - 95*v^7 + 177*v^5 - 6*v^3 - 68*v) / 42 $$\beta_{2}$$ $$=$$ $$( -\nu^{11} + 17\nu^{9} - 95\nu^{7} + 177\nu^{5} + 36\nu^{3} - 278\nu ) / 42$$ (-v^11 + 17*v^9 - 95*v^7 + 177*v^5 + 36*v^3 - 278*v) / 42 $$\beta_{3}$$ $$=$$ $$( \nu^{8} - 16\nu^{6} + 79\nu^{4} - 116\nu^{2} + 16 ) / 6$$ (v^8 - 16*v^6 + 79*v^4 - 116*v^2 + 16) / 6 $$\beta_{4}$$ $$=$$ $$( -\nu^{11} + 24\nu^{9} - 207\nu^{7} + 772\nu^{5} - 1154\nu^{3} + 464\nu ) / 84$$ (-v^11 + 24*v^9 - 207*v^7 + 772*v^5 - 1154*v^3 + 464*v) / 84 $$\beta_{5}$$ $$=$$ $$( 2\nu^{11} - 41\nu^{9} + 316\nu^{7} - 1117\nu^{5} + 1720\nu^{3} - 830\nu ) / 42$$ (2*v^11 - 41*v^9 + 316*v^7 - 1117*v^5 + 1720*v^3 - 830*v) / 42 $$\beta_{6}$$ $$=$$ $$( -\nu^{11} + 18\nu^{9} - 115\nu^{7} + 322\nu^{5} - 426\nu^{3} + 264\nu ) / 12$$ (-v^11 + 18*v^9 - 115*v^7 + 322*v^5 - 426*v^3 + 264*v) / 12 $$\beta_{7}$$ $$=$$ $$( -\nu^{10} + 18\nu^{8} - 113\nu^{6} + 286\nu^{4} - 232\nu^{2} - 26 ) / 6$$ (-v^10 + 18*v^8 - 113*v^6 + 286*v^4 - 232*v^2 - 26) / 6 $$\beta_{8}$$ $$=$$ $$( -\nu^{10} + 18\nu^{8} - 113\nu^{6} + 286\nu^{4} - 244\nu^{2} + 22 ) / 6$$ (-v^10 + 18*v^8 - 113*v^6 + 286*v^4 - 244*v^2 + 22) / 6 $$\beta_{9}$$ $$=$$ $$( \nu^{10} - 18\nu^{8} + 115\nu^{6} - 310\nu^{4} + 318\nu^{2} - 54 ) / 6$$ (v^10 - 18*v^8 + 115*v^6 - 310*v^4 + 318*v^2 - 54) / 6 $$\beta_{10}$$ $$=$$ $$( -\nu^{10} + 19\nu^{8} - 127\nu^{6} + 353\nu^{4} - 370\nu^{2} + 72 ) / 6$$ (-v^10 + 19*v^8 - 127*v^6 + 353*v^4 - 370*v^2 + 72) / 6 $$\beta_{11}$$ $$=$$ $$( 15\nu^{11} - 276\nu^{9} + 1789\nu^{7} - 4776\nu^{5} + 4402\nu^{3} + 152\nu ) / 84$$ (15*v^11 - 276*v^9 + 1789*v^7 - 4776*v^5 + 4402*v^3 + 152*v) / 84
 $$\nu$$ $$=$$ $$( \beta_{11} - \beta_{5} + \beta_{4} + 3\beta_{2} + 2\beta_1 ) / 4$$ (b11 - b5 + b4 + 3*b2 + 2*b1) / 4 $$\nu^{2}$$ $$=$$ $$( -\beta_{8} + \beta_{7} + 8 ) / 2$$ (-b8 + b7 + 8) / 2 $$\nu^{3}$$ $$=$$ $$( 5\beta_{11} - 5\beta_{5} + 5\beta_{4} + 19\beta_{2} + 6\beta_1 ) / 4$$ (5*b11 - 5*b5 + 5*b4 + 19*b2 + 6*b1) / 4 $$\nu^{4}$$ $$=$$ $$( \beta_{10} - \beta_{9} - 9\beta_{8} + 7\beta_{7} - \beta_{3} + 45 ) / 2$$ (b10 - b9 - 9*b8 + 7*b7 - b3 + 45) / 2 $$\nu^{5}$$ $$=$$ $$( 29\beta_{11} + 4\beta_{6} - 25\beta_{5} + 33\beta_{4} + 119\beta_{2} + 18\beta_1 ) / 4$$ (29*b11 + 4*b6 - 25*b5 + 33*b4 + 119*b2 + 18*b1) / 4 $$\nu^{6}$$ $$=$$ $$( 12\beta_{10} - 6\beta_{9} - 65\beta_{8} + 47\beta_{7} - 12\beta_{3} + 276 ) / 2$$ (12*b10 - 6*b9 - 65*b8 + 47*b7 - 12*b3 + 276) / 2 $$\nu^{7}$$ $$=$$ $$( 179\beta_{11} + 48\beta_{6} - 119\beta_{5} + 251\beta_{4} + 761\beta_{2} + 50\beta_1 ) / 4$$ (179*b11 + 48*b6 - 119*b5 + 251*b4 + 761*b2 + 50*b1) / 4 $$\nu^{8}$$ $$=$$ $$( 113\beta_{10} - 17\beta_{9} - 445\beta_{8} + 315\beta_{7} - 101\beta_{3} + 1757 ) / 2$$ (113*b10 - 17*b9 - 445*b8 + 315*b7 - 101*b3 + 1757) / 2 $$\nu^{9}$$ $$=$$ $$( 1143\beta_{11} + 428\beta_{6} - 523\beta_{5} + 2003\beta_{4} + 4949\beta_{2} + 78\beta_1 ) / 4$$ (1143*b11 + 428*b6 - 523*b5 + 2003*b4 + 4949*b2 + 78*b1) / 4 $$\nu^{10}$$ $$=$$ $$( 964\beta_{10} + 86\beta_{9} - 3007\beta_{8} + 2117\beta_{7} - 748\beta_{3} + 11400 ) / 2$$ (964*b10 + 86*b9 - 3007*b8 + 2117*b7 - 748*b3 + 11400) / 2 $$\nu^{11}$$ $$=$$ $$( 7461\beta_{11} + 3424\beta_{6} - 1913\beta_{5} + 15949\beta_{4} + 32583\beta_{2} - 578\beta_1 ) / 4$$ (7461*b11 + 3424*b6 - 1913*b5 + 15949*b4 + 32583*b2 - 578*b1) / 4

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.33191 −1.75510 2.66376 −2.48658 2.39251 0.377920 −0.377920 −2.39251 2.48658 −2.66376 1.75510 1.33191
0 −3.13347 0 0.555963 0 −1.00000 0 6.81864 0
1.2 0 −2.90620 0 3.85750 0 −1.00000 0 5.44602 0
1.3 0 −1.96750 0 −3.06146 0 −1.00000 0 0.871066 0
1.4 0 −0.892634 0 3.31293 0 −1.00000 0 −2.20320 0
1.5 0 −0.848497 0 1.37882 0 −1.00000 0 −2.28005 0
1.6 0 −0.589521 0 1.60045 0 −1.00000 0 −2.65247 0
1.7 0 0.589521 0 −1.60045 0 −1.00000 0 −2.65247 0
1.8 0 0.848497 0 −1.37882 0 −1.00000 0 −2.28005 0
1.9 0 0.892634 0 −3.31293 0 −1.00000 0 −2.20320 0
1.10 0 1.96750 0 3.06146 0 −1.00000 0 0.871066 0
1.11 0 2.90620 0 −3.85750 0 −1.00000 0 5.44602 0
1.12 0 3.13347 0 −0.555963 0 −1.00000 0 6.81864 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7168.2.a.bi 12
4.b odd 2 1 7168.2.a.bj 12
8.b even 2 1 inner 7168.2.a.bi 12
8.d odd 2 1 7168.2.a.bj 12
32.g even 8 2 448.2.m.d 12
32.g even 8 2 896.2.m.h 12
32.h odd 8 2 112.2.m.d 12
32.h odd 8 2 896.2.m.g 12
224.x even 8 2 784.2.m.h 12
224.be even 24 4 784.2.x.m 24
224.bf odd 24 4 784.2.x.l 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.m.d 12 32.h odd 8 2
448.2.m.d 12 32.g even 8 2
784.2.m.h 12 224.x even 8 2
784.2.x.l 24 224.bf odd 24 4
784.2.x.m 24 224.be even 24 4
896.2.m.g 12 32.h odd 8 2
896.2.m.h 12 32.g even 8 2
7168.2.a.bi 12 1.a even 1 1 trivial
7168.2.a.bi 12 8.b even 2 1 inner
7168.2.a.bj 12 4.b odd 2 1
7168.2.a.bj 12 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7168))$$:

 $$T_{3}^{12} - 24T_{3}^{10} + 196T_{3}^{8} - 632T_{3}^{6} + 772T_{3}^{4} - 384T_{3}^{2} + 64$$ T3^12 - 24*T3^10 + 196*T3^8 - 632*T3^6 + 772*T3^4 - 384*T3^2 + 64 $$T_{5}^{12} - 40T_{5}^{10} + 580T_{5}^{8} - 3688T_{5}^{6} + 9892T_{5}^{4} - 10176T_{5}^{2} + 2304$$ T5^12 - 40*T5^10 + 580*T5^8 - 3688*T5^6 + 9892*T5^4 - 10176*T5^2 + 2304 $$T_{11}^{12} - 76T_{11}^{10} + 2148T_{11}^{8} - 28736T_{11}^{6} + 190080T_{11}^{4} - 569344T_{11}^{2} + 541696$$ T11^12 - 76*T11^10 + 2148*T11^8 - 28736*T11^6 + 190080*T11^4 - 569344*T11^2 + 541696 $$T_{13}^{12} - 112T_{13}^{10} + 4548T_{13}^{8} - 82856T_{13}^{6} + 694020T_{13}^{4} - 2563072T_{13}^{2} + 3211264$$ T13^12 - 112*T13^10 + 4548*T13^8 - 82856*T13^6 + 694020*T13^4 - 2563072*T13^2 + 3211264 $$T_{17}^{6} - 4T_{17}^{5} - 44T_{17}^{4} + 200T_{17}^{3} + 40T_{17}^{2} - 288T_{17} - 96$$ T17^6 - 4*T17^5 - 44*T17^4 + 200*T17^3 + 40*T17^2 - 288*T17 - 96 $$T_{23}^{6} + 8T_{23}^{5} - 36T_{23}^{4} - 320T_{23}^{3} + 340T_{23}^{2} + 2656T_{23} - 3184$$ T23^6 + 8*T23^5 - 36*T23^4 - 320*T23^3 + 340*T23^2 + 2656*T23 - 3184 $$T_{31}^{6} - 4T_{31}^{5} - 16T_{31}^{4} + 72T_{31}^{3} - 24T_{31}^{2} - 96T_{31} + 64$$ T31^6 - 4*T31^5 - 16*T31^4 + 72*T31^3 - 24*T31^2 - 96*T31 + 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} - 24 T^{10} + 196 T^{8} + \cdots + 64$$
$5$ $$T^{12} - 40 T^{10} + 580 T^{8} + \cdots + 2304$$
$7$ $$(T + 1)^{12}$$
$11$ $$T^{12} - 76 T^{10} + 2148 T^{8} + \cdots + 541696$$
$13$ $$T^{12} - 112 T^{10} + 4548 T^{8} + \cdots + 3211264$$
$17$ $$(T^{6} - 4 T^{5} - 44 T^{4} + 200 T^{3} + \cdots - 96)^{2}$$
$19$ $$T^{12} - 128 T^{10} + 5748 T^{8} + \cdots + 2849344$$
$23$ $$(T^{6} + 8 T^{5} - 36 T^{4} - 320 T^{3} + \cdots - 3184)^{2}$$
$29$ $$T^{12} - 156 T^{10} + 8188 T^{8} + \cdots + 8620096$$
$31$ $$(T^{6} - 4 T^{5} - 16 T^{4} + 72 T^{3} + \cdots + 64)^{2}$$
$37$ $$T^{12} - 172 T^{10} + 9980 T^{8} + \cdots + 5053504$$
$41$ $$(T^{6} - 16 T^{5} + 68 T^{4} + 40 T^{3} + \cdots + 160)^{2}$$
$43$ $$T^{12} - 220 T^{10} + \cdots + 23040000$$
$47$ $$(T^{6} - 8 T^{5} - 104 T^{4} + 1032 T^{3} + \cdots + 19776)^{2}$$
$53$ $$T^{12} - 348 T^{10} + 40156 T^{8} + \cdots + 78400$$
$59$ $$T^{12} - 240 T^{10} + \cdots + 119596096$$
$61$ $$T^{12} - 264 T^{10} + 16100 T^{8} + \cdots + 256$$
$67$ $$T^{12} - 236 T^{10} + 10980 T^{8} + \cdots + 3686400$$
$71$ $$(T^{6} + 4 T^{5} - 208 T^{4} - 880 T^{3} + \cdots + 2560)^{2}$$
$73$ $$(T^{6} - 308 T^{4} - 384 T^{3} + \cdots - 55232)^{2}$$
$79$ $$(T^{6} - 12 T^{5} - 44 T^{4} + 576 T^{3} + \cdots - 2240)^{2}$$
$83$ $$T^{12} - 568 T^{10} + \cdots + 320093429824$$
$89$ $$(T^{6} - 12 T^{5} - 204 T^{4} + \cdots - 188352)^{2}$$
$97$ $$(T^{6} - 24 T^{5} - 60 T^{4} + 2568 T^{3} + \cdots - 39008)^{2}$$