# Properties

 Label 117.4.g.e Level $117$ Weight $4$ Character orbit 117.g Analytic conductor $6.903$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 117.g (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.90322347067$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 2x^{7} + 29x^{6} + 2x^{5} + 595x^{4} - 288x^{3} + 2526x^{2} + 1872x + 6084$$ x^8 - 2*x^7 + 29*x^6 + 2*x^5 + 595*x^4 - 288*x^3 + 2526*x^2 + 1872*x + 6084 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}\cdot 3$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{6} + \beta_{5} - \beta_{3} + 5 \beta_{2} + \beta_1 - 1) q^{4} + ( - \beta_{5} + \beta_{3} + 2) q^{5} + (\beta_{7} + \beta_{5} - 4 \beta_{2} + \beta_1 - 1) q^{7} + (\beta_{7} + 5 \beta_{5} - \beta_{4} - 2 \beta_{3} - 16) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b6 + b5 - b3 + 5*b2 + b1 - 1) * q^4 + (-b5 + b3 + 2) * q^5 + (b7 + b5 - 4*b2 + b1 - 1) * q^7 + (b7 + 5*b5 - b4 - 2*b3 - 16) * q^8 $$q + \beta_1 q^{2} + (\beta_{6} + \beta_{5} - \beta_{3} + 5 \beta_{2} + \beta_1 - 1) q^{4} + ( - \beta_{5} + \beta_{3} + 2) q^{5} + (\beta_{7} + \beta_{5} - 4 \beta_{2} + \beta_1 - 1) q^{7} + (\beta_{7} + 5 \beta_{5} - \beta_{4} - 2 \beta_{3} - 16) q^{8} + (2 \beta_{6} + \beta_{4} + 11 \beta_{2} + 9 \beta_1 + 11) q^{10} + ( - \beta_{6} + \beta_{4} + 8 \beta_{2} + 4 \beta_1 + 8) q^{11} + ( - \beta_{6} - \beta_{5} - \beta_{4} - 3 \beta_{3} - 25 \beta_{2} + 2 \beta_1 - 20) q^{13} + ( - 6 \beta_{5} - 7 \beta_{3} - 7) q^{14} + ( - 5 \beta_{6} - 2 \beta_{4} - 21 \beta_{2} - 19 \beta_1 - 21) q^{16} + (\beta_{7} - 19 \beta_{5} - 15 \beta_{2} - 19 \beta_1 + 19) q^{17} + (\beta_{7} - 7 \beta_{6} + 6 \beta_{5} + 7 \beta_{3} + 28 \beta_{2} + 6 \beta_1 - 6) q^{19} + (2 \beta_{7} + 9 \beta_{6} + 23 \beta_{5} - 9 \beta_{3} + 105 \beta_{2} + 23 \beta_1 - 23) q^{20} + ( - \beta_{7} + 9 \beta_{6} + 2 \beta_{5} - 9 \beta_{3} + 54 \beta_{2} + 2 \beta_1 - 2) q^{22} + (\beta_{6} + 3 \beta_{4} + 28 \beta_{2} - 4 \beta_1 + 28) q^{23} + ( - 2 \beta_{7} - 19 \beta_{5} + 2 \beta_{4} + 5 \beta_{3} - 5) q^{25} + ( - \beta_{7} - 7 \beta_{6} - 27 \beta_{5} - 3 \beta_{4} + 5 \beta_{3} + 47 \beta_{2} + \cdots + 46) q^{26}+ \cdots + (29 \beta_{7} + 84 \beta_{6} - 49 \beta_{5} - 84 \beta_{3} + 501 \beta_{2} + \cdots + 49) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b6 + b5 - b3 + 5*b2 + b1 - 1) * q^4 + (-b5 + b3 + 2) * q^5 + (b7 + b5 - 4*b2 + b1 - 1) * q^7 + (b7 + 5*b5 - b4 - 2*b3 - 16) * q^8 + (2*b6 + b4 + 11*b2 + 9*b1 + 11) * q^10 + (-b6 + b4 + 8*b2 + 4*b1 + 8) * q^11 + (-b6 - b5 - b4 - 3*b3 - 25*b2 + 2*b1 - 20) * q^13 + (-6*b5 - 7*b3 - 7) * q^14 + (-5*b6 - 2*b4 - 21*b2 - 19*b1 - 21) * q^16 + (b7 - 19*b5 - 15*b2 - 19*b1 + 19) * q^17 + (b7 - 7*b6 + 6*b5 + 7*b3 + 28*b2 + 6*b1 - 6) * q^19 + (2*b7 + 9*b6 + 23*b5 - 9*b3 + 105*b2 + 23*b1 - 23) * q^20 + (-b7 + 9*b6 + 2*b5 - 9*b3 + 54*b2 + 2*b1 - 2) * q^22 + (b6 + 3*b4 + 28*b2 - 4*b1 + 28) * q^23 + (-2*b7 - 19*b5 + 2*b4 + 5*b3 - 5) * q^25 + (-b7 - 7*b6 - 27*b5 - 3*b4 + 5*b3 + 47*b2 - 43*b1 + 46) * q^26 + (-b6 + b4 + 60*b2 - 48*b1 + 60) * q^28 + (b6 - 6*b4 + 43*b2 + 11*b1 + 43) * q^29 + (-53*b5 - 7*b3 + 33) * q^31 + (3*b7 - 20*b6 - 29*b5 + 20*b3 - 149*b2 - 29*b1 + 29) * q^32 + (-37*b5 + 13*b3 + 284) * q^34 + (-5*b7 + b6 + 44*b5 - b3 - 44*b2 + 44*b1 - 44) * q^35 + (2*b6 + b4 - 15*b2 - 21*b1 - 15) * q^37 + (-7*b7 - 18*b5 + 7*b4 - 5*b3 - 74) * q^38 + (b7 + 113*b5 - b4 - 28*b3 - 306) * q^40 + (3*b4 - 285*b2 + 43*b1 - 285) * q^41 + (-b7 - 14*b6 + 57*b5 + 14*b3 + 84*b2 + 57*b1 - 57) * q^43 + (b7 + 90*b5 - b4 - 13*b3 - 34) * q^44 + (b7 + 15*b6 + 22*b5 - 15*b3 - 54*b2 + 22*b1 - 22) * q^46 + (3*b7 - 8*b5 - 3*b4 + 47*b3 + 28) * q^47 + (29*b6 + 2*b4 - 289*b2 + 43*b1 - 289) * q^49 + (36*b6 + 5*b4 + 237*b2 + 24*b1 + 237) * q^50 + (b7 - 12*b6 - 57*b5 - 3*b4 + 42*b3 - 236*b2 - 10*b1 + 198) * q^52 + (2*b7 + 37*b5 - 2*b4 - 23*b3 - 84) * q^53 + (17*b6 - 3*b4 - 88*b2 + 74*b1 - 88) * q^55 + (-b7 + 13*b6 - 46*b5 - 13*b3 - 518*b2 - 46*b1 + 46) * q^56 + (b7 - 24*b6 + 79*b5 + 24*b3 + 141*b2 + 79*b1 - 79) * q^58 + (-14*b7 - 24*b6 - 10*b5 + 24*b3 - 72*b2 - 10*b1 + 10) * q^59 + (-3*b7 - 31*b6 + 26*b5 + 31*b3 - 245*b2 + 26*b1 - 26) * q^61 + (46*b6 - 7*b4 + 703*b2 - 16*b1 + 703) * q^62 + (-4*b7 - 175*b5 + 4*b4 - 9*b3 + 344) * q^64 + (2*b7 - 26*b6 + 14*b5 + 5*b4 - 41*b2 - 55*b1 - 307) * q^65 + (14*b6 + b4 + 348*b2 - 129*b1 + 348) * q^67 + (50*b6 + 21*b4 + 335*b2 + 223*b1 + 335) * q^68 + (b7 + 22*b5 - b4 - 15*b3 - 592) * q^70 + (-11*b7 - 49*b6 + 76*b5 + 49*b3 - 304*b2 + 76*b1 - 76) * q^71 + (2*b7 + 194*b5 - 2*b4 - 24*b3 + 141) * q^73 + (2*b7 - 13*b6 - 25*b5 + 13*b3 - 277*b2 - 25*b1 + 25) * q^74 + (-b6 + 3*b4 + 468*b2 - 82*b1 + 468) * q^76 + (16*b7 - 98*b5 - 16*b4 + 2*b3 - 578) * q^77 + (2*b7 + 21*b5 - 2*b4 - 7*b3 - 197) * q^79 + (-75*b6 - 12*b4 - 573*b2 - 315*b1 - 573) * q^80 + (61*b6 - 251*b5 - 61*b3 + 559*b2 - 251*b1 + 251) * q^82 + (5*b7 - 138*b5 - 5*b4 - b3 + 170) * q^83 + (-25*b7 - 50*b6 - 147*b5 + 50*b3 - 275*b2 - 147*b1 + 147) * q^85 + (-14*b7 + 46*b5 + 14*b4 - 37*b3 - 815) * q^86 + (-37*b6 - 21*b4 - 712*b2 - 106*b1 - 712) * q^88 + (-50*b6 - 28*b4 + 478*b2 - 146*b1 + 478) * q^89 + (b7 + 25*b6 - 202*b5 + 21*b4 - 81*b3 + 156*b2 - 149*b1 + 626) * q^91 + (-9*b7 + 102*b5 + 9*b4 - 35*b3 - 134) * q^92 + (37*b6 + 47*b4 + 10*b2 + 366*b1 + 10) * q^94 + (-7*b7 + 29*b6 + 58*b5 - 29*b3 - 580*b2 + 58*b1 - 58) * q^95 + (-22*b7 - 63*b6 + 71*b5 + 63*b3 + 506*b2 + 71*b1 - 71) * q^97 + (29*b7 + 84*b6 - 49*b5 - 84*b3 + 501*b2 - 49*b1 + 49) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2 q^{2} - 22 q^{4} + 12 q^{5} + 14 q^{7} - 108 q^{8}+O(q^{10})$$ 8 * q + 2 * q^2 - 22 * q^4 + 12 * q^5 + 14 * q^7 - 108 * q^8 $$8 q + 2 q^{2} - 22 q^{4} + 12 q^{5} + 14 q^{7} - 108 q^{8} + 62 q^{10} + 40 q^{11} - 60 q^{13} - 80 q^{14} - 122 q^{16} + 98 q^{17} - 124 q^{19} - 466 q^{20} - 220 q^{22} + 104 q^{23} - 116 q^{25} - 14 q^{26} + 144 q^{28} + 194 q^{29} + 52 q^{31} + 654 q^{32} + 2124 q^{34} + 88 q^{35} - 102 q^{37} - 664 q^{38} - 1996 q^{40} - 1054 q^{41} - 450 q^{43} + 88 q^{44} + 172 q^{46} + 192 q^{47} - 1070 q^{49} + 996 q^{50} + 2280 q^{52} - 524 q^{53} - 204 q^{55} + 2164 q^{56} - 722 q^{58} + 308 q^{59} + 928 q^{61} + 2780 q^{62} + 2052 q^{64} - 2346 q^{65} + 1134 q^{67} + 1786 q^{68} - 4648 q^{70} + 1064 q^{71} + 1904 q^{73} + 1158 q^{74} + 1708 q^{76} - 5016 q^{77} - 1492 q^{79} - 2922 q^{80} - 1734 q^{82} + 808 q^{83} + 1394 q^{85} - 6336 q^{86} - 3060 q^{88} + 1620 q^{89} + 3278 q^{91} - 664 q^{92} + 772 q^{94} + 2204 q^{95} - 2166 q^{97} - 1906 q^{98}+O(q^{100})$$ 8 * q + 2 * q^2 - 22 * q^4 + 12 * q^5 + 14 * q^7 - 108 * q^8 + 62 * q^10 + 40 * q^11 - 60 * q^13 - 80 * q^14 - 122 * q^16 + 98 * q^17 - 124 * q^19 - 466 * q^20 - 220 * q^22 + 104 * q^23 - 116 * q^25 - 14 * q^26 + 144 * q^28 + 194 * q^29 + 52 * q^31 + 654 * q^32 + 2124 * q^34 + 88 * q^35 - 102 * q^37 - 664 * q^38 - 1996 * q^40 - 1054 * q^41 - 450 * q^43 + 88 * q^44 + 172 * q^46 + 192 * q^47 - 1070 * q^49 + 996 * q^50 + 2280 * q^52 - 524 * q^53 - 204 * q^55 + 2164 * q^56 - 722 * q^58 + 308 * q^59 + 928 * q^61 + 2780 * q^62 + 2052 * q^64 - 2346 * q^65 + 1134 * q^67 + 1786 * q^68 - 4648 * q^70 + 1064 * q^71 + 1904 * q^73 + 1158 * q^74 + 1708 * q^76 - 5016 * q^77 - 1492 * q^79 - 2922 * q^80 - 1734 * q^82 + 808 * q^83 + 1394 * q^85 - 6336 * q^86 - 3060 * q^88 + 1620 * q^89 + 3278 * q^91 - 664 * q^92 + 772 * q^94 + 2204 * q^95 - 2166 * q^97 - 1906 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + 29x^{6} + 2x^{5} + 595x^{4} - 288x^{3} + 2526x^{2} + 1872x + 6084$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 5984 \nu^{7} + 26943 \nu^{6} - 182186 \nu^{5} + 346833 \nu^{4} - 3020182 \nu^{3} + 11402021 \nu^{2} - 10968108 \nu - 5315076 ) / 37839126$$ (-5984*v^7 + 26943*v^6 - 182186*v^5 + 346833*v^4 - 3020182*v^3 + 11402021*v^2 - 10968108*v - 5315076) / 37839126 $$\beta_{3}$$ $$=$$ $$( - 6325 \nu^{7} + 66824 \nu^{6} - 151547 \nu^{5} - 228206 \nu^{4} + 1218353 \nu^{3} - 1751772 \nu^{2} - 2486484 \nu - 364394082 ) / 37839126$$ (-6325*v^7 + 66824*v^6 - 151547*v^5 - 228206*v^4 + 1218353*v^3 - 1751772*v^2 - 2486484*v - 364394082) / 37839126 $$\beta_{4}$$ $$=$$ $$( - 4762 \nu^{7} + 120183 \nu^{6} - 812666 \nu^{5} + 3650909 \nu^{4} - 13471942 \nu^{3} + 50860301 \nu^{2} - 245245534 \nu + 145078050 ) / 12613042$$ (-4762*v^7 + 120183*v^6 - 812666*v^5 + 3650909*v^4 - 13471942*v^3 + 50860301*v^2 - 245245534*v + 145078050) / 12613042 $$\beta_{5}$$ $$=$$ $$( 14975 \nu^{7} - 8650 \nu^{6} + 358801 \nu^{5} + 540298 \nu^{4} + 9678629 \nu^{3} + 4147476 \nu^{2} + 5886972 \nu + 74245782 ) / 37839126$$ (14975*v^7 - 8650*v^6 + 358801*v^5 + 540298*v^4 + 9678629*v^3 + 4147476*v^2 + 5886972*v + 74245782) / 37839126 $$\beta_{6}$$ $$=$$ $$( 56492 \nu^{7} - 274785 \nu^{6} + 1858070 \nu^{5} - 5277333 \nu^{4} + 30802090 \nu^{3} - 116286395 \nu^{2} + 96372822 \nu - 331704750 ) / 37839126$$ (56492*v^7 - 274785*v^6 + 1858070*v^5 - 5277333*v^4 + 30802090*v^3 - 116286395*v^2 + 96372822*v - 331704750) / 37839126 $$\beta_{7}$$ $$=$$ $$( - 341411 \nu^{7} + 675847 \nu^{6} - 10275913 \nu^{5} - 849943 \nu^{4} - 203391203 \nu^{3} + 61980363 \nu^{2} - 864335982 \nu - 641863404 ) / 37839126$$ (-341411*v^7 + 675847*v^6 - 10275913*v^5 - 849943*v^4 - 203391203*v^3 + 61980363*v^2 - 864335982*v - 641863404) / 37839126
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} - \beta_{3} + 13\beta_{2} + \beta _1 - 1$$ b6 + b5 - b3 + 13*b2 + b1 - 1 $$\nu^{3}$$ $$=$$ $$\beta_{7} + 21\beta_{5} - \beta_{4} - 2\beta_{3} - 32$$ b7 + 21*b5 - b4 - 2*b3 - 32 $$\nu^{4}$$ $$=$$ $$-29\beta_{6} - 2\beta_{4} - 269\beta_{2} - 43\beta _1 - 269$$ -29*b6 - 2*b4 - 269*b2 - 43*b1 - 269 $$\nu^{5}$$ $$=$$ $$-29\beta_{7} - 84\beta_{6} - 509\beta_{5} + 84\beta_{3} - 501\beta_{2} - 509\beta _1 + 509$$ -29*b7 - 84*b6 - 509*b5 + 84*b3 - 501*b2 - 509*b1 + 509 $$\nu^{6}$$ $$=$$ $$-84\beta_{7} - 1511\beta_{5} + 84\beta_{4} + 767\beta_{3} + 7960$$ -84*b7 - 1511*b5 + 84*b4 + 767*b3 + 7960 $$\nu^{7}$$ $$=$$ $$2782\beta_{6} + 767\beta_{4} + 18109\beta_{2} + 13077\beta _1 + 18109$$ 2782*b6 + 767*b4 + 18109*b2 + 13077*b1 + 18109

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$$.

 $$n$$ $$28$$ $$92$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 −2.11303 − 3.65987i −0.733051 − 1.26968i 1.18088 + 2.04535i 2.66520 + 4.61626i −2.11303 + 3.65987i −0.733051 + 1.26968i 1.18088 − 2.04535i 2.66520 − 4.61626i
−2.11303 3.65987i 0 −4.92977 + 8.53861i 5.85953 0 12.0627 20.8932i 7.85849 0 −12.3814 21.4451i
55.2 −0.733051 1.26968i 0 2.92527 5.06672i −9.85055 0 −14.9698 + 25.9285i −20.3063 0 7.22095 + 12.5071i
55.3 1.18088 + 2.04535i 0 1.21104 2.09758i −6.42208 0 14.7469 25.5424i 24.6145 0 −7.58371 13.1354i
55.4 2.66520 + 4.61626i 0 −10.2065 + 17.6783i 16.4131 0 −4.83984 + 8.38285i −66.1667 0 43.7441 + 75.7670i
100.1 −2.11303 + 3.65987i 0 −4.92977 8.53861i 5.85953 0 12.0627 + 20.8932i 7.85849 0 −12.3814 + 21.4451i
100.2 −0.733051 + 1.26968i 0 2.92527 + 5.06672i −9.85055 0 −14.9698 25.9285i −20.3063 0 7.22095 12.5071i
100.3 1.18088 2.04535i 0 1.21104 + 2.09758i −6.42208 0 14.7469 + 25.5424i 24.6145 0 −7.58371 + 13.1354i
100.4 2.66520 4.61626i 0 −10.2065 17.6783i 16.4131 0 −4.83984 8.38285i −66.1667 0 43.7441 75.7670i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 100.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.4.g.e 8
3.b odd 2 1 39.4.e.c 8
12.b even 2 1 624.4.q.i 8
13.c even 3 1 inner 117.4.g.e 8
13.c even 3 1 1521.4.a.v 4
13.e even 6 1 1521.4.a.bb 4
39.h odd 6 1 507.4.a.i 4
39.i odd 6 1 39.4.e.c 8
39.i odd 6 1 507.4.a.m 4
39.k even 12 2 507.4.b.h 8
156.p even 6 1 624.4.q.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.c 8 3.b odd 2 1
39.4.e.c 8 39.i odd 6 1
117.4.g.e 8 1.a even 1 1 trivial
117.4.g.e 8 13.c even 3 1 inner
507.4.a.i 4 39.h odd 6 1
507.4.a.m 4 39.i odd 6 1
507.4.b.h 8 39.k even 12 2
624.4.q.i 8 12.b even 2 1
624.4.q.i 8 156.p even 6 1
1521.4.a.v 4 13.c even 3 1
1521.4.a.bb 4 13.e even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 2T_{2}^{7} + 29T_{2}^{6} + 2T_{2}^{5} + 595T_{2}^{4} - 288T_{2}^{3} + 2526T_{2}^{2} + 1872T_{2} + 6084$$ acting on $$S_{4}^{\mathrm{new}}(117, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 2 T^{7} + 29 T^{6} + \cdots + 6084$$
$3$ $$T^{8}$$
$5$ $$(T^{4} - 6 T^{3} - 203 T^{2} + 156 T + 6084)^{2}$$
$7$ $$T^{8} - 14 T^{7} + \cdots + 42523388944$$
$11$ $$T^{8} - 40 T^{7} + \cdots + 751198464$$
$13$ $$T^{8} + 60 T^{7} + \cdots + 23298085122481$$
$17$ $$T^{8} + \cdots + 509493017090304$$
$19$ $$T^{8} + 124 T^{7} + \cdots + 2959721107456$$
$23$ $$T^{8} - 104 T^{7} + \cdots + 6612632822016$$
$29$ $$T^{8} - 194 T^{7} + \cdots + 75\!\cdots\!24$$
$31$ $$(T^{4} - 26 T^{3} - 80975 T^{2} + \cdots + 328187792)^{2}$$
$37$ $$T^{8} + \cdots + 738573457717264$$
$41$ $$T^{8} + 1054 T^{7} + \cdots + 10\!\cdots\!04$$
$43$ $$T^{8} + 450 T^{7} + \cdots + 55\!\cdots\!84$$
$47$ $$(T^{4} - 96 T^{3} - 434600 T^{2} + \cdots + 42871452048)^{2}$$
$53$ $$(T^{4} + 262 T^{3} - 111719 T^{2} + \cdots + 744728256)^{2}$$
$59$ $$T^{8} - 308 T^{7} + \cdots + 44\!\cdots\!56$$
$61$ $$T^{8} - 928 T^{7} + \cdots + 27\!\cdots\!21$$
$67$ $$T^{8} - 1134 T^{7} + \cdots + 10\!\cdots\!44$$
$71$ $$T^{8} - 1064 T^{7} + \cdots + 98\!\cdots\!16$$
$73$ $$(T^{4} - 952 T^{3} + \cdots - 120133390247)^{2}$$
$79$ $$(T^{4} + 746 T^{3} + 184337 T^{2} + \cdots + 680937616)^{2}$$
$83$ $$(T^{4} - 404 T^{3} + \cdots + 58964273856)^{2}$$
$89$ $$T^{8} - 1620 T^{7} + \cdots + 72\!\cdots\!96$$
$97$ $$T^{8} + 2166 T^{7} + \cdots + 19\!\cdots\!96$$