# Properties

 Label 289.4.b.b Level $289$ Weight $4$ Character orbit 289.b Analytic conductor $17.052$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$289 = 17^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 289.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.0515519917$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.27793984.1 Defining polynomial: $$x^{6} - 2x^{3} + 49x^{2} - 14x + 2$$ x^6 - 2*x^3 + 49*x^2 - 14*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 17) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + ( - \beta_{5} + \beta_{4} + \beta_{2}) q^{3} + (2 \beta_{3} - \beta_1 + 8) q^{4} + ( - \beta_{5} - \beta_{4}) q^{5} + (\beta_{5} + 12 \beta_{4} + 6 \beta_{2}) q^{6} + ( - 2 \beta_{5} - 3 \beta_{4} - \beta_{2}) q^{7} + ( - 2 \beta_{3} - 7 \beta_1 + 16) q^{8} + ( - 5 \beta_{3} + 3 \beta_1 - 19) q^{9}+O(q^{10})$$ q - b1 * q^2 + (-b5 + b4 + b2) * q^3 + (2*b3 - b1 + 8) * q^4 + (-b5 - b4) * q^5 + (b5 + 12*b4 + 6*b2) * q^6 + (-2*b5 - 3*b4 - b2) * q^7 + (-2*b3 - 7*b1 + 16) * q^8 + (-5*b3 + 3*b1 - 19) * q^9 $$q - \beta_1 q^{2} + ( - \beta_{5} + \beta_{4} + \beta_{2}) q^{3} + (2 \beta_{3} - \beta_1 + 8) q^{4} + ( - \beta_{5} - \beta_{4}) q^{5} + (\beta_{5} + 12 \beta_{4} + 6 \beta_{2}) q^{6} + ( - 2 \beta_{5} - 3 \beta_{4} - \beta_{2}) q^{7} + ( - 2 \beta_{3} - 7 \beta_1 + 16) q^{8} + ( - 5 \beta_{3} + 3 \beta_1 - 19) q^{9} + (4 \beta_{5} + 8 \beta_{4}) q^{10} + ( - \beta_{5} + 5 \beta_{4} + 11 \beta_{2}) q^{11} + ( - 13 \beta_{5} + 8 \beta_{4} + 26 \beta_{2}) q^{12} + ( - 7 \beta_{3} + \beta_1 + 12) q^{13} + (10 \beta_{5} + 12 \beta_{4} - 4 \beta_{2}) q^{14} + ( - 5 \beta_{3} - 7 \beta_1 - 32) q^{15} + (2 \beta_{3} - 9 \beta_1 + 48) q^{16} + (4 \beta_{3} + 37 \beta_1 - 48) q^{18} + (7 \beta_{3} - 15 \beta_1 - 24) q^{19} + ( - 12 \beta_{5} - 24 \beta_{4} + 8 \beta_{2}) q^{20} + ( - 9 \beta_{3} - 21 \beta_1 - 54) q^{21} + ( - 13 \beta_{5} + 52 \beta_{4} + 34 \beta_{2}) q^{22} + ( - 2 \beta_{5} - 23 \beta_{4} - 39 \beta_{2}) q^{23} + ( - 3 \beta_{5} + 112 \beta_{4} + 46 \beta_{2}) q^{24} + ( - 8 \beta_{3} - 12 \beta_1 + 81) q^{25} + (12 \beta_{3} + 10 \beta_1 - 16) q^{26} + (4 \beta_{5} + 2 \beta_{4} - 40 \beta_{2}) q^{27} + ( - 22 \beta_{5} - 72 \beta_{4} + 4 \beta_{2}) q^{28} + ( - 15 \beta_{5} - 71 \beta_{4} + 16 \beta_{2}) q^{29} + (24 \beta_{3} + 40 \beta_1 + 112) q^{30} + ( - 8 \beta_{5} + 41 \beta_{4} - 39 \beta_{2}) q^{31} + (30 \beta_{3} - 7 \beta_1 + 16) q^{32} + ( - 21 \beta_{3} + 55 \beta_1 - 122) q^{33} + ( - 17 \beta_{3} - 27 \beta_1 - 96) q^{35} + ( - 42 \beta_{3} + 49 \beta_1 - 440) q^{36} + (25 \beta_{5} + 51 \beta_{4} + 28 \beta_{2}) q^{37} + (16 \beta_{3} - 12 \beta_1 + 240) q^{38} + (8 \beta_{5} + 42 \beta_{4} - 36 \beta_{2}) q^{39} + (20 \beta_{5} + 64 \beta_{4} - 8 \beta_{2}) q^{40} + ( - 30 \beta_{5} + 59 \beta_{4} - 52 \beta_{2}) q^{41} + (60 \beta_{3} + 60 \beta_1 + 336) q^{42} + (29 \beta_{3} + 27 \beta_1 - 204) q^{43} + ( - 39 \beta_{5} + 200 \beta_{4} + 110 \beta_{2}) q^{44} + (27 \beta_{5} + 55 \beta_{4} - 20 \beta_{2}) q^{45} + (68 \beta_{5} - 140 \beta_{4} - 120 \beta_{2}) q^{46} + (2 \beta_{3} - 46 \beta_1 + 228) q^{47} + ( - 45 \beta_{5} + 144 \beta_{4} + 114 \beta_{2}) q^{48} + ( - 37 \beta_{3} - 57 \beta_1 + 121) q^{49} + (40 \beta_{3} - 69 \beta_1 + 192) q^{50} + (12 \beta_{3} - 18 \beta_1 - 256) q^{52} + (54 \beta_{3} - 62 \beta_1 - 98) q^{53} + (26 \beta_{5} - 192 \beta_{4} - 84 \beta_{2}) q^{54} + ( - 7 \beta_{3} + 11 \beta_1 + 24) q^{55} + (54 \beta_{5} + 96 \beta_{4} - 60 \beta_{2}) q^{56} + (18 \beta_{5} + 114 \beta_{4} + 108 \beta_{2}) q^{57} + (100 \beta_{5} + 184 \beta_{4} - 80 \beta_{2}) q^{58} + ( - 65 \beta_{3} + 65 \beta_1 - 212) q^{59} + ( - 88 \beta_{3} - 88 \beta_1 - 384) q^{60} + (39 \beta_{5} + \beta_{4} - 64 \beta_{2}) q^{61} + (22 \beta_{5} - 92 \beta_{4} + 20 \beta_{2}) q^{62} + (48 \beta_{5} + 171 \beta_{4} - 9 \beta_{2}) q^{63} + ( - 62 \beta_{3} - 41 \beta_1 - 272) q^{64} + (12 \beta_{5} + 64 \beta_{4} - 28 \beta_{2}) q^{65} + ( - 68 \beta_{3} + 240 \beta_1 - 880) q^{66} + (54 \beta_{3} + 78 \beta_1 + 292) q^{67} + (47 \beta_{3} - 233 \beta_1 + 254) q^{69} + (88 \beta_{3} + 120 \beta_1 + 432) q^{70} + ( - 36 \beta_{5} - 55 \beta_{4} - 185 \beta_{2}) q^{71} + ( - 46 \beta_{3} + 319 \beta_1 - 400) q^{72} + (8 \beta_{5} + 137 \beta_{4} - 16 \beta_{2}) q^{73} + ( - 154 \beta_{5} - 88 \beta_{4} + 108 \beta_{2}) q^{74} + ( - 45 \beta_{5} + 273 \beta_{4} + 105 \beta_{2}) q^{75} + ( - 64 \beta_{3} - 180 \beta_1 + 384) q^{76} + ( - 9 \beta_{3} - 9 \beta_1 + 174) q^{77} + ( - 30 \beta_{5} - 208 \beta_{4} - 4 \beta_{2}) q^{78} + (90 \beta_{5} + 69 \beta_{4} - 267 \beta_{2}) q^{79} + ( - 20 \beta_{5} + 8 \beta_{2}) q^{80} + ( - 37 \beta_{3} - 57 \beta_1 - 137) q^{81} + (83 \beta_{5} + 32 \beta_{4} + 74 \beta_{2}) q^{82} + (105 \beta_{3} + 23 \beta_1 + 756) q^{83} + ( - 168 \beta_{3} - 288 \beta_1 - 528) q^{84} + ( - 112 \beta_{3} + 144 \beta_1 - 432) q^{86} + ( - 163 \beta_{3} - 209 \beta_1 - 352) q^{87} + ( - 89 \beta_{5} + 336 \beta_{4} + 426 \beta_{2}) q^{88} + (83 \beta_{3} - 193 \beta_1 - 20) q^{89} + ( - 116 \beta_{5} - 296 \beta_{4} + 16 \beta_{2}) q^{90} + (22 \beta_{5} + 162 \beta_{4} - 64 \beta_{2}) q^{91} + (72 \beta_{5} - 840 \beta_{4} - 344 \beta_{2}) q^{92} + (87 \beta_{3} - 65 \beta_1 - 218) q^{93} + (88 \beta_{3} - 280 \beta_1 + 736) q^{94} + (56 \beta_{5} + 60 \beta_{4} + 28 \beta_{2}) q^{95} + ( - 99 \beta_{5} - 80 \beta_{4} + 238 \beta_{2}) q^{96} + ( - 60 \beta_{5} - 25 \beta_{4} - 140 \beta_{2}) q^{97} + (188 \beta_{3} - 67 \beta_1 + 912) q^{98} + (103 \beta_{5} - 521 \beta_{4} - 281 \beta_{2}) q^{99}+O(q^{100})$$ q - b1 * q^2 + (-b5 + b4 + b2) * q^3 + (2*b3 - b1 + 8) * q^4 + (-b5 - b4) * q^5 + (b5 + 12*b4 + 6*b2) * q^6 + (-2*b5 - 3*b4 - b2) * q^7 + (-2*b3 - 7*b1 + 16) * q^8 + (-5*b3 + 3*b1 - 19) * q^9 + (4*b5 + 8*b4) * q^10 + (-b5 + 5*b4 + 11*b2) * q^11 + (-13*b5 + 8*b4 + 26*b2) * q^12 + (-7*b3 + b1 + 12) * q^13 + (10*b5 + 12*b4 - 4*b2) * q^14 + (-5*b3 - 7*b1 - 32) * q^15 + (2*b3 - 9*b1 + 48) * q^16 + (4*b3 + 37*b1 - 48) * q^18 + (7*b3 - 15*b1 - 24) * q^19 + (-12*b5 - 24*b4 + 8*b2) * q^20 + (-9*b3 - 21*b1 - 54) * q^21 + (-13*b5 + 52*b4 + 34*b2) * q^22 + (-2*b5 - 23*b4 - 39*b2) * q^23 + (-3*b5 + 112*b4 + 46*b2) * q^24 + (-8*b3 - 12*b1 + 81) * q^25 + (12*b3 + 10*b1 - 16) * q^26 + (4*b5 + 2*b4 - 40*b2) * q^27 + (-22*b5 - 72*b4 + 4*b2) * q^28 + (-15*b5 - 71*b4 + 16*b2) * q^29 + (24*b3 + 40*b1 + 112) * q^30 + (-8*b5 + 41*b4 - 39*b2) * q^31 + (30*b3 - 7*b1 + 16) * q^32 + (-21*b3 + 55*b1 - 122) * q^33 + (-17*b3 - 27*b1 - 96) * q^35 + (-42*b3 + 49*b1 - 440) * q^36 + (25*b5 + 51*b4 + 28*b2) * q^37 + (16*b3 - 12*b1 + 240) * q^38 + (8*b5 + 42*b4 - 36*b2) * q^39 + (20*b5 + 64*b4 - 8*b2) * q^40 + (-30*b5 + 59*b4 - 52*b2) * q^41 + (60*b3 + 60*b1 + 336) * q^42 + (29*b3 + 27*b1 - 204) * q^43 + (-39*b5 + 200*b4 + 110*b2) * q^44 + (27*b5 + 55*b4 - 20*b2) * q^45 + (68*b5 - 140*b4 - 120*b2) * q^46 + (2*b3 - 46*b1 + 228) * q^47 + (-45*b5 + 144*b4 + 114*b2) * q^48 + (-37*b3 - 57*b1 + 121) * q^49 + (40*b3 - 69*b1 + 192) * q^50 + (12*b3 - 18*b1 - 256) * q^52 + (54*b3 - 62*b1 - 98) * q^53 + (26*b5 - 192*b4 - 84*b2) * q^54 + (-7*b3 + 11*b1 + 24) * q^55 + (54*b5 + 96*b4 - 60*b2) * q^56 + (18*b5 + 114*b4 + 108*b2) * q^57 + (100*b5 + 184*b4 - 80*b2) * q^58 + (-65*b3 + 65*b1 - 212) * q^59 + (-88*b3 - 88*b1 - 384) * q^60 + (39*b5 + b4 - 64*b2) * q^61 + (22*b5 - 92*b4 + 20*b2) * q^62 + (48*b5 + 171*b4 - 9*b2) * q^63 + (-62*b3 - 41*b1 - 272) * q^64 + (12*b5 + 64*b4 - 28*b2) * q^65 + (-68*b3 + 240*b1 - 880) * q^66 + (54*b3 + 78*b1 + 292) * q^67 + (47*b3 - 233*b1 + 254) * q^69 + (88*b3 + 120*b1 + 432) * q^70 + (-36*b5 - 55*b4 - 185*b2) * q^71 + (-46*b3 + 319*b1 - 400) * q^72 + (8*b5 + 137*b4 - 16*b2) * q^73 + (-154*b5 - 88*b4 + 108*b2) * q^74 + (-45*b5 + 273*b4 + 105*b2) * q^75 + (-64*b3 - 180*b1 + 384) * q^76 + (-9*b3 - 9*b1 + 174) * q^77 + (-30*b5 - 208*b4 - 4*b2) * q^78 + (90*b5 + 69*b4 - 267*b2) * q^79 + (-20*b5 + 8*b2) * q^80 + (-37*b3 - 57*b1 - 137) * q^81 + (83*b5 + 32*b4 + 74*b2) * q^82 + (105*b3 + 23*b1 + 756) * q^83 + (-168*b3 - 288*b1 - 528) * q^84 + (-112*b3 + 144*b1 - 432) * q^86 + (-163*b3 - 209*b1 - 352) * q^87 + (-89*b5 + 336*b4 + 426*b2) * q^88 + (83*b3 - 193*b1 - 20) * q^89 + (-116*b5 - 296*b4 + 16*b2) * q^90 + (22*b5 + 162*b4 - 64*b2) * q^91 + (72*b5 - 840*b4 - 344*b2) * q^92 + (87*b3 - 65*b1 - 218) * q^93 + (88*b3 - 280*b1 + 736) * q^94 + (56*b5 + 60*b4 + 28*b2) * q^95 + (-99*b5 - 80*b4 + 238*b2) * q^96 + (-60*b5 - 25*b4 - 140*b2) * q^97 + (188*b3 - 67*b1 + 912) * q^98 + (103*b5 - 521*b4 - 281*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{2} + 50 q^{4} + 78 q^{8} - 118 q^{9}+O(q^{10})$$ 6 * q - 2 * q^2 + 50 * q^4 + 78 * q^8 - 118 * q^9 $$6 q - 2 q^{2} + 50 q^{4} + 78 q^{8} - 118 q^{9} + 60 q^{13} - 216 q^{15} + 274 q^{16} - 206 q^{18} - 160 q^{19} - 384 q^{21} + 446 q^{25} - 52 q^{26} + 800 q^{30} + 142 q^{32} - 664 q^{33} - 664 q^{35} - 2626 q^{36} + 1448 q^{38} + 2256 q^{42} - 1112 q^{43} + 1280 q^{47} + 538 q^{49} + 1094 q^{50} - 1548 q^{52} - 604 q^{53} + 152 q^{55} - 1272 q^{59} - 2656 q^{60} - 1838 q^{64} - 4936 q^{66} + 2016 q^{67} + 1152 q^{69} + 3008 q^{70} - 1854 q^{72} + 1816 q^{76} + 1008 q^{77} - 1010 q^{81} + 4792 q^{83} - 4080 q^{84} - 2528 q^{86} - 2856 q^{87} - 340 q^{89} - 1264 q^{93} + 4032 q^{94} + 5714 q^{98}+O(q^{100})$$ 6 * q - 2 * q^2 + 50 * q^4 + 78 * q^8 - 118 * q^9 + 60 * q^13 - 216 * q^15 + 274 * q^16 - 206 * q^18 - 160 * q^19 - 384 * q^21 + 446 * q^25 - 52 * q^26 + 800 * q^30 + 142 * q^32 - 664 * q^33 - 664 * q^35 - 2626 * q^36 + 1448 * q^38 + 2256 * q^42 - 1112 * q^43 + 1280 * q^47 + 538 * q^49 + 1094 * q^50 - 1548 * q^52 - 604 * q^53 + 152 * q^55 - 1272 * q^59 - 2656 * q^60 - 1838 * q^64 - 4936 * q^66 + 2016 * q^67 + 1152 * q^69 + 3008 * q^70 - 1854 * q^72 + 1816 * q^76 + 1008 * q^77 - 1010 * q^81 + 4792 * q^83 - 4080 * q^84 - 2528 * q^86 - 2856 * q^87 - 340 * q^89 - 1264 * q^93 + 4032 * q^94 + 5714 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{3} + 49x^{2} - 14x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$( 4\nu^{5} + 25\nu^{4} + 28\nu^{3} - 4\nu^{2} + 799 ) / 171$$ (4*v^5 + 25*v^4 + 28*v^3 - 4*v^2 + 799) / 171 $$\beta_{2}$$ $$=$$ $$( -7\nu^{5} - \nu^{4} - 49\nu^{3} + 7\nu^{2} - 684\nu + 98 ) / 342$$ (-7*v^5 - v^4 - 49*v^3 + 7*v^2 - 684*v + 98) / 342 $$\beta_{3}$$ $$=$$ $$( -\nu^{5} + 8\nu^{4} - 7\nu^{3} + \nu^{2} + 299 ) / 57$$ (-v^5 + 8*v^4 - 7*v^3 + v^2 + 299) / 57 $$\beta_{4}$$ $$=$$ $$( -49\nu^{5} - 7\nu^{4} - \nu^{3} + 49\nu^{2} - 2394\nu + 344 ) / 171$$ (-49*v^5 - 7*v^4 - v^3 + 49*v^2 - 2394*v + 344) / 171 $$\beta_{5}$$ $$=$$ $$( -245\nu^{5} - 35\nu^{4} - 5\nu^{3} + 587\nu^{2} - 11970\nu + 1720 ) / 171$$ (-245*v^5 - 35*v^4 - 5*v^3 + 587*v^2 - 11970*v + 1720) / 171
 $$\nu$$ $$=$$ $$( \beta_{3} - 2\beta_{2} - \beta_1 ) / 4$$ (b3 - 2*b2 - b1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{5} - 5\beta_{4} ) / 2$$ (b5 - 5*b4) / 2 $$\nu^{3}$$ $$=$$ $$( 2\beta_{4} - 7\beta_{3} - 14\beta_{2} + 7\beta _1 + 4 ) / 4$$ (2*b4 - 7*b3 - 14*b2 + 7*b1 + 4) / 4 $$\nu^{4}$$ $$=$$ $$4\beta_{3} + 3\beta _1 - 35$$ 4*b3 + 3*b1 - 35 $$\nu^{5}$$ $$=$$ $$( 2\beta_{5} - 24\beta_{4} - 51\beta_{3} + 98\beta_{2} + 47\beta _1 + 48 ) / 4$$ (2*b5 - 24*b4 - 51*b3 + 98*b2 + 47*b1 + 48) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/289\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
288.1
 0.143705 − 0.143705i 0.143705 + 0.143705i −1.93854 − 1.93854i −1.93854 + 1.93854i 1.79483 + 1.79483i 1.79483 − 1.79483i
−4.67129 7.62999i 13.8209 11.9174i 35.6419i 26.1222i −27.1912 −31.2167 55.6696i
288.2 −4.67129 7.62999i 13.8209 11.9174i 35.6419i 26.1222i −27.1912 −31.2167 55.6696i
288.3 −1.36122 3.15463i −6.14708 3.03171i 4.29415i 7.94049i 19.2573 17.0483 4.12682i
288.4 −1.36122 3.15463i −6.14708 3.03171i 4.29415i 7.94049i 19.2573 17.0483 4.12682i
288.5 5.03251 8.47535i 17.3261 0.885690i 42.6523i 3.81828i 46.9339 −44.8316 4.45724i
288.6 5.03251 8.47535i 17.3261 0.885690i 42.6523i 3.81828i 46.9339 −44.8316 4.45724i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 288.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 289.4.b.b 6
17.b even 2 1 inner 289.4.b.b 6
17.c even 4 1 17.4.a.b 3
17.c even 4 1 289.4.a.b 3
51.f odd 4 1 153.4.a.g 3
68.f odd 4 1 272.4.a.h 3
85.f odd 4 1 425.4.b.f 6
85.i odd 4 1 425.4.b.f 6
85.j even 4 1 425.4.a.g 3
119.f odd 4 1 833.4.a.d 3
136.i even 4 1 1088.4.a.v 3
136.j odd 4 1 1088.4.a.x 3
187.f odd 4 1 2057.4.a.e 3
204.l even 4 1 2448.4.a.bi 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.a.b 3 17.c even 4 1
153.4.a.g 3 51.f odd 4 1
272.4.a.h 3 68.f odd 4 1
289.4.a.b 3 17.c even 4 1
289.4.b.b 6 1.a even 1 1 trivial
289.4.b.b 6 17.b even 2 1 inner
425.4.a.g 3 85.j even 4 1
425.4.b.f 6 85.f odd 4 1
425.4.b.f 6 85.i odd 4 1
833.4.a.d 3 119.f odd 4 1
1088.4.a.v 3 136.i even 4 1
1088.4.a.x 3 136.j odd 4 1
2057.4.a.e 3 187.f odd 4 1
2448.4.a.bi 3 204.l even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + T_{2}^{2} - 24T_{2} - 32$$ acting on $$S_{4}^{\mathrm{new}}(289, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{3} + T^{2} - 24 T - 32)^{2}$$
$3$ $$T^{6} + 140 T^{4} + 5476 T^{2} + \cdots + 41616$$
$5$ $$T^{6} + 152 T^{4} + 1424 T^{2} + \cdots + 1024$$
$7$ $$T^{6} + 760 T^{4} + 53892 T^{2} + \cdots + 627264$$
$11$ $$T^{6} + 3516 T^{4} + \cdots + 22014864$$
$13$ $$(T^{3} - 30 T^{2} - 1472 T - 9392)^{2}$$
$17$ $$T^{6}$$
$19$ $$(T^{3} + 80 T^{2} - 4632 T - 340128)^{2}$$
$23$ $$T^{6} + 51704 T^{4} + \cdots + 2561741095936$$
$29$ $$T^{6} + 100120 T^{4} + \cdots + 2306218853376$$
$31$ $$T^{6} + 76072 T^{4} + \cdots + 6659865664$$
$37$ $$T^{6} + 162664 T^{4} + \cdots + 38152265269504$$
$41$ $$T^{6} + 259564 T^{4} + \cdots + 2685481897536$$
$43$ $$(T^{3} + 556 T^{2} + 51096 T - 7270272)^{2}$$
$47$ $$(T^{3} - 640 T^{2} + 85328 T - 1671168)^{2}$$
$53$ $$(T^{3} + 302 T^{2} - 153460 T - 18162072)^{2}$$
$59$ $$(T^{3} + 636 T^{2} - 101768 T - 49419072)^{2}$$
$61$ $$T^{6} + 255880 T^{4} + \cdots + 46141914470656$$
$67$ $$(T^{3} - 1008 T^{2} + 65040 T - 765952)^{2}$$
$71$ $$T^{6} + 1341352 T^{4} + \cdots + 75\!\cdots\!56$$
$73$ $$T^{6} + \cdots + 398302285230144$$
$79$ $$T^{6} + 2595384 T^{4} + \cdots + 55\!\cdots\!76$$
$83$ $$(T^{3} - 2396 T^{2} + 1488888 T - 142080704)^{2}$$
$89$ $$(T^{3} + 170 T^{2} - 1072304 T - 446571376)^{2}$$
$97$ $$T^{6} + 1245100 T^{4} + \cdots + 42\!\cdots\!00$$