# Properties

 Label 15.7.d.c Level $15$ Weight $7$ Character orbit 15.d Analytic conductor $3.451$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$15 = 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 15.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.45081125430$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 60x^{6} + 7774x^{4} + 206220x^{2} + 5736025$$ x^8 + 60*x^6 + 7774*x^4 + 206220*x^2 + 5736025 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{7}\cdot 3^{6}\cdot 5^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{2} + (\beta_{5} + \beta_{3} + \beta_1) q^{3} + (\beta_{2} + 9) q^{4} + (\beta_{6} - 7 \beta_{5} + 2 \beta_{3}) q^{5} + (\beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} - 77) q^{6} + (\beta_{4} - 3 \beta_{3} - 6 \beta_1) q^{7} + ( - 10 \beta_{5} + 8 \beta_{3}) q^{8} + ( - 3 \beta_{7} + 3 \beta_{6} - 3 \beta_{2} - 246) q^{9}+O(q^{10})$$ q - b5 * q^2 + (b5 + b3 + b1) * q^3 + (b2 + 9) * q^4 + (b6 - 7*b5 + 2*b3) * q^5 + (b7 + b6 + b5 - b3 - b2 - 77) * q^6 + (b4 - 3*b3 - 6*b1) * q^7 + (-10*b5 + 8*b3) * q^8 + (-3*b7 + 3*b6 - 3*b2 - 246) * q^9 $$q - \beta_{5} q^{2} + (\beta_{5} + \beta_{3} + \beta_1) q^{3} + (\beta_{2} + 9) q^{4} + (\beta_{6} - 7 \beta_{5} + 2 \beta_{3}) q^{5} + (\beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} - 77) q^{6} + (\beta_{4} - 3 \beta_{3} - 6 \beta_1) q^{7} + ( - 10 \beta_{5} + 8 \beta_{3}) q^{8} + ( - 3 \beta_{7} + 3 \beta_{6} - 3 \beta_{2} - 246) q^{9} + ( - \beta_{4} + 9 \beta_{3} + 10 \beta_{2} + 18 \beta_1 + 530) q^{10} + (6 \beta_{7} - 8 \beta_{6} - \beta_{5} + \beta_{3} - 3 \beta_{2} - 3) q^{11} + (45 \beta_{5} - 3 \beta_{4} - 21 \beta_{3} + 30 \beta_1) q^{12} + ( - \beta_{4} - 30 \beta_{3} - 60 \beta_1) q^{13} + ( - 18 \beta_{7} - 34 \beta_{6} - 26 \beta_{5} + 26 \beta_{3} + 9 \beta_{2} + \cdots + 9) q^{14}+ \cdots + ( - 1134 \beta_{7} - 2556 \beta_{6} - 1845 \beta_{5} + \cdots + 790857) q^{99}+O(q^{100})$$ q - b5 * q^2 + (b5 + b3 + b1) * q^3 + (b2 + 9) * q^4 + (b6 - 7*b5 + 2*b3) * q^5 + (b7 + b6 + b5 - b3 - b2 - 77) * q^6 + (b4 - 3*b3 - 6*b1) * q^7 + (-10*b5 + 8*b3) * q^8 + (-3*b7 + 3*b6 - 3*b2 - 246) * q^9 + (-b4 + 9*b3 + 10*b2 + 18*b1 + 530) * q^10 + (6*b7 - 8*b6 - b5 + b3 - 3*b2 - 3) * q^11 + (45*b5 - 3*b4 - 21*b3 + 30*b1) * q^12 + (-b4 - 30*b3 - 60*b1) * q^13 + (-18*b7 - 34*b6 - 26*b5 + 26*b3 + 9*b2 + 9) * q^14 + (5*b7 + 20*b6 + 140*b5 + 3*b4 - 62*b3 - 20*b2 + 21*b1 + 245) * q^15 + (-46*b2 + 98) * q^16 + (-187*b5 + 69*b3) * q^17 + (540*b5 + 3*b4 - 123*b3 - 174*b1) * q^18 + (b2 - 3565) * q^19 + (30*b7 - 18*b6 - 694*b5 - 86*b3 - 15*b2 - 15) * q^20 + (-3*b7 + 81*b6 + 39*b5 - 39*b3 + 162*b2 + 3030) * q^21 + (-4*b4 + 156*b3 + 312*b1) * q^22 + (-160*b5 + 609*b3) * q^23 + (2*b7 + 50*b6 + 26*b5 - 26*b3 - 50*b2 + 1862) * q^24 + (5*b4 + 330*b3 + 75*b2 + 660*b1 + 100) * q^25 + (-48*b7 - 32*b6 - 40*b5 + 40*b3 + 24*b2 + 24) * q^26 + (-1548*b5 + 27*b4 - 225*b3 + 45*b1) * q^27 + (6*b4 - 798*b3 - 1596*b1) * q^28 + (102*b7 + 28*b6 + 65*b5 - 65*b3 - 51*b2 - 51) * q^29 + (-15*b7 - 63*b6 + 851*b5 - 30*b4 + 269*b3 - 180*b2 + 740*b1 - 8880) * q^30 + (-185*b2 + 737) * q^31 + (3532*b5 - 880*b3) * q^32 + (2022*b5 - 33*b4 + 324*b3 - 816*b1) * q^33 + (256*b2 + 13168) * q^34 + (-240*b7 - 20*b6 - 3925*b5 - 890*b3 + 120*b2 + 120) * q^35 + (-18*b7 - 450*b6 - 234*b5 + 234*b3 - 279*b2 - 23319) * q^36 + (-27*b4 + 354*b3 + 708*b1) * q^37 + (3500*b5 + 8*b3) * q^38 + (102*b7 - 180*b6 - 39*b5 + 39*b3 - 63*b2 + 29145) * q^39 + (22*b4 + 402*b3 - 20*b2 + 804*b1 + 17740) * q^40 + (174*b7 + 218*b6 + 196*b5 - 196*b3 - 87*b2 - 87) * q^41 + (-13461*b5 - 75*b4 + 1899*b3 + 1230*b1) * q^42 + (16*b4 - 633*b3 - 1266*b1) * q^43 + (-24*b7 + 936*b6 + 456*b5 - 456*b3 + 12*b2 + 12) * q^44 + (-135*b7 - 126*b6 + 4572*b5 + 105*b4 - 147*b3 + 855*b2 - 240*b1 - 19845) * q^45 + (769*b2 + 7417) * q^46 + (13026*b5 + 617*b3) * q^47 + (-1558*b5 + 138*b4 + 1478*b3 - 868*b1) * q^48 + (-849*b2 - 130184) * q^49 + (600*b7 + 520*b6 - 4415*b5 + 40*b3 - 300*b2 - 300) * q^50 + (118*b7 + 532*b6 + 325*b5 - 325*b3 - 532*b2 + 8302) * q^51 + (192*b4 - 192*b3 - 384*b1) * q^52 + (-12763*b5 - 1539*b3) * q^53 + (-279*b7 - 711*b6 - 495*b5 + 495*b3 + 1440*b2 + 114876) * q^54 + (-215*b4 - 90*b3 - 1650*b2 - 180*b1 + 55350) * q^55 + (-516*b7 + 412*b6 - 52*b5 + 52*b3 + 258*b2 + 258) * q^56 + (-3529*b5 - 3*b4 - 3595*b3 - 3544*b1) * q^57 + (-232*b4 + 4128*b3 + 8256*b1) * q^58 + (-786*b7 - 3064*b6 - 1925*b5 + 1925*b3 + 393*b2 + 393) * q^59 + (780*b7 + 300*b6 + 13455*b5 - 99*b4 - 9*b3 - 300*b2 - 3618*b1 - 80220) * q^60 + (1795*b2 + 186677) * q^61 + (11288*b5 - 1480*b3) * q^62 + (17613*b5 - 234*b4 - 5760*b3 + 8424*b1) * q^63 + (-1468*b2 - 257948) * q^64 + (-90*b7 - 1300*b6 - 5315*b5 + 5675*b3 + 45*b2 + 45) * q^65 + (-420*b7 + 108*b6 - 156*b5 + 156*b3 - 1080*b2 - 152520) * q^66 + (-280*b4 - 5439*b3 - 10878*b1) * q^67 + (-17840*b5 - 2368*b3) * q^68 + (-449*b7 + 3205*b6 + 1378*b5 - 1378*b3 - 3205*b2 + 188041) * q^69 + (500*b4 - 9300*b3 + 2775*b2 - 18600*b1 + 284175) * q^70 + (1332*b7 - 2136*b6 - 402*b5 + 402*b3 - 666*b2 - 666) * q^71 + (7488*b5 + 294*b4 + 834*b3 + 1668*b1) * q^72 + (738*b4 + 7224*b3 + 14448*b1) * q^73 + (1032*b7 + 1464*b6 + 1248*b5 - 1248*b3 - 516*b2 - 516) * q^74 + (-1050*b7 + 1440*b6 + 2995*b5 - 225*b4 - 2345*b3 - 225*b2 + 1675*b1 - 321225) * q^75 + (-3556*b2 - 27396) * q^76 + (-27636*b5 + 21984*b3) * q^77 + (-28416*b5 - 24*b4 + 2160*b3 + 4512*b1) * q^78 + (8731*b2 - 160795) * q^79 + (-1380*b7 + 1340*b6 + 28340*b5 + 4980*b3 + 690*b2 + 690) * q^80 + (1404*b7 + 2808*b6 + 2106*b5 - 2106*b3 + 3753*b2 - 219726) * q^81 + (-566*b4 + 8574*b3 + 17148*b1) * q^82 + (-31216*b5 - 5487*b3) * q^83 + (2322*b7 - 1854*b6 + 234*b5 - 234*b3 + 3312*b2 + 778680) * q^84 + (89*b4 + 4374*b3 + 835*b2 + 8748*b1 + 206405) * q^85 + (-1458*b7 - 1714*b6 - 1586*b5 + 1586*b3 + 729*b2 + 729) * q^86 + (55284*b5 - 69*b4 - 2610*b3 - 10428*b1) * q^87 + (-632*b4 - 2472*b3 - 4944*b1) * q^88 + (1236*b7 + 4824*b6 + 3030*b5 - 3030*b3 - 618*b2 - 618) * q^89 + (-1500*b7 - 3180*b6 - 33615*b5 + 396*b4 + 2376*b3 - 4110*b2 - 12528*b1 - 341430) * q^90 + (-9744*b2 + 47952) * q^91 + (-47162*b5 - 32824*b3) * q^92 + (-5923*b5 + 555*b4 + 6287*b3 - 3148*b1) * q^93 + (-12409*b2 - 955217) * q^94 + (30*b7 - 3592*b6 + 24324*b5 - 7234*b3 - 15*b2 - 15) * q^95 + (-2652*b7 - 7932*b6 - 5292*b5 + 5292*b3 + 7932*b2 - 17556) * q^96 + (1732*b4 - 17052*b3 - 34104*b1) * q^97 + (185369*b5 - 6792*b3) * q^98 + (-1134*b7 - 2556*b6 - 1845*b5 + 1845*b3 - 5463*b2 + 790857) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 72 q^{4} - 612 q^{6} - 1980 q^{9}+O(q^{10})$$ 8 * q + 72 * q^4 - 612 * q^6 - 1980 * q^9 $$8 q + 72 q^{4} - 612 q^{6} - 1980 q^{9} + 4240 q^{10} + 1980 q^{15} + 784 q^{16} - 28520 q^{19} + 24228 q^{21} + 14904 q^{24} + 800 q^{25} - 71100 q^{30} + 5896 q^{31} + 105344 q^{34} - 186624 q^{36} + 233568 q^{39} + 141920 q^{40} - 159300 q^{45} + 59336 q^{46} - 1041472 q^{49} + 66888 q^{51} + 917892 q^{54} + 442800 q^{55} - 638640 q^{60} + 1493416 q^{61} - 2063584 q^{64} - 1221840 q^{66} + 1502532 q^{69} + 2273400 q^{70} - 2574000 q^{75} - 219168 q^{76} - 1286360 q^{79} - 1752192 q^{81} + 6238728 q^{84} + 1651240 q^{85} - 2737440 q^{90} + 383616 q^{91} - 7641736 q^{94} - 151056 q^{96} + 6322320 q^{99}+O(q^{100})$$ 8 * q + 72 * q^4 - 612 * q^6 - 1980 * q^9 + 4240 * q^10 + 1980 * q^15 + 784 * q^16 - 28520 * q^19 + 24228 * q^21 + 14904 * q^24 + 800 * q^25 - 71100 * q^30 + 5896 * q^31 + 105344 * q^34 - 186624 * q^36 + 233568 * q^39 + 141920 * q^40 - 159300 * q^45 + 59336 * q^46 - 1041472 * q^49 + 66888 * q^51 + 917892 * q^54 + 442800 * q^55 - 638640 * q^60 + 1493416 * q^61 - 2063584 * q^64 - 1221840 * q^66 + 1502532 * q^69 + 2273400 * q^70 - 2574000 * q^75 - 219168 * q^76 - 1286360 * q^79 - 1752192 * q^81 + 6238728 * q^84 + 1651240 * q^85 - 2737440 * q^90 + 383616 * q^91 - 7641736 * q^94 - 151056 * q^96 + 6322320 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 60x^{6} + 7774x^{4} + 206220x^{2} + 5736025$$ :

 $$\beta_{1}$$ $$=$$ $$( - 13 \nu^{7} + 1437 \nu^{6} + 1615 \nu^{5} + 64665 \nu^{4} + 198313 \nu^{3} + 11638263 \nu^{2} + 1771445 \nu + 164874195 ) / 10346400$$ (-13*v^7 + 1437*v^6 + 1615*v^5 + 64665*v^4 + 198313*v^3 + 11638263*v^2 + 1771445*v + 164874195) / 10346400 $$\beta_{2}$$ $$=$$ $$( \nu^{4} + 30\nu^{2} + 3437 ) / 36$$ (v^4 + 30*v^2 + 3437) / 36 $$\beta_{3}$$ $$=$$ $$( 13\nu^{7} - 1615\nu^{5} - 198313\nu^{3} - 1771445\nu ) / 5173200$$ (13*v^7 - 1615*v^5 - 198313*v^3 - 1771445*v) / 5173200 $$\beta_{4}$$ $$=$$ $$( -\nu^{6} - 45\nu^{4} - 4724\nu^{2} - 64110 ) / 225$$ (-v^6 - 45*v^4 - 4724*v^2 - 64110) / 225 $$\beta_{5}$$ $$=$$ $$( 73\nu^{7} + 1985\nu^{5} + 411827\nu^{3} - 606845\nu ) / 5173200$$ (73*v^7 + 1985*v^5 + 411827*v^3 - 606845*v) / 5173200 $$\beta_{6}$$ $$=$$ $$( \nu^{7} + 60\nu^{5} + 10169\nu^{3} + 666060\nu ) / 43110$$ (v^7 + 60*v^5 + 10169*v^3 + 666060*v) / 43110 $$\beta_{7}$$ $$=$$ $$( 61\nu^{7} + 6055\nu^{5} + 4790\nu^{4} + 433499\nu^{3} + 143700\nu^{2} + 20279345\nu + 16635670 ) / 344880$$ (61*v^7 + 6055*v^5 + 4790*v^4 + 433499*v^3 + 143700*v^2 + 20279345*v + 16635670) / 344880
 $$\nu$$ $$=$$ $$( \beta_{6} - 2\beta_{5} + 2\beta_{3} ) / 15$$ (b6 - 2*b5 + 2*b3) / 15 $$\nu^{2}$$ $$=$$ $$( \beta_{4} + 16\beta_{3} + 32\beta _1 - 225 ) / 15$$ (b4 + 16*b3 + 32*b1 - 225) / 15 $$\nu^{3}$$ $$=$$ $$( -10\beta_{7} + 33\beta_{6} + 119\beta_{5} - 269\beta_{3} + 5\beta_{2} + 5 ) / 15$$ (-10*b7 + 33*b6 + 119*b5 - 269*b3 + 5*b2 + 5) / 15 $$\nu^{4}$$ $$=$$ $$-2\beta_{4} - 32\beta_{3} + 36\beta_{2} - 64\beta _1 - 2987$$ -2*b4 - 32*b3 + 36*b2 - 64*b1 - 2987 $$\nu^{5}$$ $$=$$ $$( 1380\beta_{7} - 5399\beta_{6} - 7712\beta_{5} - 3988\beta_{3} - 690\beta_{2} - 690 ) / 15$$ (1380*b7 - 5399*b6 - 7712*b5 - 3988*b3 - 690*b2 - 690) / 15 $$\nu^{6}$$ $$=$$ $$( -6749\beta_{4} - 53984\beta_{3} - 24300\beta_{2} - 107968\beta _1 + 2117475 ) / 15$$ (-6749*b4 - 53984*b3 - 24300*b2 - 107968*b1 + 2117475) / 15 $$\nu^{7}$$ $$=$$ $$( 18890\beta_{7} - 31047\beta_{6} + 584729\beta_{5} + 1642621\beta_{3} - 9445\beta_{2} - 9445 ) / 15$$ (18890*b7 - 31047*b6 + 584729*b5 + 1642621*b3 - 9445*b2 - 9445) / 15

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/15\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$11$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
14.1
 −2.84382 + 4.80492i −2.84382 − 4.80492i 5.55992 − 6.77589i 5.55992 + 6.77589i −5.55992 + 6.77589i −5.55992 − 6.77589i 2.84382 − 4.80492i 2.84382 + 4.80492i
−11.8944 9.31012 25.3441i 77.4763 −102.129 + 72.0738i −110.738 + 301.452i 553.145i −160.292 −555.643 471.913i 1214.76 857.274i
14.2 −11.8944 9.31012 + 25.3441i 77.4763 −102.129 72.0738i −110.738 301.452i 553.145i −160.292 −555.643 + 471.913i 1214.76 + 857.274i
14.3 −2.12691 19.8701 18.2805i −59.4763 72.7643 101.638i −42.2619 + 38.8810i 435.542i 262.622 60.6432 726.473i −154.763 + 216.175i
14.4 −2.12691 19.8701 + 18.2805i −59.4763 72.7643 + 101.638i −42.2619 38.8810i 435.542i 262.622 60.6432 + 726.473i −154.763 216.175i
14.5 2.12691 −19.8701 18.2805i −59.4763 −72.7643 + 101.638i −42.2619 38.8810i 435.542i −262.622 60.6432 + 726.473i −154.763 + 216.175i
14.6 2.12691 −19.8701 + 18.2805i −59.4763 −72.7643 101.638i −42.2619 + 38.8810i 435.542i −262.622 60.6432 726.473i −154.763 216.175i
14.7 11.8944 −9.31012 25.3441i 77.4763 102.129 72.0738i −110.738 301.452i 553.145i 160.292 −555.643 + 471.913i 1214.76 857.274i
14.8 11.8944 −9.31012 + 25.3441i 77.4763 102.129 + 72.0738i −110.738 + 301.452i 553.145i 160.292 −555.643 471.913i 1214.76 + 857.274i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 14.8 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
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.7.d.c 8
3.b odd 2 1 inner 15.7.d.c 8
4.b odd 2 1 240.7.c.c 8
5.b even 2 1 inner 15.7.d.c 8
5.c odd 4 2 75.7.c.e 8
12.b even 2 1 240.7.c.c 8
15.d odd 2 1 inner 15.7.d.c 8
15.e even 4 2 75.7.c.e 8
20.d odd 2 1 240.7.c.c 8
60.h even 2 1 240.7.c.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.7.d.c 8 1.a even 1 1 trivial
15.7.d.c 8 3.b odd 2 1 inner
15.7.d.c 8 5.b even 2 1 inner
15.7.d.c 8 15.d odd 2 1 inner
75.7.c.e 8 5.c odd 4 2
75.7.c.e 8 15.e even 4 2
240.7.c.c 8 4.b odd 2 1
240.7.c.c 8 12.b even 2 1
240.7.c.c 8 20.d odd 2 1
240.7.c.c 8 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 146T_{2}^{2} + 640$$ acting on $$S_{7}^{\mathrm{new}}(15, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - 146 T^{2} + 640)^{2}$$
$3$ $$T^{8} + 990 T^{6} + \cdots + 282429536481$$
$5$ $$T^{8} - 400 T^{6} + \cdots + 59\!\cdots\!25$$
$7$ $$(T^{4} + 495666 T^{2} + \cdots + 58041360000)^{2}$$
$11$ $$(T^{4} + 3382560 T^{2} + \cdots + 1754154144000)^{2}$$
$13$ $$(T^{4} + 3949776 T^{2} + \cdots + 3379669401600)^{2}$$
$17$ $$(T^{4} - 10866836 T^{2} + \cdots + 28015260981760)^{2}$$
$19$ $$(T^{2} + 7130 T + 12704536)^{4}$$
$23$ $$(T^{4} - 477962246 T^{2} + \cdots + 11\!\cdots\!40)^{2}$$
$29$ $$(T^{4} + 777857040 T^{2} + \cdots + 85\!\cdots\!00)^{2}$$
$31$ $$(T^{2} - 1474 T - 159937856)^{4}$$
$37$ $$(T^{4} + 831035376 T^{2} + \cdots + 26\!\cdots\!00)^{2}$$
$41$ $$(T^{4} + 3099275460 T^{2} + \cdots + 16\!\cdots\!00)^{2}$$
$43$ $$(T^{4} + 1690395426 T^{2} + \cdots + 30\!\cdots\!00)^{2}$$
$47$ $$(T^{4} - 25487397926 T^{2} + \cdots + 56\!\cdots\!60)^{2}$$
$53$ $$(T^{4} - 27378449876 T^{2} + \cdots + 13\!\cdots\!60)^{2}$$
$59$ $$(T^{4} + 201291276960 T^{2} + \cdots + 64\!\cdots\!00)^{2}$$
$61$ $$(T^{2} - 373354 T + 19740227104)^{4}$$
$67$ $$(T^{4} + 150261817986 T^{2} + \cdots + 32\!\cdots\!00)^{2}$$
$71$ $$(T^{4} + 186412415040 T^{2} + \cdots + 37\!\cdots\!00)^{2}$$
$73$ $$(T^{4} + 449137827936 T^{2} + \cdots + 21\!\cdots\!00)^{2}$$
$79$ $$(T^{2} + 321590 T - 331589130704)^{4}$$
$83$ $$(T^{4} - 185781764486 T^{2} + \cdots + 80\!\cdots\!60)^{2}$$
$89$ $$(T^{4} + 498657072960 T^{2} + \cdots + 39\!\cdots\!00)^{2}$$
$97$ $$(T^{4} + 2533468716576 T^{2} + \cdots + 25\!\cdots\!00)^{2}$$