# Properties

 Label 72.7.b.d Level $72$ Weight $7$ Character orbit 72.b Analytic conductor $16.564$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 72.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.5638940206$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 78x^{10} + 3408x^{8} + 73216x^{6} + 13959168x^{4} + 1308622848x^{2} + 68719476736$$ x^12 + 78*x^10 + 3408*x^8 + 73216*x^6 + 13959168*x^4 + 1308622848*x^2 + 68719476736 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{30}\cdot 3^{6}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{3} - 13) q^{4} + ( - \beta_{8} - 2 \beta_1) q^{5} - \beta_{7} q^{7} + (\beta_{10} - 13 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b3 - 13) * q^4 + (-b8 - 2*b1) * q^5 - b7 * q^7 + (b10 - 13*b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{3} - 13) q^{4} + ( - \beta_{8} - 2 \beta_1) q^{5} - \beta_{7} q^{7} + (\beta_{10} - 13 \beta_1) q^{8} + ( - \beta_{9} + \beta_{7} - 2 \beta_{3} + 118) q^{10} + ( - \beta_{10} - \beta_{6} - \beta_{4} + 20 \beta_1) q^{11} + ( - \beta_{11} - \beta_{9} + \beta_{7} + \beta_{5} - 12 \beta_{3}) q^{13} + (5 \beta_{8} + 2 \beta_{6} + \beta_{2} + \beta_1) q^{14} + ( - 2 \beta_{11} - 4 \beta_{7} + \beta_{5} - 13 \beta_{3} - 122) q^{16} + ( - 6 \beta_{10} + 2 \beta_{8} + 2 \beta_{6} - \beta_{4} + 2 \beta_{2} + 65 \beta_1) q^{17} + (2 \beta_{11} - 6 \beta_{9} + 2 \beta_{7} + 22 \beta_{3} + 328) q^{19} + ( - 2 \beta_{10} + 26 \beta_{8} - 4 \beta_{6} - 6 \beta_{4} + 2 \beta_{2} + 118 \beta_1) q^{20} + (4 \beta_{11} - 2 \beta_{9} - 22 \beta_{7} - 6 \beta_{5} + 30 \beta_{3} + \cdots + 1324) q^{22}+ \cdots + (1376 \beta_{10} + 5040 \beta_{8} - 224 \beta_{6} - 3312 \beta_{4} + \cdots - 26315 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b3 - 13) * q^4 + (-b8 - 2*b1) * q^5 - b7 * q^7 + (b10 - 13*b1) * q^8 + (-b9 + b7 - 2*b3 + 118) * q^10 + (-b10 - b6 - b4 + 20*b1) * q^11 + (-b11 - b9 + b7 + b5 - 12*b3) * q^13 + (5*b8 + 2*b6 + b2 + b1) * q^14 + (-2*b11 - 4*b7 + b5 - 13*b3 - 122) * q^16 + (-6*b10 + 2*b8 + 2*b6 - b4 + 2*b2 + 65*b1) * q^17 + (2*b11 - 6*b9 + 2*b7 + 22*b3 + 328) * q^19 + (-2*b10 + 26*b8 - 4*b6 - 6*b4 + 2*b2 + 118*b1) * q^20 + (4*b11 - 2*b9 - 22*b7 - 6*b5 + 30*b3 + 1324) * q^22 + (16*b10 + 12*b8 - 10*b4 + 4*b2 - 282*b1) * q^23 + (-4*b11 + 12*b9 - 4*b7 + 8*b5 + 76*b3 - 3983) * q^25 + (-8*b10 - 66*b8 - 4*b6 - 20*b4 + 6*b2 - 22*b1) * q^26 + (-4*b11 + 8*b9 + 48*b7 - 6*b5 - 8*b3 + 938) * q^28 + (32*b10 + 19*b8 - 4*b4 + 8*b2 + 50*b1) * q^29 + (8*b11 + 8*b9 - 7*b7 + 8*b5 - 176*b3) * q^31 + (-6*b10 - 164*b8 + 8*b6 + 4*b4 + 12*b2 - 146*b1) * q^32 + (8*b11 + 4*b9 + 76*b7 - 28*b5 + 60*b3 + 4168) * q^34 + (-51*b10 + 16*b8 + 13*b6 - 43*b4 + 16*b2 + 1380*b1) * q^35 + (3*b11 + 3*b9 + 29*b7 + 29*b5 - 508*b3) * q^37 + (16*b10 + 360*b8 - 16*b6 - 72*b4 + 8*b2 + 344*b1) * q^38 + (12*b11 + 16*b9 - 120*b7 - 54*b5 + 166*b3 + 5908) * q^40 + (-38*b10 + 2*b8 - 30*b6 + 31*b4 + 2*b2 - 895*b1) * q^41 + (-10*b11 + 30*b9 - 10*b7 + 64*b5 + 850*b3 - 28392) * q^43 + (12*b10 + 540*b8 + 40*b6 + 108*b4 + 12*b2 + 1468*b1) * q^44 + (-32*b11 + 8*b9 - 72*b7 - 48*b5 - 224*b3 + 17808) * q^46 + (-16*b10 - 140*b8 - 118*b4 - 4*b2 - 4966*b1) * q^47 + (28*b11 - 84*b9 + 28*b7 + 72*b5 + 1388*b3 - 25387) * q^49 + (96*b10 - 720*b8 + 32*b6 - 112*b4 - 16*b2 - 4143*b1) * q^50 + (24*b11 - 80*b9 - 124*b5 + 112*b3 + 45668) * q^52 + (-96*b10 - 59*b8 + 268*b4 - 24*b2 + 9830*b1) * q^53 + (-72*b11 - 72*b9 + 96*b7 + 184*b5 - 2768*b3) * q^55 + (-2*b10 - 856*b8 - 80*b6 + 312*b4 - 56*b2 + 954*b1) * q^56 + (-64*b11 + 11*b9 - 139*b7 - 96*b5 + 54*b3 - 3218) * q^58 + (258*b10 - 80*b8 - 62*b6 - 294*b4 - 80*b2 + 5840*b1) * q^59 + (17*b11 + 17*b9 - 369*b7 + 143*b5 - 2516*b3) * q^61 + (-192*b10 + 523*b8 + 30*b6 - 352*b4 - 49*b2 - 81*b1) * q^62 + (-4*b11 - 160*b9 + 408*b7 - 158*b5 - 234*b3 + 17212) * q^64 + (338*b10 - 38*b8 + 186*b6 + 819*b4 - 38*b2 - 20115*b1) * q^65 + (-12*b11 + 36*b9 - 12*b7 + 192*b5 + 2748*b3 - 80176) * q^67 + (8*b10 + 232*b8 - 144*b6 + 776*b4 - 120*b2 + 4584*b1) * q^68 + (76*b11 + 26*b9 + 542*b7 - 242*b5 + 1594*b3 + 89540) * q^70 + (-256*b10 + 960*b8 - 736*b4 - 64*b2 - 28128*b1) * q^71 + (-36*b11 + 108*b9 - 36*b7 + 200*b5 + 2604*b3 - 89130) * q^73 + (-488*b10 + 38*b8 - 52*b6 - 964*b4 - 50*b2 - 478*b1) * q^74 + (320*b9 - 832*b7 - 192*b5 + 872*b3 + 88696) * q^76 + (-288*b10 - 116*b8 + 1316*b4 - 72*b2 + 49580*b1) * q^77 + (224*b11 + 224*b9 - 247*b7 - 32*b5 - 576*b3) * q^79 + (76*b10 + 1208*b8 + 272*b6 + 1608*b4 + 24*b2 + 6948*b1) * q^80 + (136*b11 - 60*b9 - 628*b7 - 220*b5 - 1028*b3 - 57912) * q^82 + (-37*b10 + 32*b8 + 91*b6 - 1557*b4 + 32*b2 + 39860*b1) * q^83 + (-56*b11 - 56*b9 + 1688*b7 + 152*b5 - 2304*b3) * q^85 + (944*b10 - 1800*b8 + 80*b6 - 1688*b4 - 40*b2 - 29496*b1) * q^86 + (-104*b11 + 608*b9 + 464*b7 + 20*b5 + 556*b3 - 256056) * q^88 + (-436*b10 - 36*b8 - 580*b6 + 1586*b4 - 36*b2 - 43090*b1) * q^89 + (130*b11 - 390*b9 + 130*b7 - 128*b5 - 490*b3 + 64584) * q^91 + (-176*b10 - 2832*b8 + 160*b6 + 2160*b4 + 176*b2 + 18448*b1) * q^92 + (32*b11 - 136*b9 + 200*b7 + 48*b5 - 4128*b3 + 312688) * q^94 + (240*b10 - 4428*b8 - 2710*b4 + 60*b2 - 113286*b1) * q^95 + (-168*b11 + 504*b9 - 168*b7 - 48*b5 - 2568*b3 - 7246) * q^97 + (1376*b10 + 5040*b8 - 224*b6 - 3312*b4 + 112*b2 - 26315*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 156 q^{4}+O(q^{10})$$ 12 * q - 156 * q^4 $$12 q - 156 q^{4} + 1416 q^{10} - 1464 q^{16} + 3936 q^{19} + 15888 q^{22} - 47796 q^{25} + 11256 q^{28} + 50016 q^{34} + 70896 q^{40} - 340704 q^{43} + 213696 q^{46} - 304644 q^{49} + 548016 q^{52} - 38616 q^{58} + 206544 q^{64} - 962112 q^{67} + 1074480 q^{70} - 1069560 q^{73} + 1064352 q^{76} - 694944 q^{82} - 3072672 q^{88} + 775008 q^{91} + 3752256 q^{94} - 86952 q^{97}+O(q^{100})$$ 12 * q - 156 * q^4 + 1416 * q^10 - 1464 * q^16 + 3936 * q^19 + 15888 * q^22 - 47796 * q^25 + 11256 * q^28 + 50016 * q^34 + 70896 * q^40 - 340704 * q^43 + 213696 * q^46 - 304644 * q^49 + 548016 * q^52 - 38616 * q^58 + 206544 * q^64 - 962112 * q^67 + 1074480 * q^70 - 1069560 * q^73 + 1064352 * q^76 - 694944 * q^82 - 3072672 * q^88 + 775008 * q^91 + 3752256 * q^94 - 86952 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 78x^{10} + 3408x^{8} + 73216x^{6} + 13959168x^{4} + 1308622848x^{2} + 68719476736$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{11} + 78\nu^{9} + 3408\nu^{7} + 73216\nu^{5} + 13959168\nu^{3} + 1308622848\nu ) / 1073741824$$ (v^11 + 78*v^9 + 3408*v^7 + 73216*v^5 + 13959168*v^3 + 1308622848*v) / 1073741824 $$\beta_{2}$$ $$=$$ $$( - 37 \nu^{11} - 17222 \nu^{9} - 720016 \nu^{7} + 22882816 \nu^{5} + 46324449280 \nu^{3} + 433556815872 \nu ) / 11811160064$$ (-37*v^11 - 17222*v^9 - 720016*v^7 + 22882816*v^5 + 46324449280*v^3 + 433556815872*v) / 11811160064 $$\beta_{3}$$ $$=$$ $$( -\nu^{10} - 78\nu^{8} - 3408\nu^{6} - 73216\nu^{4} - 13959168\nu^{2} - 1090519040 ) / 16777216$$ (-v^10 - 78*v^8 - 3408*v^6 - 73216*v^4 - 13959168*v^2 - 1090519040) / 16777216 $$\beta_{4}$$ $$=$$ $$( -7\nu^{11} - 546\nu^{9} - 23856\nu^{7} - 512512\nu^{5} - 97714176\nu^{3} + 25199378432\nu ) / 1073741824$$ (-7*v^11 - 546*v^9 - 23856*v^7 - 512512*v^5 - 97714176*v^3 + 25199378432*v) / 1073741824 $$\beta_{5}$$ $$=$$ $$( -\nu^{10} - 78\nu^{8} - 3408\nu^{6} - 73216\nu^{4} + 254476288\nu^{2} + 2399141888 ) / 16777216$$ (-v^10 - 78*v^8 - 3408*v^6 - 73216*v^4 + 254476288*v^2 + 2399141888) / 16777216 $$\beta_{6}$$ $$=$$ $$( -\nu^{11} - 1102\nu^{9} + 47792\nu^{7} - 1728000\nu^{5} - 162332672\nu^{3} - 14998831104\nu ) / 134217728$$ (-v^11 - 1102*v^9 + 47792*v^7 - 1728000*v^5 - 162332672*v^3 - 14998831104*v) / 134217728 $$\beta_{7}$$ $$=$$ $$( -15\nu^{10} + 1390\nu^{8} - 48048\nu^{6} + 2776576\nu^{4} - 125763584\nu^{2} + 8908701696 ) / 92274688$$ (-15*v^10 + 1390*v^8 - 48048*v^6 + 2776576*v^4 - 125763584*v^2 + 8908701696) / 92274688 $$\beta_{8}$$ $$=$$ $$( -43\nu^{11} + 230\nu^{9} + 1936\nu^{7} - 9546240\nu^{5} - 499318784\nu^{3} - 11408506880\nu ) / 2952790016$$ (-43*v^11 + 230*v^9 + 1936*v^7 - 9546240*v^5 - 499318784*v^3 - 11408506880*v) / 2952790016 $$\beta_{9}$$ $$=$$ $$( 71\nu^{10} + 930\nu^{8} - 51920\nu^{6} + 21869056\nu^{4} + 872873984\nu^{2} + 40214986752 ) / 92274688$$ (71*v^10 + 930*v^8 - 51920*v^6 + 21869056*v^4 + 872873984*v^2 + 40214986752) / 92274688 $$\beta_{10}$$ $$=$$ $$( -65\nu^{11} - 974\nu^{9} + 97968\nu^{7} + 9200128\nu^{5} - 607453184\nu^{3} - 27883732992\nu ) / 1073741824$$ (-65*v^11 - 974*v^9 + 97968*v^7 + 9200128*v^5 - 607453184*v^3 - 27883732992*v) / 1073741824 $$\beta_{11}$$ $$=$$ $$( -303\nu^{10} - 11346\nu^{8} + 712272\nu^{6} + 39091712\nu^{4} - 1438318592\nu^{2} - 202651992064 ) / 184549376$$ (-303*v^10 - 11346*v^8 + 712272*v^6 + 39091712*v^4 - 1438318592*v^2 - 202651992064) / 184549376
 $$\nu$$ $$=$$ $$( \beta_{4} + 7\beta_1 ) / 32$$ (b4 + 7*b1) / 32 $$\nu^{2}$$ $$=$$ $$( \beta_{5} - \beta_{3} - 208 ) / 16$$ (b5 - b3 - 208) / 16 $$\nu^{3}$$ $$=$$ $$( 4\beta_{8} - 7\beta_{4} + 4\beta_{2} + 27\beta_1 ) / 16$$ (4*b8 - 7*b4 + 4*b2 + 27*b1) / 16 $$\nu^{4}$$ $$=$$ $$( 8\beta_{11} + 24\beta_{9} + 24\beta_{7} - 7\beta_{5} + 31\beta_{3} - 976 ) / 8$$ (8*b11 + 24*b9 + 24*b7 - 7*b5 + 31*b3 - 976) / 8 $$\nu^{5}$$ $$=$$ $$( 320\beta_{10} - 1436\beta_{8} - 64\beta_{6} - 19\beta_{4} - 28\beta_{2} - 2393\beta_1 ) / 8$$ (320*b10 - 1436*b8 - 64*b6 - 19*b4 - 28*b2 - 2393*b1) / 8 $$\nu^{6}$$ $$=$$ $$( 328\beta_{11} - 296\beta_{9} - 1320\beta_{7} - 259\beta_{5} - 8997\beta_{3} + 68848 ) / 4$$ (328*b11 - 296*b9 - 1320*b7 - 259*b5 - 8997*b3 + 68848) / 4 $$\nu^{7}$$ $$=$$ $$( 5952\beta_{10} + 10612\beta_{8} + 4544\beta_{6} - 1423\beta_{4} + 756\beta_{2} + 581747\beta_1 ) / 4$$ (5952*b10 + 10612*b8 + 4544*b6 - 1423*b4 + 756*b2 + 581747*b1) / 4 $$\nu^{8}$$ $$=$$ $$( -3224\beta_{11} - 8904\beta_{9} + 63800\beta_{7} - 1279\beta_{5} - 198857\beta_{3} - 18562128 ) / 2$$ (-3224*b11 - 8904*b9 + 63800*b7 - 1279*b5 - 198857*b3 - 18562128) / 2 $$\nu^{9}$$ $$=$$ $$( 19520\beta_{10} + 772996\beta_{8} - 122688\beta_{6} - 736699\beta_{4} - 42236\beta_{2} + 7075183\beta_1 ) / 2$$ (19520*b10 + 772996*b8 - 122688*b6 - 736699*b4 - 42236*b2 + 7075183*b1) / 2 $$\nu^{10}$$ $$=$$ $$-226936\beta_{11} + 379800\beta_{9} - 1583208\beta_{7} - 537835\beta_{5} - 767613\beta_{3} - 234853008$$ -226936*b11 + 379800*b9 - 1583208*b7 - 537835*b5 - 767613*b3 - 234853008 $$\nu^{11}$$ $$=$$ $$- 8761024 \beta_{10} - 29535788 \beta_{8} + 1499072 \beta_{6} - 4669783 \beta_{4} - 2230444 \beta_{2} + 14244635 \beta_1$$ -8761024*b10 - 29535788*b8 + 1499072*b6 - 4669783*b4 - 2230444*b2 + 14244635*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$55$$ $$65$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 7.38480 − 3.07648i 7.38480 + 3.07648i 4.24448 − 6.78118i 4.24448 + 6.78118i 1.98725 − 7.74925i 1.98725 + 7.74925i −1.98725 − 7.74925i −1.98725 + 7.74925i −4.24448 − 6.78118i −4.24448 + 6.78118i −7.38480 − 3.07648i −7.38480 + 3.07648i
−7.38480 3.07648i 0 45.0705 + 45.4384i 70.4379i 0 87.7768i −193.046 474.212i 0 −216.701 + 520.170i
19.2 −7.38480 + 3.07648i 0 45.0705 45.4384i 70.4379i 0 87.7768i −193.046 + 474.212i 0 −216.701 520.170i
19.3 −4.24448 6.78118i 0 −27.9688 + 57.5651i 206.098i 0 210.403i 509.073 54.6722i 0 1397.59 874.779i
19.4 −4.24448 + 6.78118i 0 −27.9688 57.5651i 206.098i 0 210.403i 509.073 + 54.6722i 0 1397.59 + 874.779i
19.5 −1.98725 7.74925i 0 −56.1017 + 30.7994i 106.706i 0 614.112i 350.160 + 373.539i 0 −826.887 + 212.051i
19.6 −1.98725 + 7.74925i 0 −56.1017 30.7994i 106.706i 0 614.112i 350.160 373.539i 0 −826.887 212.051i
19.7 1.98725 7.74925i 0 −56.1017 30.7994i 106.706i 0 614.112i −350.160 + 373.539i 0 −826.887 212.051i
19.8 1.98725 + 7.74925i 0 −56.1017 + 30.7994i 106.706i 0 614.112i −350.160 373.539i 0 −826.887 + 212.051i
19.9 4.24448 6.78118i 0 −27.9688 57.5651i 206.098i 0 210.403i −509.073 54.6722i 0 1397.59 + 874.779i
19.10 4.24448 + 6.78118i 0 −27.9688 + 57.5651i 206.098i 0 210.403i −509.073 + 54.6722i 0 1397.59 874.779i
19.11 7.38480 3.07648i 0 45.0705 45.4384i 70.4379i 0 87.7768i 193.046 474.212i 0 −216.701 520.170i
19.12 7.38480 + 3.07648i 0 45.0705 + 45.4384i 70.4379i 0 87.7768i 193.046 + 474.212i 0 −216.701 + 520.170i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 19.12 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
8.d odd 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.7.b.d 12
3.b odd 2 1 inner 72.7.b.d 12
4.b odd 2 1 288.7.b.c 12
8.b even 2 1 288.7.b.c 12
8.d odd 2 1 inner 72.7.b.d 12
12.b even 2 1 288.7.b.c 12
24.f even 2 1 inner 72.7.b.d 12
24.h odd 2 1 288.7.b.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.7.b.d 12 1.a even 1 1 trivial
72.7.b.d 12 3.b odd 2 1 inner
72.7.b.d 12 8.d odd 2 1 inner
72.7.b.d 12 24.f even 2 1 inner
288.7.b.c 12 4.b odd 2 1
288.7.b.c 12 8.b even 2 1
288.7.b.c 12 12.b even 2 1
288.7.b.c 12 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + 58824T_{5}^{4} + 750878400T_{5}^{2} + 2399578560000$$ acting on $$S_{7}^{\mathrm{new}}(72, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + 78 T^{10} + \cdots + 68719476736$$
$3$ $$T^{12}$$
$5$ $$(T^{6} + 58824 T^{4} + \cdots + 2399578560000)^{2}$$
$7$ $$(T^{6} + 429108 T^{4} + \cdots + 128634653983680)^{2}$$
$11$ $$(T^{6} - 7021536 T^{4} + \cdots - 14\!\cdots\!20)^{2}$$
$13$ $$(T^{6} + 16498128 T^{4} + \cdots + 17\!\cdots\!20)^{2}$$
$17$ $$(T^{6} - 85317504 T^{4} + \cdots - 51\!\cdots\!80)^{2}$$
$19$ $$(T^{3} - 984 T^{2} + \cdots + 40086711808)^{4}$$
$23$ $$(T^{6} + 405312000 T^{4} + \cdots + 17\!\cdots\!04)^{2}$$
$29$ $$(T^{6} + 1501983624 T^{4} + \cdots + 41\!\cdots\!00)^{2}$$
$31$ $$(T^{6} + 1586576052 T^{4} + \cdots + 28\!\cdots\!80)^{2}$$
$37$ $$(T^{6} + 6158858832 T^{4} + \cdots + 91\!\cdots\!80)^{2}$$
$41$ $$(T^{6} - 7294871424 T^{4} + \cdots - 12\!\cdots\!80)^{2}$$
$43$ $$(T^{3} + 85176 T^{2} + \cdots - 657572380382720)^{4}$$
$47$ $$(T^{6} + 9235517952 T^{4} + \cdots + 81\!\cdots\!04)^{2}$$
$53$ $$(T^{6} + 43574252040 T^{4} + \cdots + 44\!\cdots\!04)^{2}$$
$59$ $$(T^{6} - 149144563584 T^{4} + \cdots - 35\!\cdots\!80)^{2}$$
$61$ $$(T^{6} + 212406839760 T^{4} + \cdots + 18\!\cdots\!00)^{2}$$
$67$ $$(T^{3} + 240528 T^{2} + \cdots - 18\!\cdots\!40)^{4}$$
$71$ $$(T^{6} + 381073489920 T^{4} + \cdots + 91\!\cdots\!00)^{2}$$
$73$ $$(T^{3} + 267390 T^{2} + \cdots - 19\!\cdots\!00)^{4}$$
$79$ $$(T^{6} + 692559891828 T^{4} + \cdots + 38\!\cdots\!20)^{2}$$
$83$ $$(T^{6} - 844970997984 T^{4} + \cdots - 11\!\cdots\!20)^{2}$$
$89$ $$(T^{6} - 3016772427264 T^{4} + \cdots - 62\!\cdots\!20)^{2}$$
$97$ $$(T^{3} + 21738 T^{2} + \cdots - 71\!\cdots\!00)^{4}$$