# Properties

 Label 4140.2.f.c Level $4140$ Weight $2$ Character orbit 4140.f Analytic conductor $33.058$ Analytic rank $0$ Dimension $14$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4140.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$33.0580664368$$ Analytic rank: $$0$$ Dimension: $$14$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} + \cdots)$$ Defining polynomial: $$x^{14} + 27x^{12} + 283x^{10} + 1441x^{8} + 3596x^{6} + 3740x^{4} + 772x^{2} + 4$$ x^14 + 27*x^12 + 283*x^10 + 1441*x^8 + 3596*x^6 + 3740*x^4 + 772*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 1380) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{10} q^{5} + \beta_{13} q^{7}+O(q^{10})$$ q - b10 * q^5 + b13 * q^7 $$q - \beta_{10} q^{5} + \beta_{13} q^{7} + (\beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{2} - 1) q^{11} + (\beta_{12} + \beta_{10} + \beta_{7} - \beta_1) q^{13} + (\beta_{13} + \beta_{12} - \beta_{9} - \beta_{8} - \beta_1) q^{17} + ( - \beta_{4} + \beta_{3}) q^{19} - \beta_{6} q^{23} + (\beta_{13} - \beta_{6} - \beta_{5} + \beta_{2} - 1) q^{25} + (\beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{4} - \beta_{3} - \beta_{2} + 2) q^{29} + ( - 2 \beta_{10} + 2 \beta_{7} + \beta_{5} - \beta_{3} + 1) q^{31} + ( - \beta_{12} - 2 \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + 1) q^{35} + (\beta_{11} + 3 \beta_{6} + \beta_1) q^{37} + ( - \beta_{9} + \beta_{8} + \beta_{5} + \beta_{4} - 3) q^{41} + ( - \beta_{13} - 2 \beta_{12} + \beta_{11} + \beta_{6} + \beta_1) q^{43} + ( - \beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6}) q^{47} + ( - \beta_{10} + \beta_{7} - 2 \beta_{3} - \beta_{2} - 1) q^{49} + (\beta_{12} - \beta_{11} - 2 \beta_{9} - 2 \beta_{8} + \beta_{6}) q^{53} + ( - \beta_{12} + 2 \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 - 1) q^{55} + (\beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} + 2 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 2) q^{59} + ( - \beta_{9} + \beta_{8} + \beta_{4} + \beta_{3} + 2 \beta_{2} + 4) q^{61} + ( - \beta_{13} + \beta_{11} - \beta_{10} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} + \beta_1 + 3) q^{65} + (2 \beta_{12} - \beta_{11} + 2 \beta_{10} - \beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_1) q^{67} + ( - \beta_{9} + \beta_{8} + \beta_{5} - \beta_{4} - 3) q^{71} + ( - 2 \beta_{13} + \beta_{12} + \beta_{10} + \beta_{7} + 2 \beta_{6} - \beta_1) q^{73} + (\beta_{13} - 3 \beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - 3 \beta_{6} + 2 \beta_1) q^{77} + ( - \beta_{9} + \beta_{8} + 2 \beta_{4} - 2 \beta_{2} - 2) q^{79} + (\beta_{13} + \beta_{12} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + 4 \beta_{6} - \beta_1) q^{83} + (\beta_{13} - \beta_{12} + 2 \beta_{11} - \beta_{10} - \beta_{7} - 3 \beta_{6} + 2 \beta_{4} + \beta_{3} + \cdots - 3) q^{85}+ \cdots + ( - \beta_{13} - 2 \beta_{12} + \beta_{11} - \beta_{9} - \beta_{8} + \beta_{6} + 3 \beta_1) q^{97}+O(q^{100})$$ q - b10 * q^5 + b13 * q^7 + (b10 - b9 + b8 - b7 - b5 - b4 + b2 - 1) * q^11 + (b12 + b10 + b7 - b1) * q^13 + (b13 + b12 - b9 - b8 - b1) * q^17 + (-b4 + b3) * q^19 - b6 * q^23 + (b13 - b6 - b5 + b2 - 1) * q^25 + (b10 - b9 + b8 - b7 - b4 - b3 - b2 + 2) * q^29 + (-2*b10 + 2*b7 + b5 - b3 + 1) * q^31 + (-b12 - 2*b7 + b6 + b4 + b3 + 1) * q^35 + (b11 + 3*b6 + b1) * q^37 + (-b9 + b8 + b5 + b4 - 3) * q^41 + (-b13 - 2*b12 + b11 + b6 + b1) * q^43 + (-b13 + b12 + b11 - b10 + b9 + b8 - b7 - b6) * q^47 + (-b10 + b7 - 2*b3 - b2 - 1) * q^49 + (b12 - b11 - 2*b9 - 2*b8 + b6) * q^53 + (-b12 + 2*b7 - 2*b6 + b5 + b3 + b2 - b1 - 1) * q^55 + (b10 + b9 - b8 - b7 + 2*b5 + b4 + b3 - b2 + 2) * q^59 + (-b9 + b8 + b4 + b3 + 2*b2 + 4) * q^61 + (-b13 + b11 - b10 + b7 + b6 + b5 + b4 + b2 + b1 + 3) * q^65 + (2*b12 - b11 + 2*b10 - b9 - b8 + 2*b7 + b6 - b1) * q^67 + (-b9 + b8 + b5 - b4 - 3) * q^71 + (-2*b13 + b12 + b10 + b7 + 2*b6 - b1) * q^73 + (b13 - 3*b12 + b11 - b10 - b9 - b8 - b7 - 3*b6 + 2*b1) * q^77 + (-b9 + b8 + 2*b4 - 2*b2 - 2) * q^79 + (b13 + b12 - b10 - b9 - b8 - b7 + 4*b6 - b1) * q^83 + (b13 - b12 + 2*b11 - b10 - b7 - 3*b6 + 2*b4 + b3 + b2 + b1 - 3) * q^85 + (b10 - 2*b9 + 2*b8 - b7 - 2*b5 - 2*b4 + 2) * q^89 + (-3*b9 + 3*b8 - 2*b5 - 3*b4 - b3) * q^91 + (2*b13 - b12 + 2*b11 + b9 + 3*b8 - 2*b7 + b6 - b5 - 2*b4 - b3 + 2*b2 - 2*b1 + 2) * q^95 + (-b13 - 2*b12 + b11 - b9 - b8 + b6 + 3*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14 q+O(q^{10})$$ 14 * q $$14 q + 4 q^{19} - 6 q^{25} + 30 q^{29} + 6 q^{31} + 14 q^{35} - 46 q^{41} - 20 q^{49} - 16 q^{55} + 10 q^{59} + 64 q^{61} + 36 q^{65} - 42 q^{71} - 32 q^{79} - 42 q^{85} + 52 q^{89} + 28 q^{91} + 44 q^{95}+O(q^{100})$$ 14 * q + 4 * q^19 - 6 * q^25 + 30 * q^29 + 6 * q^31 + 14 * q^35 - 46 * q^41 - 20 * q^49 - 16 * q^55 + 10 * q^59 + 64 * q^61 + 36 * q^65 - 42 * q^71 - 32 * q^79 - 42 * q^85 + 52 * q^89 + 28 * q^91 + 44 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} + 27x^{12} + 283x^{10} + 1441x^{8} + 3596x^{6} + 3740x^{4} + 772x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} + 6\nu$$ v^3 + 6*v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 4$$ v^2 + 4 $$\beta_{3}$$ $$=$$ $$( \nu^{12} + 6\nu^{10} - 153\nu^{8} - 1732\nu^{6} - 5726\nu^{4} - 5036\nu^{2} + 136 ) / 372$$ (v^12 + 6*v^10 - 153*v^8 - 1732*v^6 - 5726*v^4 - 5036*v^2 + 136) / 372 $$\beta_{4}$$ $$=$$ $$( 15\nu^{12} + 338\nu^{10} + 2851\nu^{8} + 11282\nu^{6} + 21122\nu^{4} + 15228\nu^{2} + 800 ) / 372$$ (15*v^12 + 338*v^10 + 2851*v^8 + 11282*v^6 + 21122*v^4 + 15228*v^2 + 800) / 372 $$\beta_{5}$$ $$=$$ $$( -4\nu^{12} - 86\nu^{10} - 659\nu^{8} - 2124\nu^{6} - 2423\nu^{4} - 6\nu^{2} + 200 ) / 62$$ (-4*v^12 - 86*v^10 - 659*v^8 - 2124*v^6 - 2423*v^4 - 6*v^2 + 200) / 62 $$\beta_{6}$$ $$=$$ $$( 6\nu^{13} + 160\nu^{11} + 1655\nu^{9} + 8332\nu^{7} + 20731\nu^{5} + 22050\nu^{3} + 5218\nu ) / 372$$ (6*v^13 + 160*v^11 + 1655*v^9 + 8332*v^7 + 20731*v^5 + 22050*v^3 + 5218*v) / 372 $$\beta_{7}$$ $$=$$ $$( - 21 \nu^{13} + 2 \nu^{12} - 591 \nu^{11} + 12 \nu^{10} - 6459 \nu^{9} - 306 \nu^{8} - 34215 \nu^{7} - 3464 \nu^{6} - 88260 \nu^{5} - 11824 \nu^{4} - 93078 \nu^{3} - 12676 \nu^{2} + \cdots - 1216 ) / 744$$ (-21*v^13 + 2*v^12 - 591*v^11 + 12*v^10 - 6459*v^9 - 306*v^8 - 34215*v^7 - 3464*v^6 - 88260*v^5 - 11824*v^4 - 93078*v^3 - 12676*v^2 - 16992*v - 1216) / 744 $$\beta_{8}$$ $$=$$ $$( - 21 \nu^{13} - 2 \nu^{12} - 591 \nu^{11} - 74 \nu^{10} - 6459 \nu^{9} - 934 \nu^{8} - 34215 \nu^{7} - 5154 \nu^{6} - 88260 \nu^{5} - 12604 \nu^{4} - 93078 \nu^{3} - 11504 \nu^{2} + \cdots - 1264 ) / 744$$ (-21*v^13 - 2*v^12 - 591*v^11 - 74*v^10 - 6459*v^9 - 934*v^8 - 34215*v^7 - 5154*v^6 - 88260*v^5 - 12604*v^4 - 93078*v^3 - 11504*v^2 - 16248*v - 1264) / 744 $$\beta_{9}$$ $$=$$ $$( - 21 \nu^{13} + 2 \nu^{12} - 591 \nu^{11} + 74 \nu^{10} - 6459 \nu^{9} + 934 \nu^{8} - 34215 \nu^{7} + 5154 \nu^{6} - 88260 \nu^{5} + 12604 \nu^{4} - 93078 \nu^{3} + 11504 \nu^{2} + \cdots + 1264 ) / 744$$ (-21*v^13 + 2*v^12 - 591*v^11 + 74*v^10 - 6459*v^9 + 934*v^8 - 34215*v^7 + 5154*v^6 - 88260*v^5 + 12604*v^4 - 93078*v^3 + 11504*v^2 - 16248*v + 1264) / 744 $$\beta_{10}$$ $$=$$ $$( - 21 \nu^{13} - 2 \nu^{12} - 591 \nu^{11} - 12 \nu^{10} - 6459 \nu^{9} + 306 \nu^{8} - 34215 \nu^{7} + 3464 \nu^{6} - 88260 \nu^{5} + 11824 \nu^{4} - 93078 \nu^{3} + 12676 \nu^{2} + \cdots + 1216 ) / 744$$ (-21*v^13 - 2*v^12 - 591*v^11 - 12*v^10 - 6459*v^9 + 306*v^8 - 34215*v^7 + 3464*v^6 - 88260*v^5 + 11824*v^4 - 93078*v^3 + 12676*v^2 - 16992*v + 1216) / 744 $$\beta_{11}$$ $$=$$ $$( 16\nu^{13} + 468\nu^{11} + 5271\nu^{9} + 28460\nu^{7} + 74017\nu^{5} + 78826\nu^{3} + 16870\nu ) / 372$$ (16*v^13 + 468*v^11 + 5271*v^9 + 28460*v^7 + 74017*v^5 + 78826*v^3 + 16870*v) / 372 $$\beta_{12}$$ $$=$$ $$( -27\nu^{13} - 689\nu^{11} - 6781\nu^{9} - 32255\nu^{7} - 74798\nu^{5} - 71046\nu^{3} - 10244\nu ) / 372$$ (-27*v^13 - 689*v^11 - 6781*v^9 - 32255*v^7 - 74798*v^5 - 71046*v^3 - 10244*v) / 372 $$\beta_{13}$$ $$=$$ $$( -22\nu^{13} - 597\nu^{11} - 6306\nu^{9} - 32483\nu^{7} - 82534\nu^{5} - 88786\nu^{3} - 20848\nu ) / 372$$ (-22*v^13 - 597*v^11 - 6306*v^9 - 32483*v^7 - 82534*v^5 - 88786*v^3 - 20848*v) / 372
 $$\nu$$ $$=$$ $$( -\beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} ) / 2$$ (-b10 + b9 + b8 - b7) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 4$$ b2 - 4 $$\nu^{3}$$ $$=$$ $$3\beta_{10} - 3\beta_{9} - 3\beta_{8} + 3\beta_{7} + \beta_1$$ 3*b10 - 3*b9 - 3*b8 + 3*b7 + b1 $$\nu^{4}$$ $$=$$ $$\beta_{10} - \beta_{7} + 2\beta_{3} - 7\beta_{2} + 24$$ b10 - b7 + 2*b3 - 7*b2 + 24 $$\nu^{5}$$ $$=$$ $$-2\beta_{13} + \beta_{11} - 19\beta_{10} + 21\beta_{9} + 21\beta_{8} - 19\beta_{7} - 3\beta_{6} - 11\beta_1$$ -2*b13 + b11 - 19*b10 + 21*b9 + 21*b8 - 19*b7 - 3*b6 - 11*b1 $$\nu^{6}$$ $$=$$ $$-17\beta_{10} + 4\beta_{9} - 4\beta_{8} + 17\beta_{7} + 2\beta_{5} + 2\beta_{4} - 24\beta_{3} + 49\beta_{2} - 156$$ -17*b10 + 4*b9 - 4*b8 + 17*b7 + 2*b5 + 2*b4 - 24*b3 + 49*b2 - 156 $$\nu^{7}$$ $$=$$ $$30 \beta_{13} + 2 \beta_{12} - 15 \beta_{11} + 128 \beta_{10} - 156 \beta_{9} - 156 \beta_{8} + 128 \beta_{7} + 61 \beta_{6} + 99 \beta_1$$ 30*b13 + 2*b12 - 15*b11 + 128*b10 - 156*b9 - 156*b8 + 128*b7 + 61*b6 + 99*b1 $$\nu^{8}$$ $$=$$ $$197 \beta_{10} - 68 \beta_{9} + 68 \beta_{8} - 197 \beta_{7} - 30 \beta_{5} - 28 \beta_{4} + 230 \beta_{3} - 353 \beta_{2} + 1072$$ 197*b10 - 68*b9 + 68*b8 - 197*b7 - 30*b5 - 28*b4 + 230*b3 - 353*b2 + 1072 $$\nu^{9}$$ $$=$$ $$- 338 \beta_{13} - 32 \beta_{12} + 157 \beta_{11} - 904 \beta_{10} + 1194 \beta_{9} + 1194 \beta_{8} - 904 \beta_{7} - 787 \beta_{6} - 841 \beta_1$$ -338*b13 - 32*b12 + 157*b11 - 904*b10 + 1194*b9 + 1194*b8 - 904*b7 - 787*b6 - 841*b1 $$\nu^{10}$$ $$=$$ $$- 1965 \beta_{10} + 810 \beta_{9} - 810 \beta_{8} + 1965 \beta_{7} + 322 \beta_{5} + 282 \beta_{4} - 2052 \beta_{3} + 2617 \beta_{2} - 7692$$ -1965*b10 + 810*b9 - 810*b8 + 1965*b7 + 322*b5 + 282*b4 - 2052*b3 + 2617*b2 - 7692 $$\nu^{11}$$ $$=$$ $$3390 \beta_{13} + 362 \beta_{12} - 1437 \beta_{11} + 6624 \beta_{10} - 9320 \beta_{9} - 9320 \beta_{8} + 6624 \beta_{7} + 8455 \beta_{6} + 7003 \beta_1$$ 3390*b13 + 362*b12 - 1437*b11 + 6624*b10 - 9320*b9 - 9320*b8 + 6624*b7 + 8455*b6 + 7003*b1 $$\nu^{12}$$ $$=$$ $$18213 \beta_{10} - 8336 \beta_{9} + 8336 \beta_{8} - 18213 \beta_{7} - 3058 \beta_{5} - 2512 \beta_{4} + 17758 \beta_{3} - 19889 \beta_{2} + 57120$$ 18213*b10 - 8336*b9 + 8336*b8 - 18213*b7 - 3058*b5 - 2512*b4 + 17758*b3 - 19889*b2 + 57120 $$\nu^{13}$$ $$=$$ $$- 31918 \beta_{13} - 3604 \beta_{12} + 12389 \beta_{11} - 49978 \beta_{10} + 73852 \beta_{9} + 73852 \beta_{8} - 49978 \beta_{7} - 82667 \beta_{6} - 57917 \beta_1$$ -31918*b13 - 3604*b12 + 12389*b11 - 49978*b10 + 73852*b9 + 73852*b8 - 49978*b7 - 82667*b6 - 57917*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$$.

 $$n$$ $$461$$ $$1657$$ $$2071$$ $$3961$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
829.1
 1.52925i − 1.52925i − 0.0729221i 0.0729221i − 2.88373i 2.88373i − 2.04690i 2.04690i − 2.47928i 2.47928i 0.511427i − 0.511427i − 2.39625i 2.39625i
0 0 0 −2.14859 0.619333i 0 1.66138i 0 0 0
829.2 0 0 0 −2.14859 + 0.619333i 0 1.66138i 0 0 0
829.3 0 0 0 −1.54426 1.61718i 0 3.99468i 0 0 0
829.4 0 0 0 −1.54426 + 1.61718i 0 3.99468i 0 0 0
829.5 0 0 0 −0.792997 2.09073i 0 4.31589i 0 0 0
829.6 0 0 0 −0.792997 + 2.09073i 0 4.31589i 0 0 0
829.7 0 0 0 0.181772 2.22867i 0 0.189781i 0 0 0
829.8 0 0 0 0.181772 + 2.22867i 0 0.189781i 0 0 0
829.9 0 0 0 0.258163 2.22111i 0 2.14682i 0 0 0
829.10 0 0 0 0.258163 + 2.22111i 0 2.14682i 0 0 0
829.11 0 0 0 1.81604 1.30461i 0 3.73844i 0 0 0
829.12 0 0 0 1.81604 + 1.30461i 0 3.73844i 0 0 0
829.13 0 0 0 2.22987 0.166381i 0 1.74202i 0 0 0
829.14 0 0 0 2.22987 + 0.166381i 0 1.74202i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 829.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4140.2.f.c 14
3.b odd 2 1 1380.2.f.b 14
5.b even 2 1 inner 4140.2.f.c 14
15.d odd 2 1 1380.2.f.b 14
15.e even 4 1 6900.2.a.bc 7
15.e even 4 1 6900.2.a.bd 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.f.b 14 3.b odd 2 1
1380.2.f.b 14 15.d odd 2 1
4140.2.f.c 14 1.a even 1 1 trivial
4140.2.f.c 14 5.b even 2 1 inner
6900.2.a.bc 7 15.e even 4 1
6900.2.a.bd 7 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{14} + 59T_{7}^{12} + 1323T_{7}^{10} + 14065T_{7}^{8} + 72984T_{7}^{6} + 178488T_{7}^{4} + 166704T_{7}^{2} + 5776$$ acting on $$S_{2}^{\mathrm{new}}(4140, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{14}$$
$3$ $$T^{14}$$
$5$ $$T^{14} + 3 T^{12} - 8 T^{11} + \cdots + 78125$$
$7$ $$T^{14} + 59 T^{12} + 1323 T^{10} + \cdots + 5776$$
$11$ $$(T^{7} - 64 T^{5} - 62 T^{4} + 1200 T^{3} + \cdots - 9216)^{2}$$
$13$ $$T^{14} + 104 T^{12} + 4268 T^{10} + \cdots + 6071296$$
$17$ $$T^{14} + 167 T^{12} + 10683 T^{10} + \cdots + 10000$$
$19$ $$(T^{7} - 2 T^{6} - 118 T^{5} + 158 T^{4} + \cdots - 36680)^{2}$$
$23$ $$(T^{2} + 1)^{7}$$
$29$ $$(T^{7} - 15 T^{6} + 17 T^{5} + 595 T^{4} + \cdots + 12272)^{2}$$
$31$ $$(T^{7} - 3 T^{6} - 141 T^{5} + 499 T^{4} + \cdots + 54720)^{2}$$
$37$ $$T^{14} + 243 T^{12} + \cdots + 16126968064$$
$41$ $$(T^{7} + 23 T^{6} + 103 T^{5} + \cdots - 15248)^{2}$$
$43$ $$T^{14} + 336 T^{12} + \cdots + 389193984$$
$47$ $$T^{14} + 492 T^{12} + \cdots + 1973013529600$$
$53$ $$T^{14} + 251 T^{12} + \cdots + 374345104$$
$59$ $$(T^{7} - 5 T^{6} - 255 T^{5} + \cdots + 1483200)^{2}$$
$61$ $$(T^{7} - 32 T^{6} + 274 T^{5} + \cdots + 377248)^{2}$$
$67$ $$T^{14} + 315 T^{12} + 37075 T^{10} + \cdots + 300304$$
$71$ $$(T^{7} + 21 T^{6} + 55 T^{5} - 875 T^{4} + \cdots + 784)^{2}$$
$73$ $$T^{14} + 440 T^{12} + \cdots + 47127199744$$
$79$ $$(T^{7} + 16 T^{6} - 264 T^{5} + \cdots - 4755584)^{2}$$
$83$ $$T^{14} + 415 T^{12} + 47443 T^{10} + \cdots + 1638400$$
$89$ $$(T^{7} - 26 T^{6} + 12 T^{5} + \cdots + 526112)^{2}$$
$97$ $$T^{14} + 804 T^{12} + \cdots + 56070144$$