# Properties

 Label 441.4.c.a Level $441$ Weight $4$ Character orbit 441.c Analytic conductor $26.020$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 441.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$26.0198423125$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 48 x^{14} + 1647 x^{12} - 27620 x^{10} + 336765 x^{8} - 1200006 x^{6} + 3242464 x^{4} - 1762200 x^{2} + 810000$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{12}\cdot 3^{8}\cdot 7^{4}$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + ( -4 - \beta_{1} ) q^{4} + \beta_{10} q^{5} + ( -5 \beta_{4} + \beta_{13} ) q^{8} +O(q^{10})$$ $$q + \beta_{4} q^{2} + ( -4 - \beta_{1} ) q^{4} + \beta_{10} q^{5} + ( -5 \beta_{4} + \beta_{13} ) q^{8} + \beta_{3} q^{10} + ( -2 \beta_{4} + 3 \beta_{9} + \beta_{15} ) q^{11} + ( -\beta_{5} - \beta_{7} + \beta_{11} ) q^{13} + ( 23 + \beta_{2} + \beta_{6} ) q^{16} + ( -2 \beta_{10} + \beta_{14} ) q^{17} + ( -2 \beta_{3} + 3 \beta_{5} - \beta_{7} + \beta_{11} ) q^{19} + ( -2 \beta_{8} + \beta_{12} + \beta_{14} ) q^{20} + ( 28 + \beta_{1} + 10 \beta_{2} + 2 \beta_{6} ) q^{22} + ( -17 \beta_{4} - 14 \beta_{9} + 3 \beta_{13} + \beta_{15} ) q^{23} + ( -4 \beta_{1} + 5 \beta_{2} - \beta_{6} ) q^{25} + ( -3 \beta_{8} - 2 \beta_{10} - 3 \beta_{12} - \beta_{14} ) q^{26} + ( -34 \beta_{4} - 6 \beta_{9} - 4 \beta_{13} + \beta_{15} ) q^{29} + ( -\beta_{3} + 5 \beta_{5} - 3 \beta_{11} ) q^{31} + ( -17 \beta_{4} - 14 \beta_{9} + 5 \beta_{13} - 2 \beta_{15} ) q^{32} + ( -7 \beta_{3} - 6 \beta_{5} - 4 \beta_{7} - \beta_{11} ) q^{34} + ( 148 - 5 \beta_{1} + 3 \beta_{2} - 4 \beta_{6} ) q^{37} + ( 5 \beta_{8} + 14 \beta_{10} - 5 \beta_{12} - 3 \beta_{14} ) q^{38} + ( -4 \beta_{3} + 18 \beta_{5} - 3 \beta_{7} + 3 \beta_{11} ) q^{40} + ( -4 \beta_{8} - 9 \beta_{10} + \beta_{12} - 3 \beta_{14} ) q^{41} + ( 14 - 11 \beta_{1} + 13 \beta_{2} - 10 \beta_{6} ) q^{43} + ( 21 \beta_{4} - 100 \beta_{9} - 15 \beta_{13} - 4 \beta_{15} ) q^{44} + ( 176 + 25 \beta_{1} - 4 \beta_{2} + 5 \beta_{6} ) q^{46} + ( 9 \beta_{10} - 3 \beta_{12} + 4 \beta_{14} ) q^{47} + ( -36 \beta_{4} - 58 \beta_{9} + \beta_{13} - 4 \beta_{15} ) q^{50} + ( 6 \beta_{3} + 34 \beta_{5} + 11 \beta_{7} + 6 \beta_{11} ) q^{52} + ( -6 \beta_{4} - 34 \beta_{9} - 20 \beta_{13} - 3 \beta_{15} ) q^{53} + ( -10 \beta_{3} + 3 \beta_{7} + 7 \beta_{11} ) q^{55} + ( 423 + 21 \beta_{1} - 3 \beta_{2} - 2 \beta_{6} ) q^{58} + ( -8 \beta_{8} - 8 \beta_{10} - 9 \beta_{12} ) q^{59} + ( 7 \beta_{3} + 8 \beta_{5} - 4 \beta_{7} - 9 \beta_{11} ) q^{61} + ( 7 \beta_{8} + 8 \beta_{10} + 2 \beta_{12} - \beta_{14} ) q^{62} + ( 347 + 34 \beta_{1} - 15 \beta_{2} + 9 \beta_{6} ) q^{64} + ( 18 \beta_{4} - 91 \beta_{9} + 18 \beta_{13} + 4 \beta_{15} ) q^{65} + ( -48 - 8 \beta_{1} + 19 \beta_{2} - 5 \beta_{6} ) q^{67} + ( 32 \beta_{10} - 14 \beta_{12} - 3 \beta_{14} ) q^{68} + ( 40 \beta_{4} + 13 \beta_{9} + 12 \beta_{13} + 2 \beta_{15} ) q^{71} + ( -11 \beta_{3} + 15 \beta_{5} - 17 \beta_{7} - 2 \beta_{11} ) q^{73} + ( 103 \beta_{4} - 28 \beta_{9} + 10 \beta_{13} + \beta_{15} ) q^{74} + ( 38 \beta_{3} - 18 \beta_{5} + 9 \beta_{7} - 4 \beta_{11} ) q^{76} + ( 267 - 23 \beta_{1} + 10 \beta_{2} + 25 \beta_{6} ) q^{79} + ( 4 \beta_{8} + 26 \beta_{10} - 5 \beta_{12} + \beta_{14} ) q^{80} + ( -5 \beta_{3} + 66 \beta_{5} + 17 \beta_{7} + 9 \beta_{11} ) q^{82} + ( 19 \beta_{10} + 12 \beta_{12} + 5 \beta_{14} ) q^{83} + ( -272 - 25 \beta_{1} - 32 \beta_{2} + 23 \beta_{6} ) q^{85} + ( -85 \beta_{4} - 136 \beta_{9} + 18 \beta_{13} - 3 \beta_{15} ) q^{86} + ( -57 - 54 \beta_{1} - 63 \beta_{2} - 7 \beta_{6} ) q^{88} + ( 8 \beta_{8} - 66 \beta_{10} - 8 \beta_{12} - 2 \beta_{14} ) q^{89} + ( 265 \beta_{4} - 74 \beta_{9} - 7 \beta_{13} + 7 \beta_{15} ) q^{92} + ( -2 \beta_{3} - 24 \beta_{5} - 7 \beta_{7} - 10 \beta_{11} ) q^{94} + ( -188 \beta_{4} + 53 \beta_{9} + 44 \beta_{13} + 12 \beta_{15} ) q^{95} + ( 7 \beta_{3} - 62 \beta_{5} - 11 \beta_{7} - 18 \beta_{11} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 64q^{4} + O(q^{10})$$ $$16q - 64q^{4} + 376q^{16} + 528q^{22} + 40q^{25} + 2392q^{37} + 328q^{43} + 2784q^{46} + 6744q^{58} + 5432q^{64} - 616q^{67} + 4352q^{79} - 4608q^{85} - 1416q^{88} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 48 x^{14} + 1647 x^{12} - 27620 x^{10} + 336765 x^{8} - 1200006 x^{6} + 3242464 x^{4} - 1762200 x^{2} + 810000$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-10313805 \nu^{14} + 461882826 \nu^{12} - 15525573175 \nu^{10} + 234921068610 \nu^{8} - 2717580554397 \nu^{6} + 2415597870280 \nu^{4} - 1315137294000 \nu^{2} - 278327304980712$$$$)/ 24355954733376$$ $$\beta_{2}$$ $$=$$ $$($$$$-131851797 \nu^{14} + 6000144674 \nu^{12} - 198479098895 \nu^{10} + 3003233534994 \nu^{8} - 33064412041565 \nu^{6} + 30881029845512 \nu^{4} - 16812729687600 \nu^{2} - 301359467694888$$$$)/ 73067864200128$$ $$\beta_{3}$$ $$=$$ $$($$$$5684446703 \nu^{14} - 263336690854 \nu^{12} + 8955427519661 \nu^{10} - 142516478289350 \nu^{8} + 1704993863099095 \nu^{6} - 4338407303422808 \nu^{4} + 20752104691997712 \nu^{2} - 6209552911033800$$$$)/ 1096017963001920$$ $$\beta_{4}$$ $$=$$ $$($$$$-2582250337 \nu^{15} + 122400945426 \nu^{13} - 4183683881139 \nu^{11} + 68992918331690 \nu^{9} - 834373374448305 \nu^{7} + 2691078814742472 \nu^{5} - 8010514076168368 \nu^{3} + 699777739755000 \nu$$$$)/ 3653393210006400$$ $$\beta_{5}$$ $$=$$ $$($$$$2582250337 \nu^{14} - 122400945426 \nu^{12} + 4183683881139 \nu^{10} - 68992918331690 \nu^{8} + 834373374448305 \nu^{6} - 2691078814742472 \nu^{4} + 8010514076168368 \nu^{2} - 2526474344758200$$$$)/ 260956657857600$$ $$\beta_{6}$$ $$=$$ $$($$$$774906315 \nu^{14} - 34871917166 \nu^{12} + 1166481691025 \nu^{10} - 17650306515630 \nu^{8} + 198847121920499 \nu^{6} - 181490928341240 \nu^{4} + 98810108802000 \nu^{2} + 2318225515024536$$$$)/ 73067864200128$$ $$\beta_{7}$$ $$=$$ $$($$$$-49963897373 \nu^{14} + 2354075825554 \nu^{12} - 80336766247031 \nu^{10} + 1317072190119410 \nu^{8} - 15915802670641045 \nu^{6} + 50391134075700488 \nu^{4} - 156345640146407472 \nu^{2} + 49038788329507800$$$$)/ 1826696605003200$$ $$\beta_{8}$$ $$=$$ $$($$$$-2582250337 \nu^{15} + 122400945426 \nu^{13} - 4183683881139 \nu^{11} + 68992918331690 \nu^{9} - 834373374448305 \nu^{7} + 2691078814742472 \nu^{5} - 8010514076168368 \nu^{3} + 8006564159767800 \nu$$$$)/ 521913315715200$$ $$\beta_{9}$$ $$=$$ $$($$$$-253797 \nu^{15} + 12085406 \nu^{13} - 413081109 \nu^{11} + 6837155440 \nu^{9} - 82382868455 \nu^{7} + 265706935032 \nu^{5} - 646432035208 \nu^{3} + 69093405000 \nu$$$$)/ 49371246000$$ $$\beta_{10}$$ $$=$$ $$($$$$144886104737 \nu^{15} - 6952944162226 \nu^{13} + 238669929280139 \nu^{11} - 4004343941493890 \nu^{9} + 48919377548687305 \nu^{7} - 176406493692653672 \nu^{5} + 507004002338951568 \nu^{3} - 505456539561487800 \nu$$$$)/ 27400449075048000$$ $$\beta_{11}$$ $$=$$ $$($$$$-277854990781 \nu^{14} + 13359750934538 \nu^{12} - 458260687617607 \nu^{10} + 7688616812915170 \nu^{8} - 93437146332960365 \nu^{6} + 328616417180980936 \nu^{4} - 794262060747822384 \nu^{2} + 258381236675496600$$$$)/ 5480089815009600$$ $$\beta_{12}$$ $$=$$ $$($$$$-8720041741 \nu^{15} + 445313626418 \nu^{13} - 15626457197827 \nu^{11} + 283608035305270 \nu^{9} - 3627435038369465 \nu^{7} + 18513569620759696 \nu^{5} - 48538920567702024 \nu^{3} + 48046470365885400 \nu$$$$)/ 685011226876200$$ $$\beta_{13}$$ $$=$$ $$($$$$-2582250337 \nu^{15} + 122400945426 \nu^{13} - 4183683881139 \nu^{11} + 68992918331690 \nu^{9} - 834373374448305 \nu^{7} + 2691078814742472 \nu^{5} - 7836542970929968 \nu^{3} + 699777739755000 \nu$$$$)/ 173971105238400$$ $$\beta_{14}$$ $$=$$ $$($$$$546527094713 \nu^{15} - 27021548295874 \nu^{13} + 936917477650511 \nu^{11} - 16349356524516110 \nu^{9} + 204380334860676445 \nu^{7} - 898993649933543528 \nu^{5} + 2443910245958009232 \nu^{3} - 2426112663369442200 \nu$$$$)/ 13700224537524000$$ $$\beta_{15}$$ $$=$$ $$($$$$658131337369 \nu^{15} - 31288398347762 \nu^{13} + 1069442457152643 \nu^{11} - 17681301490112130 \nu^{9} + 213284353479834785 \nu^{7} - 687899473715995464 \nu^{5} + 1754382912933196016 \nu^{3} - 178878721893435000 \nu$$$$)/ 9133483025016000$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{8} - 7 \beta_{4}$$$$)/14$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{11} + 2 \beta_{7} + 12 \beta_{5} - 2 \beta_{3} + 7 \beta_{1} + 84$$$$)/14$$ $$\nu^{3}$$ $$=$$ $$\beta_{13} - 21 \beta_{4}$$ $$\nu^{4}$$ $$=$$ $$($$$$18 \beta_{11} + 43 \beta_{7} - 7 \beta_{6} + 240 \beta_{5} - 57 \beta_{3} - 7 \beta_{2} - 168 \beta_{1} - 1729$$$$)/14$$ $$\nu^{5}$$ $$=$$ $$($$$$14 \beta_{15} + 14 \beta_{14} + 189 \beta_{13} - 11 \beta_{12} - 542 \beta_{10} + 98 \beta_{9} - 440 \beta_{8} - 3241 \beta_{4}$$$$)/14$$ $$\nu^{6}$$ $$=$$ $$-31 \beta_{6} - 55 \beta_{2} - 542 \beta_{1} - 5437$$ $$\nu^{7}$$ $$=$$ $$($$$$-602 \beta_{15} + 602 \beta_{14} - 4613 \beta_{13} + 149 \beta_{12} - 13386 \beta_{10} - 5054 \beta_{9} - 9828 \beta_{8} + 72205 \beta_{4}$$$$)/14$$ $$\nu^{8}$$ $$=$$ $$($$$$-9524 \beta_{11} - 16801 \beta_{7} - 5817 \beta_{6} - 114324 \beta_{5} + 36499 \beta_{3} - 13881 \beta_{2} - 85442 \beta_{1} - 849051$$$$)/14$$ $$\nu^{9}$$ $$=$$ $$-2814 \beta_{15} - 15851 \beta_{13} - 25458 \beta_{9} + 231147 \beta_{4}$$ $$\nu^{10}$$ $$=$$ $$($$$$-226068 \beta_{11} - 325793 \beta_{7} + 150353 \beta_{6} - 2532900 \beta_{5} + 903195 \beta_{3} + 427049 \beta_{2} + 1931202 \beta_{1} + 19059467$$$$)/14$$ $$\nu^{11}$$ $$=$$ $$($$$$-577402 \beta_{15} - 577402 \beta_{14} - 2658957 \beta_{13} - 477677 \beta_{12} + 7877146 \beta_{10} - 5425294 \beta_{9} + 4990876 \beta_{8} + 36440285 \beta_{4}$$$$)/14$$ $$\nu^{12}$$ $$=$$ $$544823 \beta_{6} + 1732295 \beta_{2} + 6262822 \beta_{1} + 61427333$$ $$\nu^{13}$$ $$=$$ $$($$$$15939826 \beta_{15} - 15939826 \beta_{14} + 63593341 \beta_{13} - 15069541 \beta_{12} + 189909738 \beta_{10} + 153140302 \beta_{9} + 113262204 \beta_{8} - 824549117 \beta_{4}$$$$)/14$$ $$\nu^{14}$$ $$=$$ $$($$$$127461460 \beta_{11} + 117556481 \beta_{7} + 95472993 \beta_{6} + 1263507492 \beta_{5} - 541341899 \beta_{3} + 328312425 \beta_{2} + 999389314 \beta_{1} + 9745702827$$$$)/14$$ $$\nu^{15}$$ $$=$$ $$60540774 \beta_{15} + 216949675 \beta_{13} + 590099298 \beta_{9} - 2677172379 \beta_{4}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
440.1
 −4.21355 + 2.43270i 4.21355 + 2.43270i 3.91663 + 2.26127i −3.91663 + 2.26127i 1.57646 + 0.910170i −1.57646 + 0.910170i 0.648633 + 0.374489i −0.648633 + 0.374489i 0.648633 − 0.374489i −0.648633 − 0.374489i 1.57646 − 0.910170i −1.57646 − 0.910170i 3.91663 − 2.26127i −3.91663 − 2.26127i −4.21355 − 2.43270i 4.21355 − 2.43270i
4.86539i 0 −15.6720 −12.7643 0 0 37.3274i 0 62.1035i
440.2 4.86539i 0 −15.6720 12.7643 0 0 37.3274i 0 62.1035i
440.3 4.52254i 0 −12.4534 −1.26570 0 0 20.1405i 0 5.72419i
440.4 4.52254i 0 −12.4534 1.26570 0 0 20.1405i 0 5.72419i
440.5 1.82034i 0 4.68636 −15.0874 0 0 23.0935i 0 27.4643i
440.6 1.82034i 0 4.68636 15.0874 0 0 23.0935i 0 27.4643i
440.7 0.748977i 0 7.43903 −10.8554 0 0 11.5635i 0 8.13042i
440.8 0.748977i 0 7.43903 10.8554 0 0 11.5635i 0 8.13042i
440.9 0.748977i 0 7.43903 −10.8554 0 0 11.5635i 0 8.13042i
440.10 0.748977i 0 7.43903 10.8554 0 0 11.5635i 0 8.13042i
440.11 1.82034i 0 4.68636 −15.0874 0 0 23.0935i 0 27.4643i
440.12 1.82034i 0 4.68636 15.0874 0 0 23.0935i 0 27.4643i
440.13 4.52254i 0 −12.4534 −1.26570 0 0 20.1405i 0 5.72419i
440.14 4.52254i 0 −12.4534 1.26570 0 0 20.1405i 0 5.72419i
440.15 4.86539i 0 −15.6720 −12.7643 0 0 37.3274i 0 62.1035i
440.16 4.86539i 0 −15.6720 12.7643 0 0 37.3274i 0 62.1035i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 440.16 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
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.c.a 16
3.b odd 2 1 inner 441.4.c.a 16
7.b odd 2 1 inner 441.4.c.a 16
7.c even 3 1 63.4.p.a 16
7.c even 3 1 441.4.p.c 16
7.d odd 6 1 63.4.p.a 16
7.d odd 6 1 441.4.p.c 16
21.c even 2 1 inner 441.4.c.a 16
21.g even 6 1 63.4.p.a 16
21.g even 6 1 441.4.p.c 16
21.h odd 6 1 63.4.p.a 16
21.h odd 6 1 441.4.p.c 16
28.f even 6 1 1008.4.bt.a 16
28.g odd 6 1 1008.4.bt.a 16
84.j odd 6 1 1008.4.bt.a 16
84.n even 6 1 1008.4.bt.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.p.a 16 7.c even 3 1
63.4.p.a 16 7.d odd 6 1
63.4.p.a 16 21.g even 6 1
63.4.p.a 16 21.h odd 6 1
441.4.c.a 16 1.a even 1 1 trivial
441.4.c.a 16 3.b odd 2 1 inner
441.4.c.a 16 7.b odd 2 1 inner
441.4.c.a 16 21.c even 2 1 inner
441.4.p.c 16 7.c even 3 1
441.4.p.c 16 7.d odd 6 1
441.4.p.c 16 21.g even 6 1
441.4.p.c 16 21.h odd 6 1
1008.4.bt.a 16 28.f even 6 1
1008.4.bt.a 16 28.g odd 6 1
1008.4.bt.a 16 84.j odd 6 1
1008.4.bt.a 16 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 48 T_{2}^{6} + 657 T_{2}^{4} + 1958 T_{2}^{2} + 900$$ acting on $$S_{4}^{\mathrm{new}}(441, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 900 + 1958 T^{2} + 657 T^{4} + 48 T^{6} + T^{8} )^{2}$$
$3$ $$T^{16}$$
$5$ $$( 7001316 - 4503492 T^{2} + 83925 T^{4} - 510 T^{6} + T^{8} )^{2}$$
$7$ $$T^{16}$$
$11$ $$( 1333068831396 + 15349205660 T^{2} + 18315405 T^{4} + 7446 T^{6} + T^{8} )^{2}$$
$13$ $$( 2752420357764 + 38728522980 T^{2} + 47288277 T^{4} + 13074 T^{6} + T^{8} )^{2}$$
$17$ $$( 938202780969216 - 684682262208 T^{2} + 186189732 T^{4} - 22356 T^{6} + T^{8} )^{2}$$
$19$ $$( 567106596 + 240360422652 T^{2} + 173991213 T^{4} + 32994 T^{6} + T^{8} )^{2}$$
$23$ $$( 10073619800395776 + 4204404921728 T^{2} + 643125444 T^{4} + 42372 T^{6} + T^{8} )^{2}$$
$29$ $$( 253130837240241216 + 47051469656336 T^{2} + 3179040801 T^{4} + 93078 T^{6} + T^{8} )^{2}$$
$31$ $$( 20564942686042041 + 8708603301216 T^{2} + 1260944406 T^{4} + 69648 T^{6} + T^{8} )^{2}$$
$37$ $$( 192019982 - 7684124 T + 107391 T^{2} - 598 T^{3} + T^{4} )^{4}$$
$41$ $$( 32324337336074547456 - 2566451699751744 T^{2} + 53591867172 T^{4} - 402972 T^{6} + T^{8} )^{2}$$
$43$ $$( 144774182 - 27889316 T - 195789 T^{2} - 82 T^{3} + T^{4} )^{4}$$
$47$ $$( 1272038665930711056 - 481114740294720 T^{2} + 30230312760 T^{4} - 472272 T^{6} + T^{8} )^{2}$$
$53$ $$($$$$49\!\cdots\!04$$$$+ 15290262807651920 T^{2} + 161546992449 T^{4} + 684342 T^{6} + T^{8} )^{2}$$
$59$ $$($$$$14\!\cdots\!44$$$$- 46517349102889392 T^{2} + 420948267993 T^{4} - 1243590 T^{6} + T^{8} )^{2}$$
$61$ $$( 345703894859082816 + 2018690658550944 T^{2} + 125233457364 T^{4} + 806748 T^{6} + T^{8} )^{2}$$
$67$ $$( 4555177972 - 8473892 T - 170031 T^{2} + 154 T^{3} + T^{4} )^{4}$$
$71$ $$( 2630939135312864016 + 579610290699200 T^{2} + 28173681528 T^{4} + 317232 T^{6} + T^{8} )^{2}$$
$73$ $$($$$$91\!\cdots\!00$$$$+ 718246908615512808 T^{2} + 1992938070537 T^{4} + 2349198 T^{6} + T^{8} )^{2}$$
$79$ $$( 91919817907 + 264129988 T - 519384 T^{2} - 1088 T^{3} + T^{4} )^{4}$$
$83$ $$($$$$75\!\cdots\!64$$$$- 194675614013208132 T^{2} + 1143650214837 T^{4} - 1941810 T^{6} + T^{8} )^{2}$$
$89$ $$( 689036047036130304 - 1830458747190528 T^{2} + 1192346527824 T^{4} - 3090888 T^{6} + T^{8} )^{2}$$
$97$ $$($$$$64\!\cdots\!56$$$$+ 3466733814045027288 T^{2} + 6247433928969 T^{4} + 4322814 T^{6} + T^{8} )^{2}$$