# Properties

 Label 37.4.b.a Level $37$ Weight $4$ Character orbit 37.b Analytic conductor $2.183$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$37$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 37.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.18307067021$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 4 x^{7} + 8 x^{6} + 58 x^{5} + 197 x^{4} + 26 x^{3} + 2 x^{2} + 28 x + 196$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}\cdot 3^{5}\cdot 7$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{4} q^{2} + ( -1 - \beta_{3} ) q^{3} + ( -4 + \beta_{5} ) q^{4} + ( -\beta_{2} - \beta_{4} ) q^{5} -\beta_{1} q^{6} + ( -\beta_{3} + \beta_{7} ) q^{7} + ( \beta_{1} + 6 \beta_{4} + \beta_{6} ) q^{8} + ( 1 + 2 \beta_{3} - \beta_{7} ) q^{9} +O(q^{10})$$ $$q -\beta_{4} q^{2} + ( -1 - \beta_{3} ) q^{3} + ( -4 + \beta_{5} ) q^{4} + ( -\beta_{2} - \beta_{4} ) q^{5} -\beta_{1} q^{6} + ( -\beta_{3} + \beta_{7} ) q^{7} + ( \beta_{1} + 6 \beta_{4} + \beta_{6} ) q^{8} + ( 1 + 2 \beta_{3} - \beta_{7} ) q^{9} + ( -7 + \beta_{5} - \beta_{7} ) q^{10} + ( 10 - 2 \beta_{5} + \beta_{7} ) q^{11} + ( -6 + 3 \beta_{3} + \beta_{5} - 2 \beta_{7} ) q^{12} + ( \beta_{1} + 3 \beta_{2} - 7 \beta_{4} - \beta_{6} ) q^{13} + ( -3 \beta_{1} - 2 \beta_{2} - 3 \beta_{4} - \beta_{6} ) q^{14} + ( 3 \beta_{2} - \beta_{6} ) q^{15} + ( 29 - 5 \beta_{3} - 6 \beta_{5} + 5 \beta_{7} ) q^{16} + ( 3 \beta_{1} + \beta_{2} + 4 \beta_{4} ) q^{17} + ( 4 \beta_{1} + 2 \beta_{2} + 3 \beta_{4} + \beta_{6} ) q^{18} + ( -\beta_{1} - \beta_{2} - 4 \beta_{4} - 2 \beta_{6} ) q^{19} + ( 3 \beta_{1} - 6 \beta_{2} + 11 \beta_{4} + 2 \beta_{6} ) q^{20} + ( 30 + 13 \beta_{3} - 6 \beta_{5} - 3 \beta_{7} ) q^{21} + ( -4 \beta_{1} - 2 \beta_{2} - 32 \beta_{4} - 3 \beta_{6} ) q^{22} + ( -4 \beta_{1} + 4 \beta_{2} + 19 \beta_{4} + \beta_{6} ) q^{23} + ( 4 \beta_{2} + 23 \beta_{4} + 3 \beta_{6} ) q^{24} + ( 14 - 4 \beta_{3} - 8 \beta_{5} - 7 \beta_{7} ) q^{25} + ( -92 - 17 \beta_{3} + 13 \beta_{5} + 2 \beta_{7} ) q^{26} + ( -31 + 12 \beta_{3} + 6 \beta_{5} + 4 \beta_{7} ) q^{27} + ( -11 + 19 \beta_{3} + 13 \beta_{5} - 3 \beta_{7} ) q^{28} + ( -4 \beta_{1} + 8 \beta_{2} + 17 \beta_{4} + 3 \beta_{6} ) q^{29} + ( -6 - 6 \beta_{3} + 7 \beta_{5} ) q^{30} + ( 13 \beta_{1} - 12 \beta_{2} + 9 \beta_{4} + 4 \beta_{6} ) q^{31} + ( -13 \beta_{1} - 10 \beta_{2} - 56 \beta_{4} - 3 \beta_{6} ) q^{32} + ( 13 + 4 \beta_{3} - 8 \beta_{5} + 2 \beta_{7} ) q^{33} + ( 37 - 33 \beta_{3} - 7 \beta_{5} + 7 \beta_{7} ) q^{34} + ( \beta_{1} - 15 \beta_{2} - 28 \beta_{4} + 2 \beta_{6} ) q^{35} + ( 17 - 22 \beta_{3} - 14 \beta_{5} + 5 \beta_{7} ) q^{36} + ( -44 + 4 \beta_{1} + 9 \beta_{2} - 17 \beta_{3} + 16 \beta_{4} + 14 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{37} + ( -23 - \beta_{3} + 19 \beta_{5} - 9 \beta_{7} ) q^{38} + ( -12 \beta_{1} - 17 \beta_{2} + 25 \beta_{4} - 2 \beta_{6} ) q^{39} + ( 82 - 21 \beta_{3} - 20 \beta_{5} - 2 \beta_{7} ) q^{40} + ( 89 + 19 \beta_{3} + 2 \beta_{5} + 12 \beta_{7} ) q^{41} + ( 13 \beta_{1} + 6 \beta_{2} - 71 \beta_{4} - 3 \beta_{6} ) q^{42} + ( \beta_{1} + 25 \beta_{2} - 30 \beta_{4} ) q^{43} + ( -259 + 26 \beta_{3} + 41 \beta_{5} - 11 \beta_{7} ) q^{44} + ( -\beta_{1} + 12 \beta_{2} + 28 \beta_{4} - \beta_{6} ) q^{45} + ( 207 + 50 \beta_{3} - 22 \beta_{5} - \beta_{7} ) q^{46} + ( -184 - 55 \beta_{3} + 8 \beta_{5} + \beta_{7} ) q^{47} + ( 181 + 42 \beta_{3} - 36 \beta_{5} - 3 \beta_{7} ) q^{48} + ( -43 + 33 \beta_{3} - 6 \beta_{5} - \beta_{7} ) q^{49} + ( 2 \beta_{1} + 14 \beta_{2} - 84 \beta_{4} - \beta_{6} ) q^{50} + ( -9 \beta_{1} - 9 \beta_{2} + 72 \beta_{4} - 2 \beta_{6} ) q^{51} + ( 20 \beta_{2} + 145 \beta_{4} + 3 \beta_{6} ) q^{52} + ( 92 + 11 \beta_{3} - 22 \beta_{5} - 17 \beta_{7} ) q^{53} + ( 10 \beta_{1} - 8 \beta_{2} + 95 \beta_{4} + 2 \beta_{6} ) q^{54} + ( -5 \beta_{1} - 7 \beta_{2} - 51 \beta_{4} - \beta_{6} ) q^{55} + ( 14 \beta_{1} - 10 \beta_{2} + 142 \beta_{4} + 8 \beta_{6} ) q^{56} + ( 5 \beta_{1} - 7 \beta_{2} - 22 \beta_{4} - 8 \beta_{6} ) q^{57} + ( 145 + 62 \beta_{3} - 34 \beta_{5} + 9 \beta_{7} ) q^{58} + ( 8 \beta_{1} - 14 \beta_{2} - 78 \beta_{4} + 12 \beta_{6} ) q^{59} + ( \beta_{1} + 24 \beta_{2} + 70 \beta_{4} - \beta_{6} ) q^{60} + ( -13 \beta_{1} + 4 \beta_{2} - 55 \beta_{4} - 10 \beta_{6} ) q^{61} + ( 106 - 119 \beta_{3} - 50 \beta_{5} + 26 \beta_{7} ) q^{62} + ( -330 - 46 \beta_{3} + 12 \beta_{5} + 4 \beta_{7} ) q^{63} + ( -337 + 85 \beta_{3} + 42 \beta_{5} - 5 \beta_{7} ) q^{64} + ( 235 + 45 \beta_{3} + 34 \beta_{5} + 19 \beta_{7} ) q^{65} + ( -8 \beta_{1} - 4 \beta_{2} - 93 \beta_{4} - 10 \beta_{6} ) q^{66} + ( 156 + 5 \beta_{3} + 30 \beta_{5} - 8 \beta_{7} ) q^{67} + ( -30 \beta_{1} - 6 \beta_{2} - 122 \beta_{4} - 14 \beta_{6} ) q^{68} + ( 29 \beta_{1} + 2 \beta_{2} - 97 \beta_{4} + 12 \beta_{6} ) q^{69} + ( -281 + \beta_{3} + 13 \beta_{5} - 7 \beta_{7} ) q^{70} + ( 4 - 71 \beta_{3} + 28 \beta_{5} - 37 \beta_{7} ) q^{71} + ( -14 \beta_{1} + 6 \beta_{2} - 165 \beta_{4} - 11 \beta_{6} ) q^{72} + ( -206 - 74 \beta_{3} - 20 \beta_{5} - 11 \beta_{7} ) q^{73} + ( 166 - 5 \beta_{1} - 2 \beta_{2} - 62 \beta_{3} + 165 \beta_{4} + \beta_{5} + 13 \beta_{6} + 8 \beta_{7} ) q^{74} + ( 153 - 86 \beta_{3} + 34 \beta_{5} + 26 \beta_{7} ) q^{75} + ( 28 \beta_{1} + 10 \beta_{2} + 198 \beta_{4} + 12 \beta_{6} ) q^{76} + ( 292 - 19 \beta_{3} - 26 \beta_{5} + 9 \beta_{7} ) q^{77} + ( 427 + 120 \beta_{3} + \beta_{5} - 47 \beta_{7} ) q^{78} + ( -29 \beta_{1} - 27 \beta_{2} - 107 \beta_{4} + 15 \beta_{6} ) q^{79} + ( -13 \beta_{1} - 44 \beta_{2} - 211 \beta_{4} - 2 \beta_{6} ) q^{80} + ( -368 + 7 \beta_{3} - 18 \beta_{5} + 19 \beta_{7} ) q^{81} + ( -3 \beta_{1} - 24 \beta_{2} - 74 \beta_{4} - 10 \beta_{6} ) q^{82} + ( 242 + 87 \beta_{3} + 20 \beta_{5} + 5 \beta_{7} ) q^{83} + ( -641 - 57 \beta_{3} + 31 \beta_{5} - \beta_{7} ) q^{84} + ( 150 + 22 \beta_{3} - 16 \beta_{5} + 10 \beta_{7} ) q^{85} + ( -487 - 11 \beta_{3} + 29 \beta_{5} + 27 \beta_{7} ) q^{86} + ( 19 \beta_{1} + 2 \beta_{2} - 99 \beta_{4} + 24 \beta_{6} ) q^{87} + ( 57 \beta_{1} + 6 \beta_{2} + 461 \beta_{4} + 28 \beta_{6} ) q^{88} + ( 13 \beta_{1} - 43 \beta_{2} + 126 \beta_{4} ) q^{89} + ( 287 + 5 \beta_{3} - 20 \beta_{5} + 7 \beta_{7} ) q^{90} + ( -55 \beta_{1} + 35 \beta_{2} + 72 \beta_{4} - 12 \beta_{6} ) q^{91} + ( -2 \beta_{1} + 34 \beta_{2} - 223 \beta_{4} - 13 \beta_{6} ) q^{92} + ( -39 \beta_{1} + 34 \beta_{2} + 308 \beta_{4} - 9 \beta_{6} ) q^{93} + ( -49 \beta_{1} - 2 \beta_{2} + 207 \beta_{4} + 7 \beta_{6} ) q^{94} + ( -206 + 44 \beta_{3} + 16 \beta_{5} + 6 \beta_{7} ) q^{95} + ( 12 \beta_{1} + 38 \beta_{2} - 309 \beta_{4} - 9 \beta_{6} ) q^{96} + ( 8 \beta_{1} - 78 \beta_{2} + 78 \beta_{4} - 16 \beta_{6} ) q^{97} + ( 29 \beta_{1} + 2 \beta_{2} + 18 \beta_{4} - 5 \beta_{6} ) q^{98} + ( -305 + 15 \beta_{3} + 34 \beta_{5} - 11 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 6q^{3} - 36q^{4} + 4q^{7} + 2q^{9} + O(q^{10})$$ $$8q - 6q^{3} - 36q^{4} + 4q^{7} + 2q^{9} - 62q^{10} + 90q^{11} - 62q^{12} + 276q^{16} + 232q^{21} + 138q^{25} - 750q^{26} - 288q^{27} - 184q^{28} - 64q^{30} + 132q^{33} + 404q^{34} + 246q^{36} - 372q^{37} - 276q^{38} + 774q^{40} + 690q^{41} - 2310q^{44} + 1642q^{46} - 1392q^{47} + 1502q^{48} - 388q^{49} + 768q^{53} + 1190q^{58} + 1338q^{62} - 2588q^{63} - 3044q^{64} + 1692q^{65} + 1102q^{67} - 2316q^{70} - 12q^{71} - 1442q^{73} + 1464q^{74} + 1312q^{75} + 2496q^{77} + 3078q^{78} - 2848q^{81} + 1692q^{83} - 5140q^{84} + 1240q^{85} - 3936q^{86} + 2380q^{90} - 1788q^{95} - 2628q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} + 8 x^{6} + 58 x^{5} + 197 x^{4} + 26 x^{3} + 2 x^{2} + 28 x + 196$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$14585 \nu^{7} - 398747 \nu^{6} + 1478205 \nu^{5} - 1813577 \nu^{4} - 17599612 \nu^{3} - 62547314 \nu^{2} - 16096416 \nu - 685412$$$$)/2793216$$ $$\beta_{2}$$ $$=$$ $$($$$$-18131 \nu^{7} + 44297 \nu^{6} - 62271 \nu^{5} - 1177789 \nu^{4} - 5224748 \nu^{3} - 10517722 \nu^{2} - 5081376 \nu - 804916$$$$)/2793216$$ $$\beta_{3}$$ $$=$$ $$($$$$-12445 \nu^{7} + 48263 \nu^{6} - 88337 \nu^{5} - 735795 \nu^{4} - 2595316 \nu^{3} - 145702 \nu^{2} + 2541280 \nu + 638388$$$$)/698304$$ $$\beta_{4}$$ $$=$$ $$($$$$54955 \nu^{7} - 254737 \nu^{6} + 635895 \nu^{5} + 2705381 \nu^{4} + 9009676 \nu^{3} - 1181014 \nu^{2} + 9081888 \nu + 2022356$$$$)/2793216$$ $$\beta_{5}$$ $$=$$ $$($$$$-41609 \nu^{7} + 152779 \nu^{6} - 196621 \nu^{5} - 3002535 \nu^{4} - 7694276 \nu^{3} - 431342 \nu^{2} + 7595168 \nu - 15235548$$$$)/1396608$$ $$\beta_{6}$$ $$=$$ $$($$$$118039 \nu^{7} - 587029 \nu^{6} + 1393299 \nu^{5} + 5757881 \nu^{4} + 18933052 \nu^{3} - 31296430 \nu^{2} + 19119264 \nu + 4318244$$$$)/2793216$$ $$\beta_{7}$$ $$=$$ $$($$$$1205 \nu^{7} - 4489 \nu^{6} + 6436 \nu^{5} + 81594 \nu^{4} + 230213 \nu^{3} + 12911 \nu^{2} - 226730 \nu + 918$$$$)/21822$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$3 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + 9 \beta_{3} - 6 \beta_{2} + 21$$$$)/36$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{6} - 14 \beta_{4} - 27 \beta_{2} + 3 \beta_{1}$$$$)/18$$ $$\nu^{3}$$ $$=$$ $$($$$$-87 \beta_{7} + 34 \beta_{6} - 99 \beta_{5} - 142 \beta_{4} - 105 \beta_{3} - 192 \beta_{2} + 18 \beta_{1} - 981$$$$)/36$$ $$\nu^{4}$$ $$=$$ $$($$$$-51 \beta_{7} - 70 \beta_{5} - 41 \beta_{3} - 724$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-2469 \beta_{7} - 878 \beta_{6} - 3093 \beta_{5} + 4010 \beta_{4} - 2427 \beta_{3} + 5802 \beta_{2} - 612 \beta_{1} - 31443$$$$)/36$$ $$\nu^{6}$$ $$=$$ $$($$$$-3194 \beta_{6} + 14942 \beta_{4} + 22509 \beta_{2} - 2469 \beta_{1}$$$$)/18$$ $$\nu^{7}$$ $$=$$ $$($$$$72465 \beta_{7} - 25270 \beta_{6} + 92877 \beta_{5} + 116962 \beta_{4} + 68031 \beta_{3} + 172848 \beta_{2} - 18630 \beta_{1} + 947571$$$$)/36$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/37\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
36.1
 −0.759936 + 0.759936i 0.676150 + 0.676150i −1.76829 − 1.76829i 3.85207 − 3.85207i 3.85207 + 3.85207i −1.76829 + 1.76829i 0.676150 − 0.676150i −0.759936 − 0.759936i
5.39603i 2.81527 −21.1172 6.77552i 15.1913i 16.2590 70.7808i −19.0743 −36.5609
36.2 3.73687i −7.61586 −5.96420 2.08874i 28.4595i −23.3854 7.60752i 31.0013 7.80537
36.3 2.49424i 5.45287 1.77875 5.37972i 13.6008i −8.18671 24.3906i 2.73384 13.4183
36.4 0.835079i −3.65228 7.30264 18.7560i 3.04994i 17.3131 12.7789i −13.6608 −15.6628
36.5 0.835079i −3.65228 7.30264 18.7560i 3.04994i 17.3131 12.7789i −13.6608 −15.6628
36.6 2.49424i 5.45287 1.77875 5.37972i 13.6008i −8.18671 24.3906i 2.73384 13.4183
36.7 3.73687i −7.61586 −5.96420 2.08874i 28.4595i −23.3854 7.60752i 31.0013 7.80537
36.8 5.39603i 2.81527 −21.1172 6.77552i 15.1913i 16.2590 70.7808i −19.0743 −36.5609
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 36.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.4.b.a 8
3.b odd 2 1 333.4.c.d 8
4.b odd 2 1 592.4.g.c 8
37.b even 2 1 inner 37.4.b.a 8
37.d odd 4 1 1369.4.a.a 4
37.d odd 4 1 1369.4.a.b 4
111.d odd 2 1 333.4.c.d 8
148.b odd 2 1 592.4.g.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.4.b.a 8 1.a even 1 1 trivial
37.4.b.a 8 37.b even 2 1 inner
333.4.c.d 8 3.b odd 2 1
333.4.c.d 8 111.d odd 2 1
592.4.g.c 8 4.b odd 2 1
592.4.g.c 8 148.b odd 2 1
1369.4.a.a 4 37.d odd 4 1
1369.4.a.b 4 37.d odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(37, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1764 + 3000 T^{2} + 709 T^{4} + 50 T^{6} + T^{8}$$
$3$ $$( 427 - 57 T - 50 T^{2} + 3 T^{3} + T^{4} )^{2}$$
$5$ $$2039184 + 588072 T^{2} + 29521 T^{4} + 431 T^{6} + T^{8}$$
$7$ $$( 53892 + 2460 T - 587 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$11$ $$( 19197 + 4995 T - 756 T^{2} - 45 T^{3} + T^{4} )^{2}$$
$13$ $$19901912856336 + 51736460616 T^{2} + 39127761 T^{4} + 10911 T^{6} + T^{8}$$
$17$ $$7376047628544 + 57129962304 T^{2} + 48479680 T^{4} + 12500 T^{6} + T^{8}$$
$19$ $$3733922416896 + 37782635328 T^{2} + 67948992 T^{4} + 15924 T^{6} + T^{8}$$
$23$ $$7687465352388864 + 3830163000672 T^{2} + 664097401 T^{4} + 46151 T^{6} + T^{8}$$
$29$ $$170722326825421056 + 38394620963616 T^{2} + 3057677593 T^{4} + 98807 T^{6} + T^{8}$$
$31$ $$4860442609091887104 + 888273184398336 T^{2} + 26262320265 T^{4} + 278715 T^{6} + T^{8}$$
$37$ $$6582952005840035281 + 48345767203768644 T + 142018088190968 T^{2} + 1092583386492 T^{3} + 8269304862 T^{4} + 21569964 T^{5} + 55352 T^{6} + 372 T^{7} + T^{8}$$
$41$ $$( -99666693 + 10950795 T - 49374 T^{2} - 345 T^{3} + T^{4} )^{2}$$
$43$ $$12111354485366282496 + 1360408529161728 T^{2} + 35733923424 T^{4} + 337440 T^{6} + T^{8}$$
$47$ $$( -2544743628 - 29206602 T + 24345 T^{2} + 696 T^{3} + T^{4} )^{2}$$
$53$ $$( -7068105036 + 81571914 T - 190143 T^{2} - 384 T^{3} + T^{4} )^{2}$$
$59$ $$33486653516872421376 + 21186730028951040 T^{2} + 234781635856 T^{4} + 850844 T^{6} + T^{8}$$
$61$ $$19613833790220665856 + 1847112003726336 T^{2} + 57405891105 T^{4} + 623439 T^{6} + T^{8}$$
$67$ $$( -18172915056 + 126590808 T - 143039 T^{2} - 551 T^{3} + T^{4} )^{2}$$
$71$ $$( 46886442732 - 362436984 T - 1089243 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$73$ $$( 35735285943 - 121671339 T - 300830 T^{2} + 721 T^{3} + T^{4} )^{2}$$
$79$ $$12\!\cdots\!16$$$$+ 944975240457619464 T^{2} + 2586585683097 T^{4} + 2855991 T^{6} + T^{8}$$
$83$ $$( 56854414512 + 205812036 T - 309015 T^{2} - 846 T^{3} + T^{4} )^{2}$$
$89$ $$28\!\cdots\!44$$$$+ 369171384380461056 T^{2} + 1364455982560 T^{4} + 1978112 T^{6} + T^{8}$$
$97$ $$46\!\cdots\!96$$$$+ 1176817854840657408 T^{2} + 5225107544208 T^{4} + 4671084 T^{6} + T^{8}$$