# Properties

 Label 3750.2.a.u Level 3750 Weight 2 Character orbit 3750.a Self dual yes Analytic conductor 29.944 Analytic rank 0 Dimension 8 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$3750 = 2 \cdot 3 \cdot 5^{4}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 3750.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$29.9439007580$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.8.71684000000.1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$5^{2}$$ Twist minimal: no (minimal twist has level 150) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} + ( -1 + \beta_{6} ) q^{7} - q^{8} + q^{9} +O(q^{10})$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} + ( -1 + \beta_{6} ) q^{7} - q^{8} + q^{9} + ( 1 - \beta_{3} - \beta_{5} ) q^{11} - q^{12} + \beta_{1} q^{13} + ( 1 - \beta_{6} ) q^{14} + q^{16} + ( -1 - \beta_{3} + \beta_{4} + \beta_{7} ) q^{17} - q^{18} + ( 1 + \beta_{1} + \beta_{2} + \beta_{6} ) q^{19} + ( 1 - \beta_{6} ) q^{21} + ( -1 + \beta_{3} + \beta_{5} ) q^{22} + ( -2 - \beta_{4} - \beta_{5} ) q^{23} + q^{24} -\beta_{1} q^{26} - q^{27} + ( -1 + \beta_{6} ) q^{28} + ( 1 + \beta_{3} + \beta_{4} + \beta_{7} ) q^{29} + ( 2 - \beta_{2} - \beta_{3} + \beta_{6} ) q^{31} - q^{32} + ( -1 + \beta_{3} + \beta_{5} ) q^{33} + ( 1 + \beta_{3} - \beta_{4} - \beta_{7} ) q^{34} + q^{36} + ( 1 - \beta_{3} - \beta_{5} - \beta_{7} ) q^{37} + ( -1 - \beta_{1} - \beta_{2} - \beta_{6} ) q^{38} -\beta_{1} q^{39} + ( 2 + \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{41} + ( -1 + \beta_{6} ) q^{42} + ( -1 - \beta_{1} + \beta_{2} + \beta_{6} ) q^{43} + ( 1 - \beta_{3} - \beta_{5} ) q^{44} + ( 2 + \beta_{4} + \beta_{5} ) q^{46} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{47} - q^{48} + ( 2 - 2 \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{7} ) q^{49} + ( 1 + \beta_{3} - \beta_{4} - \beta_{7} ) q^{51} + \beta_{1} q^{52} + ( -3 - \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{7} ) q^{53} + q^{54} + ( 1 - \beta_{6} ) q^{56} + ( -1 - \beta_{1} - \beta_{2} - \beta_{6} ) q^{57} + ( -1 - \beta_{3} - \beta_{4} - \beta_{7} ) q^{58} + ( -2 - \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{5} + \beta_{6} ) q^{59} + ( 3 + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{61} + ( -2 + \beta_{2} + \beta_{3} - \beta_{6} ) q^{62} + ( -1 + \beta_{6} ) q^{63} + q^{64} + ( 1 - \beta_{3} - \beta_{5} ) q^{66} + ( 1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{67} + ( -1 - \beta_{3} + \beta_{4} + \beta_{7} ) q^{68} + ( 2 + \beta_{4} + \beta_{5} ) q^{69} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{6} ) q^{71} - q^{72} + ( 1 + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{73} + ( -1 + \beta_{3} + \beta_{5} + \beta_{7} ) q^{74} + ( 1 + \beta_{1} + \beta_{2} + \beta_{6} ) q^{76} + ( -2 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{77} + \beta_{1} q^{78} + ( -1 + 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{79} + q^{81} + ( -2 - \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{82} + ( -3 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{83} + ( 1 - \beta_{6} ) q^{84} + ( 1 + \beta_{1} - \beta_{2} - \beta_{6} ) q^{86} + ( -1 - \beta_{3} - \beta_{4} - \beta_{7} ) q^{87} + ( -1 + \beta_{3} + \beta_{5} ) q^{88} + ( \beta_{1} + \beta_{2} + 5 \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{89} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{91} + ( -2 - \beta_{4} - \beta_{5} ) q^{92} + ( -2 + \beta_{2} + \beta_{3} - \beta_{6} ) q^{93} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{94} + q^{96} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{97} + ( -2 + 2 \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{7} ) q^{98} + ( 1 - \beta_{3} - \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{2} - 8q^{3} + 8q^{4} + 8q^{6} - 4q^{7} - 8q^{8} + 8q^{9} + O(q^{10})$$ $$8q - 8q^{2} - 8q^{3} + 8q^{4} + 8q^{6} - 4q^{7} - 8q^{8} + 8q^{9} + 6q^{11} - 8q^{12} - 2q^{13} + 4q^{14} + 8q^{16} - 14q^{17} - 8q^{18} + 10q^{19} + 4q^{21} - 6q^{22} - 12q^{23} + 8q^{24} + 2q^{26} - 8q^{27} - 4q^{28} + 10q^{29} + 16q^{31} - 8q^{32} - 6q^{33} + 14q^{34} + 8q^{36} + 6q^{37} - 10q^{38} + 2q^{39} + 6q^{41} - 4q^{42} - 2q^{43} + 6q^{44} + 12q^{46} - 14q^{47} - 8q^{48} + 26q^{49} + 14q^{51} - 2q^{52} - 12q^{53} + 8q^{54} + 4q^{56} - 10q^{57} - 10q^{58} + 16q^{61} - 16q^{62} - 4q^{63} + 8q^{64} + 6q^{66} + 6q^{67} - 14q^{68} + 12q^{69} + 6q^{71} - 8q^{72} + 8q^{73} - 6q^{74} + 10q^{76} - 8q^{77} - 2q^{78} + 10q^{79} + 8q^{81} - 6q^{82} - 22q^{83} + 4q^{84} + 2q^{86} - 10q^{87} - 6q^{88} + 20q^{89} + 6q^{91} - 12q^{92} - 16q^{93} + 14q^{94} + 8q^{96} + 16q^{97} - 26q^{98} + 6q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} - 18 x^{6} + 10 x^{5} + 101 x^{4} + 40 x^{3} - 132 x^{2} - 96 x - 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-3 \nu^{7} + 11 \nu^{6} + 32 \nu^{5} - 70 \nu^{4} - 145 \nu^{3} + 21 \nu^{2} + 208 \nu + 106$$$$)/20$$ $$\beta_{2}$$ $$=$$ $$($$$$4 \nu^{7} - 19 \nu^{6} - 26 \nu^{5} + 144 \nu^{4} + 48 \nu^{3} - 247 \nu^{2} + 10 \nu + 86$$$$)/20$$ $$\beta_{3}$$ $$=$$ $$($$$$7 \nu^{7} - 27 \nu^{6} - 78 \nu^{5} + 222 \nu^{4} + 319 \nu^{3} - 361 \nu^{2} - 340 \nu + 18$$$$)/20$$ $$\beta_{4}$$ $$=$$ $$($$$$-14 \nu^{7} + 47 \nu^{6} + 186 \nu^{5} - 386 \nu^{4} - 862 \nu^{3} + 569 \nu^{2} + 998 \nu + 22$$$$)/20$$ $$\beta_{5}$$ $$=$$ $$($$$$16 \nu^{7} - 57 \nu^{6} - 194 \nu^{5} + 450 \nu^{4} + 840 \nu^{3} - 587 \nu^{2} - 866 \nu - 162$$$$)/20$$ $$\beta_{6}$$ $$=$$ $$($$$$9 \nu^{7} - 32 \nu^{6} - 111 \nu^{5} + 256 \nu^{4} + 502 \nu^{3} - 364 \nu^{2} - 578 \nu - 42$$$$)/10$$ $$\beta_{7}$$ $$=$$ $$($$$$28 \nu^{7} - 103 \nu^{6} - 332 \nu^{5} + 838 \nu^{4} + 1446 \nu^{3} - 1309 \nu^{2} - 1650 \nu - 18$$$$)/20$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + 2 \beta_{1} + 4$$$$)/5$$ $$\nu^{2}$$ $$=$$ $$($$$$5 \beta_{7} - 2 \beta_{6} + 3 \beta_{5} + 6 \beta_{4} - 7 \beta_{3} - 4 \beta_{2} + \beta_{1} + 32$$$$)/5$$ $$\nu^{3}$$ $$=$$ $$($$$$27 \beta_{7} - 3 \beta_{6} + 12 \beta_{5} + 29 \beta_{4} - 58 \beta_{3} - 10 \beta_{2} + 14 \beta_{1} + 98$$$$)/5$$ $$\nu^{4}$$ $$=$$ $$($$$$119 \beta_{7} - 50 \beta_{6} + 55 \beta_{5} + 110 \beta_{4} - 205 \beta_{3} - 58 \beta_{2} + 35 \beta_{1} + 470$$$$)/5$$ $$\nu^{5}$$ $$=$$ $$($$$$549 \beta_{7} - 199 \beta_{6} + 236 \beta_{5} + 487 \beta_{4} - 1054 \beta_{3} - 206 \beta_{2} + 182 \beta_{1} + 1904$$$$)/5$$ $$\nu^{6}$$ $$=$$ $$($$$$2411 \beta_{7} - 1082 \beta_{6} + 1063 \beta_{5} + 2016 \beta_{4} - 4417 \beta_{3} - 956 \beta_{2} + 691 \beta_{1} + 8442$$$$)/5$$ $$\nu^{7}$$ $$=$$ $$($$$$10719 \beta_{7} - 4723 \beta_{6} + 4642 \beta_{5} + 8799 \beta_{4} - 20178 \beta_{3} - 3894 \beta_{2} + 3094 \beta_{1} + 36238$$$$)/5$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.20224 −1.65651 −0.0444111 4.37243 1.37243 −1.74919 −0.852282 2.75978
−1.00000 −1.00000 1.00000 0 1.00000 −4.63137 −1.00000 1.00000 0
1.2 −1.00000 −1.00000 1.00000 0 1.00000 −3.23143 −1.00000 1.00000 0
1.3 −1.00000 −1.00000 1.00000 0 1.00000 −2.70913 −1.00000 1.00000 0
1.4 −1.00000 −1.00000 1.00000 0 1.00000 −2.61995 −1.00000 1.00000 0
1.5 −1.00000 −1.00000 1.00000 0 1.00000 0.329315 −1.00000 1.00000 0
1.6 −1.00000 −1.00000 1.00000 0 1.00000 0.533559 −1.00000 1.00000 0
1.7 −1.00000 −1.00000 1.00000 0 1.00000 3.52206 −1.00000 1.00000 0
1.8 −1.00000 −1.00000 1.00000 0 1.00000 4.80694 −1.00000 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3750.2.a.u 8
5.b even 2 1 3750.2.a.v 8
5.c odd 4 2 3750.2.c.k 16
25.d even 5 2 750.2.g.g 16
25.e even 10 2 750.2.g.f 16
25.f odd 20 2 150.2.h.b 16
25.f odd 20 2 750.2.h.d 16
75.l even 20 2 450.2.l.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.h.b 16 25.f odd 20 2
450.2.l.c 16 75.l even 20 2
750.2.g.f 16 25.e even 10 2
750.2.g.g 16 25.d even 5 2
750.2.h.d 16 25.f odd 20 2
3750.2.a.u 8 1.a even 1 1 trivial
3750.2.a.v 8 5.b even 2 1
3750.2.c.k 16 5.c odd 4 2

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3750))$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{8}$$
$3$ $$( 1 + T )^{8}$$
$5$ 
$7$ $$1 + 4 T + 23 T^{2} + 48 T^{3} + 191 T^{4} + 264 T^{5} + 1285 T^{6} + 2084 T^{7} + 10564 T^{8} + 14588 T^{9} + 62965 T^{10} + 90552 T^{11} + 458591 T^{12} + 806736 T^{13} + 2705927 T^{14} + 3294172 T^{15} + 5764801 T^{16}$$
$11$ $$1 - 6 T + 45 T^{2} - 240 T^{3} + 1205 T^{4} - 4658 T^{5} + 20073 T^{6} - 68300 T^{7} + 239780 T^{8} - 751300 T^{9} + 2428833 T^{10} - 6199798 T^{11} + 17642405 T^{12} - 38652240 T^{13} + 79720245 T^{14} - 116923026 T^{15} + 214358881 T^{16}$$
$13$ $$1 + 2 T + 47 T^{2} + 176 T^{3} + 1191 T^{4} + 5212 T^{5} + 25325 T^{6} + 85818 T^{7} + 405224 T^{8} + 1115634 T^{9} + 4279925 T^{10} + 11450764 T^{11} + 34016151 T^{12} + 65347568 T^{13} + 226860023 T^{14} + 125497034 T^{15} + 815730721 T^{16}$$
$17$ $$1 + 14 T + 153 T^{2} + 1158 T^{3} + 7331 T^{4} + 38214 T^{5} + 179335 T^{6} + 760774 T^{7} + 3198344 T^{8} + 12933158 T^{9} + 51827815 T^{10} + 187745382 T^{11} + 612292451 T^{12} + 1644194406 T^{13} + 3693048057 T^{14} + 5744741422 T^{15} + 6975757441 T^{16}$$
$19$ $$1 - 10 T + 102 T^{2} - 670 T^{3} + 4268 T^{4} - 22710 T^{5} + 117514 T^{6} - 560450 T^{7} + 2521030 T^{8} - 10648550 T^{9} + 42422554 T^{10} - 155767890 T^{11} + 556210028 T^{12} - 1658986330 T^{13} + 4798679862 T^{14} - 8938717390 T^{15} + 16983563041 T^{16}$$
$23$ $$1 + 12 T + 132 T^{2} + 716 T^{3} + 4016 T^{4} + 10172 T^{5} + 50860 T^{6} + 53308 T^{7} + 886334 T^{8} + 1226084 T^{9} + 26904940 T^{10} + 123762724 T^{11} + 1123841456 T^{12} + 4608421588 T^{13} + 19540737348 T^{14} + 40857905364 T^{15} + 78310985281 T^{16}$$
$29$ $$1 - 10 T + 197 T^{2} - 1510 T^{3} + 17343 T^{4} - 107610 T^{5} + 913319 T^{6} - 4694150 T^{7} + 32002680 T^{8} - 136130350 T^{9} + 768101279 T^{10} - 2624500290 T^{11} + 12266374383 T^{12} - 30971834990 T^{13} + 117180194237 T^{14} - 172498763090 T^{15} + 500246412961 T^{16}$$
$31$ $$1 - 16 T + 245 T^{2} - 2700 T^{3} + 25605 T^{4} - 210758 T^{5} + 1537653 T^{6} - 9946750 T^{7} + 58857900 T^{8} - 308349250 T^{9} + 1477684533 T^{10} - 6278691578 T^{11} + 23646755205 T^{12} - 77298707700 T^{13} + 217438401845 T^{14} - 440201825776 T^{15} + 852891037441 T^{16}$$
$37$ $$1 - 6 T + 243 T^{2} - 1412 T^{3} + 27071 T^{4} - 147396 T^{5} + 1822865 T^{6} - 8822506 T^{7} + 81640584 T^{8} - 326432722 T^{9} + 2495502185 T^{10} - 7466049588 T^{11} + 50735412431 T^{12} - 97913667284 T^{13} + 623471517387 T^{14} - 569591262798 T^{15} + 3512479453921 T^{16}$$
$41$ $$1 - 6 T + 145 T^{2} - 850 T^{3} + 10255 T^{4} - 69718 T^{5} + 526063 T^{6} - 4134450 T^{7} + 23274800 T^{8} - 169512450 T^{9} + 884311903 T^{10} - 4805034278 T^{11} + 28978179055 T^{12} - 98477770850 T^{13} + 688765114945 T^{14} - 1168525643286 T^{15} + 7984925229121 T^{16}$$
$43$ $$1 + 2 T + 182 T^{2} + 166 T^{3} + 15756 T^{4} - 5778 T^{5} + 902810 T^{6} - 1211542 T^{7} + 41601734 T^{8} - 52096306 T^{9} + 1669295690 T^{10} - 459391446 T^{11} + 53866628556 T^{12} + 24403401538 T^{13} + 1150488074918 T^{14} + 543637222214 T^{15} + 11688200277601 T^{16}$$
$47$ $$1 + 14 T + 158 T^{2} + 1418 T^{3} + 16156 T^{4} + 125194 T^{5} + 1006370 T^{6} + 7336014 T^{7} + 59392774 T^{8} + 344792658 T^{9} + 2223071330 T^{10} + 12998016662 T^{11} + 78836126236 T^{12} + 325211219926 T^{13} + 1703116021982 T^{14} + 7092723686482 T^{15} + 23811286661761 T^{16}$$
$53$ $$1 + 12 T + 352 T^{2} + 3946 T^{3} + 57861 T^{4} + 577532 T^{5} + 5731520 T^{6} + 48919988 T^{7} + 371517029 T^{8} + 2592759364 T^{9} + 16099839680 T^{10} + 85981231564 T^{11} + 456551121141 T^{12} + 1650199415378 T^{13} + 7801855117408 T^{14} + 14096533678044 T^{15} + 62259690411361 T^{16}$$
$59$ $$1 + 247 T^{2} + 500 T^{3} + 28643 T^{4} + 123000 T^{5} + 2182549 T^{6} + 13248500 T^{7} + 135310720 T^{8} + 781661500 T^{9} + 7597453069 T^{10} + 25261617000 T^{11} + 347077571123 T^{12} + 357462149500 T^{13} + 10418591809327 T^{14} + 146830437604321 T^{16}$$
$61$ $$1 - 16 T + 335 T^{2} - 3580 T^{3} + 40995 T^{4} - 303668 T^{5} + 2560793 T^{6} - 14411280 T^{7} + 131714360 T^{8} - 879088080 T^{9} + 9528710753 T^{10} - 68926866308 T^{11} + 567610251795 T^{12} - 3023654757580 T^{13} + 17259325410935 T^{14} - 50283885376336 T^{15} + 191707312997281 T^{16}$$
$67$ $$1 - 6 T + 238 T^{2} - 1202 T^{3} + 35076 T^{4} - 154266 T^{5} + 3495250 T^{6} - 13666446 T^{7} + 270524214 T^{8} - 915651882 T^{9} + 15690177250 T^{10} - 46397504958 T^{11} + 706820720196 T^{12} - 1622850378614 T^{13} + 21529094956222 T^{14} - 36364269631938 T^{15} + 406067677556641 T^{16}$$
$71$ $$1 - 6 T + 390 T^{2} - 2050 T^{3} + 74900 T^{4} - 339378 T^{5} + 9085578 T^{6} - 35263990 T^{7} + 765200950 T^{8} - 2503743290 T^{9} + 45800398698 T^{10} - 121467119358 T^{11} + 1903334906900 T^{12} - 3698670169550 T^{13} + 49959110729190 T^{14} - 54570720950346 T^{15} + 645753531245761 T^{16}$$
$73$ $$1 - 8 T + 327 T^{2} - 3024 T^{3} + 60051 T^{4} - 506008 T^{5} + 7383005 T^{6} - 54348912 T^{7} + 633329824 T^{8} - 3967470576 T^{9} + 39344033645 T^{10} - 196845714136 T^{11} + 1705342770291 T^{12} - 6268968497232 T^{13} + 49486291996503 T^{14} - 88379188152776 T^{15} + 806460091894081 T^{16}$$
$79$ $$1 - 10 T + 227 T^{2} - 1720 T^{3} + 32163 T^{4} - 256360 T^{5} + 3728769 T^{6} - 26100650 T^{7} + 323096880 T^{8} - 2061951350 T^{9} + 23271247329 T^{10} - 126395478040 T^{11} + 1252751455203 T^{12} - 5292537006280 T^{13} + 55180852403267 T^{14} - 192039089861590 T^{15} + 1517108809906561 T^{16}$$
$83$ $$1 + 22 T + 487 T^{2} + 7796 T^{3} + 106551 T^{4} + 1298832 T^{5} + 14259445 T^{6} + 141748378 T^{7} + 1361172004 T^{8} + 11765115374 T^{9} + 98233316605 T^{10} + 742655252784 T^{11} + 5056731560871 T^{12} + 30708760852828 T^{13} + 159219961830703 T^{14} + 596993121771794 T^{15} + 2252292232139041 T^{16}$$
$89$ $$1 - 20 T + 537 T^{2} - 8210 T^{3} + 124663 T^{4} - 1568070 T^{5} + 17732339 T^{6} - 192143300 T^{7} + 1812907320 T^{8} - 17100753700 T^{9} + 140457857219 T^{10} - 1105440739830 T^{11} + 7821635989783 T^{12} - 45845128076290 T^{13} + 266878953246057 T^{14} - 884626697910580 T^{15} + 3936588805702081 T^{16}$$
$97$ $$1 - 16 T + 458 T^{2} - 4612 T^{3} + 72331 T^{4} - 417776 T^{5} + 4642080 T^{6} - 3332736 T^{7} + 188812609 T^{8} - 323275392 T^{9} + 43677330720 T^{10} - 381292875248 T^{11} + 6403411424011 T^{12} - 39604813265284 T^{13} + 381501178257482 T^{14} - 1292772551649808 T^{15} + 7837433594376961 T^{16}$$