# Properties

 Label 2268.2.l.n Level $2268$ Weight $2$ Character orbit 2268.l Analytic conductor $18.110$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2268 = 2^{2} \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2268.l (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.1100711784$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 9 x^{14} + 31 x^{12} - 282 x^{10} + 1695 x^{8} - 3318 x^{6} + 4606 x^{4} - 4116 x^{2} + 2401$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$3^{7}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{15} q^{5} + ( -1 + \beta_{1} - \beta_{4} ) q^{7} +O(q^{10})$$ $$q + \beta_{15} q^{5} + ( -1 + \beta_{1} - \beta_{4} ) q^{7} -\beta_{11} q^{11} + ( -\beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{8} + \beta_{9} ) q^{13} + ( -\beta_{7} + \beta_{12} - \beta_{15} ) q^{17} + ( 1 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{8} ) q^{19} + ( -\beta_{6} - \beta_{7} - \beta_{13} - \beta_{14} ) q^{23} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{8} + \beta_{9} ) q^{25} + ( \beta_{6} + 2 \beta_{10} + 2 \beta_{12} + 2 \beta_{13} ) q^{29} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{31} + ( -\beta_{6} + \beta_{7} - \beta_{10} - \beta_{11} - 2 \beta_{12} ) q^{35} + ( -1 + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{8} ) q^{37} + ( -3 \beta_{6} - \beta_{7} + \beta_{10} + \beta_{11} - 3 \beta_{14} ) q^{41} + ( -\beta_{3} - 2 \beta_{5} - \beta_{8} ) q^{43} + ( -\beta_{6} - 2 \beta_{7} - \beta_{10} - \beta_{11} - 3 \beta_{12} - \beta_{14} + 3 \beta_{15} ) q^{47} + ( -\beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{9} ) q^{49} + ( \beta_{6} - 3 \beta_{12} + \beta_{14} + 3 \beta_{15} ) q^{53} + ( -4 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{8} + \beta_{9} ) q^{55} + ( \beta_{6} + 2 \beta_{10} + \beta_{12} ) q^{59} + ( -\beta_{2} + \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - \beta_{8} - 2 \beta_{9} ) q^{61} + ( -3 \beta_{6} - \beta_{7} - 4 \beta_{12} - 3 \beta_{14} + 4 \beta_{15} ) q^{65} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - \beta_{5} - 2 \beta_{8} ) q^{67} + ( 3 \beta_{6} + 3 \beta_{7} - \beta_{11} + 3 \beta_{13} + 2 \beta_{14} ) q^{71} + ( 1 - \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{5} + 3 \beta_{8} - \beta_{9} ) q^{73} + ( -3 \beta_{6} - 2 \beta_{7} + 2 \beta_{10} + \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{77} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{8} - \beta_{9} ) q^{79} + ( -2 \beta_{6} + 3 \beta_{10} - 2 \beta_{12} + 3 \beta_{13} ) q^{83} + ( 2 - 2 \beta_{1} - 2 \beta_{3} + 5 \beta_{4} + \beta_{5} + 2 \beta_{8} - \beta_{9} ) q^{85} + ( -\beta_{6} + 2 \beta_{10} + 3 \beta_{12} + 2 \beta_{13} ) q^{89} + ( 2 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + \beta_{8} - \beta_{9} ) q^{91} + ( 5 \beta_{6} + 2 \beta_{10} + 5 \beta_{12} + \beta_{13} ) q^{95} + ( 8 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 8 \beta_{4} + \beta_{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + O(q^{10})$$ $$16 q + 10 q^{13} + 8 q^{19} - 8 q^{31} - 4 q^{37} - 10 q^{43} - 20 q^{49} - 32 q^{55} + 28 q^{61} + 18 q^{67} - 20 q^{79} - 38 q^{85} - 2 q^{91} + 42 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 9 x^{14} + 31 x^{12} - 282 x^{10} + 1695 x^{8} - 3318 x^{6} + 4606 x^{4} - 4116 x^{2} + 2401$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-92249 \nu^{14} + 22952 \nu^{12} + 2800495 \nu^{10} + 21569239 \nu^{8} + 19347710 \nu^{6} - 586034029 \nu^{4} - 1081706164 \nu^{2} + 811586706$$$$)/ 982062165$$ $$\beta_{2}$$ $$=$$ $$($$$$-244351 \nu^{14} + 3103286 \nu^{12} - 15156266 \nu^{10} + 92194120 \nu^{8} - 665006344 \nu^{6} + 2126767580 \nu^{4} - 3126524037 \nu^{2} + 4446117606$$$$)/ 982062165$$ $$\beta_{3}$$ $$=$$ $$($$$$-50804 \nu^{14} + 431490 \nu^{12} - 910001 \nu^{10} + 11273818 \nu^{8} - 75102634 \nu^{6} + 34785107 \nu^{4} + 145125260 \nu^{2} + 188652058$$$$)/ 140294595$$ $$\beta_{4}$$ $$=$$ $$($$$$95779 \nu^{14} - 763661 \nu^{12} + 2336349 \nu^{10} - 25351497 \nu^{8} + 138865326 \nu^{6} - 211337511 \nu^{4} + 367714228 \nu^{2} - 326083583$$$$)/ 196412433$$ $$\beta_{5}$$ $$=$$ $$($$$$650081 \nu^{14} - 2142857 \nu^{12} - 7665412 \nu^{10} - 102476293 \nu^{8} + 182625112 \nu^{6} + 2834904373 \nu^{4} - 3275996381 \nu^{2} + 2857821120$$$$)/ 982062165$$ $$\beta_{6}$$ $$=$$ $$($$$$-202439 \nu^{15} + 3779851 \nu^{13} - 20969988 \nu^{11} + 90097992 \nu^{9} - 810563151 \nu^{7} + 3169832022 \nu^{5} - 2394515438 \nu^{3} + 2189789518 \nu$$$$)/ 1374887031$$ $$\beta_{7}$$ $$=$$ $$($$$$-1070099 \nu^{15} + 19590400 \nu^{13} - 113896481 \nu^{11} + 569013813 \nu^{9} - 4402741114 \nu^{7} + 18268910772 \nu^{5} - 31221084265 \nu^{3} + 21365317708 \nu$$$$)/ 6874435155$$ $$\beta_{8}$$ $$=$$ $$($$$$1945252 \nu^{14} - 17009491 \nu^{12} + 54869965 \nu^{10} - 532124987 \nu^{8} + 3143382740 \nu^{6} - 5341228438 \nu^{4} + 7262233307 \nu^{2} - 4847832003$$$$)/ 982062165$$ $$\beta_{9}$$ $$=$$ $$($$$$-47620 \nu^{14} + 437764 \nu^{12} - 1505123 \nu^{10} + 13183462 \nu^{8} - 81699127 \nu^{6} + 160523888 \nu^{4} - 143228568 \nu^{2} + 91036316$$$$)/22838655$$ $$\beta_{10}$$ $$=$$ $$($$$$-1980807 \nu^{15} + 23809589 \nu^{13} - 99962851 \nu^{11} + 647103666 \nu^{9} - 4655869109 \nu^{7} + 12867260469 \nu^{5} - 11700535903 \nu^{3} + 6814080875 \nu$$$$)/ 6874435155$$ $$\beta_{11}$$ $$=$$ $$($$$$5475730 \nu^{15} - 45854286 \nu^{13} + 136146167 \nu^{11} - 1401072738 \nu^{9} + 8191148113 \nu^{7} - 11791630242 \nu^{5} + 7725904137 \nu^{3} - 3523415364 \nu$$$$)/ 6874435155$$ $$\beta_{12}$$ $$=$$ $$($$$$6460385 \nu^{15} - 43459642 \nu^{13} + 90349759 \nu^{11} - 1535538006 \nu^{9} + 7245892466 \nu^{7} - 2015258469 \nu^{5} + 11091500154 \nu^{3} + 8594048162 \nu$$$$)/ 6874435155$$ $$\beta_{13}$$ $$=$$ $$($$$$9038101 \nu^{15} - 95345786 \nu^{13} + 377201131 \nu^{11} - 2746111680 \nu^{9} + 18463783844 \nu^{7} - 46118841720 \nu^{5} + 44386540142 \nu^{3} - 21891697856 \nu$$$$)/ 6874435155$$ $$\beta_{14}$$ $$=$$ $$($$$$9523 \nu^{15} - 82214 \nu^{13} + 251526 \nu^{11} - 2489934 \nu^{9} + 14949924 \nu^{7} - 22752114 \nu^{5} + 18278176 \nu^{3} - 13960100 \nu$$$$)/7123767$$ $$\beta_{15}$$ $$=$$ $$($$$$-1394152 \nu^{15} + 12302738 \nu^{13} - 39852109 \nu^{11} + 378604518 \nu^{9} - 2284375361 \nu^{7} + 3938082987 \nu^{5} - 4375137536 \nu^{3} + 4957150856 \nu$$$$)/ 982062165$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} + \beta_{14} + \beta_{13} - \beta_{11} + \beta_{10} + \beta_{6}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{9} + \beta_{8} - \beta_{5} + 6 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_{1} + 5$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{15} + 8 \beta_{14} - 5 \beta_{13} + \beta_{12} + \beta_{11} + 2 \beta_{10} - 6 \beta_{7} - 6 \beta_{6}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$6 \beta_{9} + 13 \beta_{8} + 6 \beta_{5} - 31 \beta_{4} + 8 \beta_{3} - 7 \beta_{2} + 10 \beta_{1} - 16$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-2 \beta_{15} + 16 \beta_{14} - 17 \beta_{13} + 23 \beta_{12} - 44 \beta_{11} - 22 \beta_{10} + 10 \beta_{7} - 46 \beta_{6}$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$-11 \beta_{9} - 49 \beta_{8} + 22 \beta_{5} + 21 \beta_{4} - 53 \beta_{3} - 109 \beta_{2} - 32 \beta_{1} + 364$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$23 \beta_{15} + 119 \beta_{14} + 84 \beta_{13} - 33 \beta_{12} - 178 \beta_{11} + 178 \beta_{10} + 14 \beta_{7} + 72 \beta_{6}$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$($$$$234 \beta_{9} - 65 \beta_{8} - 117 \beta_{5} + 1292 \beta_{4} - 304 \beta_{3} + 26 \beta_{2} + 421 \beta_{1} + 1175$$$$)/3$$ $$\nu^{9}$$ $$=$$ $$($$$$927 \beta_{15} + 1470 \beta_{14} - 804 \beta_{13} - 25 \beta_{12} + 467 \beta_{11} + 934 \beta_{10} - 1025 \beta_{7} - 1148 \beta_{6}$$$$)/3$$ $$\nu^{10}$$ $$=$$ $$($$$$1148 \beta_{9} + 2191 \beta_{8} + 1148 \beta_{5} - 3861 \beta_{4} + 2165 \beta_{3} - 1043 \beta_{2} + 3182 \beta_{1} - 4330$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$-156 \beta_{15} + 2061 \beta_{14} - 4591 \beta_{13} + 5330 \beta_{12} - 5600 \beta_{11} - 2800 \beta_{10} + 1322 \beta_{7} - 12972 \beta_{6}$$$$)/3$$ $$\nu^{12}$$ $$=$$ $$-1284 \beta_{9} - 3388 \beta_{8} + 2568 \beta_{5} - 197 \beta_{4} - 890 \beta_{3} - 8060 \beta_{2} - 1087 \beta_{1} + 19179$$ $$\nu^{13}$$ $$=$$ $$($$$$-5929 \beta_{15} + 3854 \beta_{14} + 16382 \beta_{13} + 4017 \beta_{12} - 34676 \beta_{11} + 34676 \beta_{10} + 13164 \beta_{7} - 5293 \beta_{6}$$$$)/3$$ $$\nu^{14}$$ $$=$$ $$($$$$24730 \beta_{9} - 55801 \beta_{8} - 12365 \beta_{5} + 315714 \beta_{4} - 56231 \beta_{3} - 21718 \beta_{2} + 68596 \beta_{1} + 303349$$$$)/3$$ $$\nu^{15}$$ $$=$$ $$($$$$146296 \beta_{15} + 202957 \beta_{14} - 108952 \beta_{13} - 7798 \beta_{12} + 141578 \beta_{11} + 283156 \beta_{10} - 175842 \beta_{7} - 213186 \beta_{6}$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$1135$$ $$1541$$ $$\chi(n)$$ $$-1 - \beta_{4}$$ $$1$$ $$\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 −1.04556 − 0.339889i 1.30887 + 2.01944i −0.817131 − 0.735533i −2.40332 − 0.123797i 2.40332 + 0.123797i 0.817131 + 0.735533i −1.30887 − 2.01944i 1.04556 + 0.339889i −1.04556 + 0.339889i 1.30887 − 2.01944i −0.817131 + 0.735533i −2.40332 + 0.123797i 2.40332 − 0.123797i 0.817131 − 0.735533i −1.30887 + 2.01944i 1.04556 − 0.339889i
0 0 0 −3.67687 0 −1.07542 2.41733i 0 0 0
109.2 0 0 0 −2.30201 0 −2.14324 + 1.55130i 0 0 0
109.3 0 0 0 −1.03112 0 1.07542 2.41733i 0 0 0
109.4 0 0 0 −0.343737 0 2.14324 + 1.55130i 0 0 0
109.5 0 0 0 0.343737 0 2.14324 + 1.55130i 0 0 0
109.6 0 0 0 1.03112 0 1.07542 2.41733i 0 0 0
109.7 0 0 0 2.30201 0 −2.14324 + 1.55130i 0 0 0
109.8 0 0 0 3.67687 0 −1.07542 2.41733i 0 0 0
541.1 0 0 0 −3.67687 0 −1.07542 + 2.41733i 0 0 0
541.2 0 0 0 −2.30201 0 −2.14324 1.55130i 0 0 0
541.3 0 0 0 −1.03112 0 1.07542 + 2.41733i 0 0 0
541.4 0 0 0 −0.343737 0 2.14324 1.55130i 0 0 0
541.5 0 0 0 0.343737 0 2.14324 1.55130i 0 0 0
541.6 0 0 0 1.03112 0 1.07542 + 2.41733i 0 0 0
541.7 0 0 0 2.30201 0 −2.14324 1.55130i 0 0 0
541.8 0 0 0 3.67687 0 −1.07542 + 2.41733i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 541.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
63.g even 3 1 inner
63.n odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.l.n 16
3.b odd 2 1 inner 2268.2.l.n 16
7.c even 3 1 2268.2.i.n 16
9.c even 3 1 2268.2.i.n 16
9.c even 3 1 2268.2.k.g 16
9.d odd 6 1 2268.2.i.n 16
9.d odd 6 1 2268.2.k.g 16
21.h odd 6 1 2268.2.i.n 16
63.g even 3 1 inner 2268.2.l.n 16
63.h even 3 1 2268.2.k.g 16
63.j odd 6 1 2268.2.k.g 16
63.n odd 6 1 inner 2268.2.l.n 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2268.2.i.n 16 7.c even 3 1
2268.2.i.n 16 9.c even 3 1
2268.2.i.n 16 9.d odd 6 1
2268.2.i.n 16 21.h odd 6 1
2268.2.k.g 16 9.c even 3 1
2268.2.k.g 16 9.d odd 6 1
2268.2.k.g 16 63.h even 3 1
2268.2.k.g 16 63.j odd 6 1
2268.2.l.n 16 1.a even 1 1 trivial
2268.2.l.n 16 3.b odd 2 1 inner
2268.2.l.n 16 63.g even 3 1 inner
2268.2.l.n 16 63.n odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2268, [\chi])$$:

 $$T_{5}^{8} - 20 T_{5}^{6} + 94 T_{5}^{4} - 87 T_{5}^{2} + 9$$ $$T_{13}^{8} - \cdots$$ $$T_{19}^{8} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$( 9 - 87 T^{2} + 94 T^{4} - 20 T^{6} + T^{8} )^{2}$$
$7$ $$( 2401 + 245 T^{2} + 57 T^{4} + 5 T^{6} + T^{8} )^{2}$$
$11$ $$( 441 - 1428 T^{2} + 610 T^{4} - 47 T^{6} + T^{8} )^{2}$$
$13$ $$( 21609 - 18522 T + 12936 T^{2} - 3990 T^{3} + 1177 T^{4} - 152 T^{5} + 45 T^{6} - 5 T^{7} + T^{8} )^{2}$$
$17$ $$194481 + 3685878 T^{2} + 69302709 T^{4} + 10434606 T^{6} + 1056388 T^{8} + 61094 T^{10} + 2589 T^{12} + 62 T^{14} + T^{16}$$
$19$ $$( 49 - 49 T + 217 T^{2} + 224 T^{3} + 541 T^{4} + 110 T^{5} + 40 T^{6} - 4 T^{7} + T^{8} )^{2}$$
$23$ $$( 225 - 555 T^{2} + 346 T^{4} - 44 T^{6} + T^{8} )^{2}$$
$29$ $$10016218555281 + 1387681503588 T^{2} + 143040909474 T^{4} + 5476284816 T^{6} + 145682443 T^{8} + 2419664 T^{10} + 29394 T^{12} + 212 T^{14} + T^{16}$$
$31$ $$( 3969 - 2646 T + 2835 T^{2} + 210 T^{3} + 394 T^{4} + 16 T^{5} + 33 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$37$ $$( 1 - 187 T + 35029 T^{2} + 11216 T^{3} + 3973 T^{4} + 254 T^{5} + 64 T^{6} + 2 T^{7} + T^{8} )^{2}$$
$41$ $$9845600625 + 4125775500 T^{2} + 1359184050 T^{4} + 132899130 T^{6} + 9168471 T^{8} + 330426 T^{10} + 8595 T^{12} + 111 T^{14} + T^{16}$$
$43$ $$( 3200521 - 833674 T + 421102 T^{2} + 35234 T^{3} + 13537 T^{4} + 362 T^{5} + 139 T^{6} + 5 T^{7} + T^{8} )^{2}$$
$47$ $$3916747167734241 + 220418128989081 T^{2} + 8490148282260 T^{4} + 166569958683 T^{6} + 2337871491 T^{8} + 19786167 T^{10} + 121500 T^{12} + 429 T^{14} + T^{16}$$
$53$ $$1315703055681 + 261666434043 T^{2} + 38722957119 T^{4} + 2208044286 T^{6} + 89845443 T^{8} + 1772874 T^{10} + 25254 T^{12} + 192 T^{14} + T^{16}$$
$59$ $$10709131895361 + 1262179559295 T^{2} + 103135702923 T^{4} + 4068367290 T^{6} + 113967883 T^{8} + 2017010 T^{10} + 26058 T^{12} + 200 T^{14} + T^{16}$$
$61$ $$( 7017201 - 4227804 T + 2295561 T^{2} - 225792 T^{3} + 34018 T^{4} - 1862 T^{5} + 291 T^{6} - 14 T^{7} + T^{8} )^{2}$$
$67$ $$( 840889 - 616224 T + 394730 T^{2} - 58170 T^{3} + 10809 T^{4} - 786 T^{5} + 143 T^{6} - 9 T^{7} + T^{8} )^{2}$$
$71$ $$( 505521 - 419085 T^{2} + 22657 T^{4} - 305 T^{6} + T^{8} )^{2}$$
$73$ $$( 7458361 + 286755 T + 360593 T^{2} - 13440 T^{3} + 13653 T^{4} - 210 T^{5} + 128 T^{6} + T^{8} )^{2}$$
$79$ $$( 2241009 + 1001493 T + 372711 T^{2} + 63390 T^{3} + 10687 T^{4} + 838 T^{5} + 150 T^{6} + 10 T^{7} + T^{8} )^{2}$$
$83$ $$603060481873556721 + 15981999895181628 T^{2} + 276217524125802 T^{4} + 2769099363618 T^{6} + 20172185623 T^{8} + 97523354 T^{10} + 344643 T^{12} + 731 T^{14} + T^{16}$$
$89$ $$4631487063921 + 1485257767083 T^{2} + 421566649983 T^{4} + 16261945398 T^{6} + 437692167 T^{8} + 6249906 T^{10} + 64566 T^{12} + 300 T^{14} + T^{16}$$
$97$ $$( 8614225 + 3698100 T + 1611080 T^{2} + 113190 T^{3} + 23589 T^{4} - 2352 T^{5} + 449 T^{6} - 21 T^{7} + T^{8} )^{2}$$