# Properties

 Label 804.2.s.a Level 804 Weight 2 Character orbit 804.s Analytic conductor 6.420 Analytic rank 0 Dimension 20 CM discriminant -3 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$804 = 2^{2} \cdot 3 \cdot 67$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 804.s (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.41997232251$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$2$$ over $$\Q(\zeta_{22})$$ Coefficient field: $$\Q(\zeta_{33})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{22}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{33}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{33}^{3} + 2 \zeta_{33}^{14} ) q^{3} + ( -3 \zeta_{33}^{5} + 2 \zeta_{33}^{7} - \zeta_{33}^{16} - \zeta_{33}^{18} ) q^{7} -3 \zeta_{33}^{6} q^{9} +O(q^{10})$$ $$q + ( \zeta_{33}^{3} + 2 \zeta_{33}^{14} ) q^{3} + ( -3 \zeta_{33}^{5} + 2 \zeta_{33}^{7} - \zeta_{33}^{16} - \zeta_{33}^{18} ) q^{7} -3 \zeta_{33}^{6} q^{9} + ( 3 - 4 \zeta_{33} + \zeta_{33}^{2} + 3 \zeta_{33}^{3} - 4 \zeta_{33}^{4} + \zeta_{33}^{5} + 3 \zeta_{33}^{6} - 4 \zeta_{33}^{7} + \zeta_{33}^{8} + 2 \zeta_{33}^{9} + 4 \zeta_{33}^{11} - \zeta_{33}^{12} - 3 \zeta_{33}^{13} + 4 \zeta_{33}^{14} - \zeta_{33}^{15} - 3 \zeta_{33}^{16} + 4 \zeta_{33}^{17} - \zeta_{33}^{18} - 3 \zeta_{33}^{19} ) q^{13} + ( 3 \zeta_{33}^{2} + 5 \zeta_{33}^{8} + 5 \zeta_{33}^{13} + 3 \zeta_{33}^{19} ) q^{19} + ( -5 + 5 \zeta_{33}^{2} - 5 \zeta_{33}^{3} + 5 \zeta_{33}^{5} - 5 \zeta_{33}^{6} + 4 \zeta_{33}^{8} - 5 \zeta_{33}^{9} + 4 \zeta_{33}^{10} - 5 \zeta_{33}^{12} + 5 \zeta_{33}^{13} - 5 \zeta_{33}^{15} + 5 \zeta_{33}^{16} - 5 \zeta_{33}^{18} ) q^{21} -5 \zeta_{33}^{15} q^{25} + ( 6 - 6 \zeta_{33} + 6 \zeta_{33}^{3} - 6 \zeta_{33}^{4} + 6 \zeta_{33}^{6} - 6 \zeta_{33}^{7} + 3 \zeta_{33}^{9} - 6 \zeta_{33}^{10} + 6 \zeta_{33}^{11} - 6 \zeta_{33}^{13} + 6 \zeta_{33}^{14} - 6 \zeta_{33}^{16} + 6 \zeta_{33}^{17} - 6 \zeta_{33}^{19} ) q^{27} + ( 6 - 6 \zeta_{33} + 6 \zeta_{33}^{3} - 6 \zeta_{33}^{4} - \zeta_{33}^{5} + 6 \zeta_{33}^{6} - 6 \zeta_{33}^{7} + 5 \zeta_{33}^{9} - 6 \zeta_{33}^{10} + 6 \zeta_{33}^{11} - 6 \zeta_{33}^{13} + 6 \zeta_{33}^{14} - \zeta_{33}^{16} + 6 \zeta_{33}^{17} - 6 \zeta_{33}^{19} ) q^{31} + ( -4 \zeta_{33}^{4} + 4 \zeta_{33}^{7} + 3 \zeta_{33}^{15} + 7 \zeta_{33}^{18} ) q^{37} + ( -2 \zeta_{33} - 7 \zeta_{33}^{2} + 5 \zeta_{33}^{12} - 5 \zeta_{33}^{13} ) q^{39} + ( -\zeta_{33}^{3} + 7 \zeta_{33}^{6} + 6 \zeta_{33}^{14} + 6 \zeta_{33}^{17} ) q^{43} + ( -5 + 5 \zeta_{33}^{2} + 5 \zeta_{33}^{5} - 5 \zeta_{33}^{6} + 5 \zeta_{33}^{8} - 5 \zeta_{33}^{9} + 8 \zeta_{33}^{10} - 12 \zeta_{33}^{12} + 5 \zeta_{33}^{13} + 8 \zeta_{33}^{14} - 5 \zeta_{33}^{15} + 5 \zeta_{33}^{16} - 5 \zeta_{33}^{18} + 5 \zeta_{33}^{19} ) q^{49} + ( -7 - 7 \zeta_{33}^{5} - 8 \zeta_{33}^{11} + \zeta_{33}^{16} ) q^{57} + ( 4 \zeta_{33} + 5 \zeta_{33}^{6} - 5 \zeta_{33}^{12} - 4 \zeta_{33}^{17} ) q^{61} + ( -3 - 3 \zeta_{33}^{2} + 6 \zeta_{33}^{11} - 9 \zeta_{33}^{13} ) q^{63} + ( 2 \zeta_{33}^{4} + 9 \zeta_{33}^{15} ) q^{67} + ( 1 - 8 \zeta_{33}^{7} - 8 \zeta_{33}^{11} + \zeta_{33}^{18} ) q^{73} + ( 10 \zeta_{33}^{7} + 5 \zeta_{33}^{18} ) q^{75} + ( 7 + 7 \zeta_{33}^{8} + 10 \zeta_{33}^{11} - 3 \zeta_{33}^{19} ) q^{79} + 9 \zeta_{33}^{12} q^{81} + ( -9 \zeta_{33}^{3} + 6 \zeta_{33}^{4} + 11 \zeta_{33}^{5} + \zeta_{33}^{6} - 10 \zeta_{33}^{14} - 5 \zeta_{33}^{15} + 5 \zeta_{33}^{16} + 10 \zeta_{33}^{17} ) q^{91} + ( -4 \zeta_{33} - 11 \zeta_{33}^{8} + 7 \zeta_{33}^{12} - 7 \zeta_{33}^{19} ) q^{93} + ( 3 \zeta_{33}^{3} + 3 \zeta_{33}^{8} - 8 \zeta_{33}^{14} + 11 \zeta_{33}^{19} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20q + 6q^{9} + O(q^{10})$$ $$20q + 6q^{9} + 16q^{19} - 12q^{21} + 10q^{25} - 20q^{37} - 24q^{39} + 10q^{49} - 66q^{57} - 132q^{63} - 16q^{67} + 90q^{73} + 44q^{79} - 18q^{81} + 48q^{91} - 36q^{93} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/804\mathbb{Z}\right)^\times$$.

 $$n$$ $$269$$ $$337$$ $$403$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{33}^{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 −0.786053 + 0.618159i 0.928368 + 0.371662i 0.981929 + 0.189251i −0.327068 − 0.945001i −0.995472 − 0.0950560i 0.580057 − 0.814576i 0.0475819 − 0.998867i −0.888835 + 0.458227i −0.786053 − 0.618159i 0.928368 − 0.371662i 0.723734 − 0.690079i 0.235759 + 0.971812i 0.723734 + 0.690079i 0.235759 − 0.971812i −0.995472 + 0.0950560i 0.580057 + 0.814576i 0.0475819 + 0.998867i −0.888835 − 0.458227i 0.981929 − 0.189251i −0.327068 + 0.945001i
0 −1.57553 + 0.719520i 0 0 0 −3.36481 2.91562i 0 1.96458 2.26725i 0
5.2 0 1.57553 0.719520i 0 0 0 −2.61965 2.26994i 0 1.96458 2.26725i 0
53.1 0 −0.936417 + 1.45709i 0 0 0 0.686312 0.313428i 0 −1.24625 2.72890i 0
53.2 0 0.936417 1.45709i 0 0 0 4.81332 2.19817i 0 −1.24625 2.72890i 0
125.1 0 −0.487975 + 1.66189i 0 0 0 1.18913 1.85033i 0 −2.52376 1.62192i 0
125.2 0 0.487975 1.66189i 0 0 0 2.74514 4.27152i 0 −2.52376 1.62192i 0
137.1 0 −1.71442 0.246497i 0 0 0 −1.43029 + 4.87111i 0 2.87848 + 0.845198i 0
137.2 0 1.71442 + 0.246497i 0 0 0 0.211757 0.721178i 0 2.87848 + 0.845198i 0
161.1 0 −1.57553 0.719520i 0 0 0 −3.36481 + 2.91562i 0 1.96458 + 2.26725i 0
161.2 0 1.57553 + 0.719520i 0 0 0 −2.61965 + 2.26994i 0 1.96458 + 2.26725i 0
209.1 0 −1.30900 + 1.13425i 0 0 0 2.17449 + 0.312644i 0 0.426945 2.96946i 0
209.2 0 1.30900 1.13425i 0 0 0 −4.40541 0.633402i 0 0.426945 2.96946i 0
377.1 0 −1.30900 1.13425i 0 0 0 2.17449 0.312644i 0 0.426945 + 2.96946i 0
377.2 0 1.30900 + 1.13425i 0 0 0 −4.40541 + 0.633402i 0 0.426945 + 2.96946i 0
521.1 0 −0.487975 1.66189i 0 0 0 1.18913 + 1.85033i 0 −2.52376 + 1.62192i 0
521.2 0 0.487975 + 1.66189i 0 0 0 2.74514 + 4.27152i 0 −2.52376 + 1.62192i 0
581.1 0 −1.71442 + 0.246497i 0 0 0 −1.43029 4.87111i 0 2.87848 0.845198i 0
581.2 0 1.71442 0.246497i 0 0 0 0.211757 + 0.721178i 0 2.87848 0.845198i 0
713.1 0 −0.936417 1.45709i 0 0 0 0.686312 + 0.313428i 0 −1.24625 + 2.72890i 0
713.2 0 0.936417 + 1.45709i 0 0 0 4.81332 + 2.19817i 0 −1.24625 + 2.72890i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 713.2 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 CM by $$\Q(\sqrt{-3})$$
67.f odd 22 1 inner
201.j even 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 804.2.s.a 20
3.b odd 2 1 CM 804.2.s.a 20
67.f odd 22 1 inner 804.2.s.a 20
201.j even 22 1 inner 804.2.s.a 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
804.2.s.a 20 1.a even 1 1 trivial
804.2.s.a 20 3.b odd 2 1 CM
804.2.s.a 20 67.f odd 22 1 inner
804.2.s.a 20 201.j even 22 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}$$ acting on $$S_{2}^{\mathrm{new}}(804, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 - 3 T^{2} + 9 T^{4} - 27 T^{6} + 81 T^{8} - 243 T^{10} + 729 T^{12} - 2187 T^{14} + 6561 T^{16} - 19683 T^{18} + 59049 T^{20}$$
$5$ $$( 1 - 5 T^{2} + 25 T^{4} - 125 T^{6} + 625 T^{8} - 3125 T^{10} + 15625 T^{12} - 78125 T^{14} + 390625 T^{16} - 1953125 T^{18} + 9765625 T^{20} )^{2}$$
$7$ $$( 1 - 4 T + 9 T^{2} - 8 T^{3} - 31 T^{4} + 180 T^{5} - 503 T^{6} + 752 T^{7} + 513 T^{8} - 7316 T^{9} + 25673 T^{10} - 51212 T^{11} + 25137 T^{12} + 257936 T^{13} - 1207703 T^{14} + 3025260 T^{15} - 3647119 T^{16} - 6588344 T^{17} + 51883209 T^{18} - 161414428 T^{19} + 282475249 T^{20} )( 1 + 4 T + 9 T^{2} + 8 T^{3} - 31 T^{4} - 180 T^{5} - 503 T^{6} - 752 T^{7} + 513 T^{8} + 7316 T^{9} + 25673 T^{10} + 51212 T^{11} + 25137 T^{12} - 257936 T^{13} - 1207703 T^{14} - 3025260 T^{15} - 3647119 T^{16} + 6588344 T^{17} + 51883209 T^{18} + 161414428 T^{19} + 282475249 T^{20} )$$
$11$ $$( 1 - 11 T + 55 T^{2} - 121 T^{3} - 121 T^{4} + 1331 T^{5} - 1331 T^{6} - 14641 T^{7} + 73205 T^{8} - 161051 T^{9} + 161051 T^{10} )^{2}( 1 + 11 T + 55 T^{2} + 121 T^{3} - 121 T^{4} - 1331 T^{5} - 1331 T^{6} + 14641 T^{7} + 73205 T^{8} + 161051 T^{9} + 161051 T^{10} )^{2}$$
$13$ $$( 1 - 2 T - 9 T^{2} + 44 T^{3} + 29 T^{4} - 630 T^{5} + 883 T^{6} + 6424 T^{7} - 24327 T^{8} - 34858 T^{9} + 385967 T^{10} - 453154 T^{11} - 4111263 T^{12} + 14113528 T^{13} + 25219363 T^{14} - 233914590 T^{15} + 139977461 T^{16} + 2760934748 T^{17} - 7341576489 T^{18} - 21208998746 T^{19} + 137858491849 T^{20} )( 1 + 2 T - 9 T^{2} - 44 T^{3} + 29 T^{4} + 630 T^{5} + 883 T^{6} - 6424 T^{7} - 24327 T^{8} + 34858 T^{9} + 385967 T^{10} + 453154 T^{11} - 4111263 T^{12} - 14113528 T^{13} + 25219363 T^{14} + 233914590 T^{15} + 139977461 T^{16} - 2760934748 T^{17} - 7341576489 T^{18} + 21208998746 T^{19} + 137858491849 T^{20} )$$
$17$ $$( 1 + 17 T^{2} + 289 T^{4} + 4913 T^{6} + 83521 T^{8} + 1419857 T^{10} + 24137569 T^{12} + 410338673 T^{14} + 6975757441 T^{16} + 118587876497 T^{18} + 2015993900449 T^{20} )^{2}$$
$19$ $$( 1 - 8 T + 45 T^{2} - 208 T^{3} + 809 T^{4} - 2520 T^{5} + 4789 T^{6} + 9568 T^{7} - 167535 T^{8} + 1158488 T^{9} - 6084739 T^{10} + 22011272 T^{11} - 60480135 T^{12} + 65626912 T^{13} + 624107269 T^{14} - 6239769480 T^{15} + 38060117729 T^{16} - 185925321712 T^{17} + 764260336845 T^{18} - 2581501582232 T^{19} + 6131066257801 T^{20} )^{2}$$
$23$ $$( 1 + 23 T^{2} + 529 T^{4} + 12167 T^{6} + 279841 T^{8} + 6436343 T^{10} + 148035889 T^{12} + 3404825447 T^{14} + 78310985281 T^{16} + 1801152661463 T^{18} + 41426511213649 T^{20} )^{2}$$
$29$ $$( 1 - 29 T^{2} )^{20}$$
$31$ $$( 1 - 4 T - 15 T^{2} + 184 T^{3} - 271 T^{4} - 4620 T^{5} + 26881 T^{6} + 35696 T^{7} - 976095 T^{8} + 2797804 T^{9} + 19067729 T^{10} + 86731924 T^{11} - 938027295 T^{12} + 1063419536 T^{13} + 24825168001 T^{14} - 132266677620 T^{15} - 240513497551 T^{16} + 5062320996424 T^{17} - 12793365561615 T^{18} - 105758488642684 T^{19} + 819628286980801 T^{20} )( 1 + 4 T - 15 T^{2} - 184 T^{3} - 271 T^{4} + 4620 T^{5} + 26881 T^{6} - 35696 T^{7} - 976095 T^{8} - 2797804 T^{9} + 19067729 T^{10} - 86731924 T^{11} - 938027295 T^{12} - 1063419536 T^{13} + 24825168001 T^{14} + 132266677620 T^{15} - 240513497551 T^{16} - 5062320996424 T^{17} - 12793365561615 T^{18} + 105758488642684 T^{19} + 819628286980801 T^{20} )$$
$37$ $$( 1 + 10 T + 63 T^{2} + 260 T^{3} + 269 T^{4} - 6930 T^{5} - 79253 T^{6} - 536120 T^{7} - 2428839 T^{8} - 4451950 T^{9} + 45347543 T^{10} - 164722150 T^{11} - 3325080591 T^{12} - 27156086360 T^{13} - 148532881733 T^{14} - 480553622010 T^{15} + 690180404021 T^{16} + 24682288054580 T^{17} + 221286205597023 T^{18} + 1299617397950770 T^{19} + 4808584372417849 T^{20} )^{2}$$
$41$ $$( 1 - 41 T^{2} + 1681 T^{4} - 68921 T^{6} + 2825761 T^{8} - 115856201 T^{10} + 4750104241 T^{12} - 194754273881 T^{14} + 7984925229121 T^{16} - 327381934393961 T^{18} + 13422659310152401 T^{20} )^{2}$$
$43$ $$( 1 - 8 T + 21 T^{2} + 176 T^{3} - 2311 T^{4} + 10920 T^{5} + 12013 T^{6} - 565664 T^{7} + 4008753 T^{8} - 7746472 T^{9} - 110404603 T^{10} - 333098296 T^{11} + 7412184297 T^{12} - 44974247648 T^{13} + 41070056413 T^{14} + 1605332197560 T^{15} - 14608670006239 T^{16} + 47840075554832 T^{17} + 245452205829621 T^{18} - 4020740895494744 T^{19} + 21611482313284249 T^{20} )( 1 + 8 T + 21 T^{2} - 176 T^{3} - 2311 T^{4} - 10920 T^{5} + 12013 T^{6} + 565664 T^{7} + 4008753 T^{8} + 7746472 T^{9} - 110404603 T^{10} + 333098296 T^{11} + 7412184297 T^{12} + 44974247648 T^{13} + 41070056413 T^{14} - 1605332197560 T^{15} - 14608670006239 T^{16} - 47840075554832 T^{17} + 245452205829621 T^{18} + 4020740895494744 T^{19} + 21611482313284249 T^{20} )$$
$47$ $$( 1 + 47 T^{2} + 2209 T^{4} + 103823 T^{6} + 4879681 T^{8} + 229345007 T^{10} + 10779215329 T^{12} + 506623120463 T^{14} + 23811286661761 T^{16} + 1119130473102767 T^{18} + 52599132235830049 T^{20} )^{2}$$
$53$ $$( 1 - 53 T^{2} + 2809 T^{4} - 148877 T^{6} + 7890481 T^{8} - 418195493 T^{10} + 22164361129 T^{12} - 1174711139837 T^{14} + 62259690411361 T^{16} - 3299763591802133 T^{18} + 174887470365513049 T^{20} )^{2}$$
$59$ $$( 1 + 59 T^{2} + 3481 T^{4} + 205379 T^{6} + 12117361 T^{8} + 714924299 T^{10} + 42180533641 T^{12} + 2488651484819 T^{14} + 146830437604321 T^{16} + 8662995818654939 T^{18} + 511116753300641401 T^{20} )^{2}$$
$61$ $$( 1 - 14 T + 135 T^{2} - 1036 T^{3} + 6269 T^{4} - 24570 T^{5} - 38429 T^{6} + 2036776 T^{7} - 26170695 T^{8} + 242146394 T^{9} - 1793637121 T^{10} + 14770930034 T^{11} - 97381156095 T^{12} + 462309453256 T^{13} - 532081823789 T^{14} - 20751731115570 T^{15} + 322981226869109 T^{16} - 3255881578117756 T^{17} + 25880487254632935 T^{18} - 163718045299677974 T^{19} + 713342911662882601 T^{20} )( 1 + 14 T + 135 T^{2} + 1036 T^{3} + 6269 T^{4} + 24570 T^{5} - 38429 T^{6} - 2036776 T^{7} - 26170695 T^{8} - 242146394 T^{9} - 1793637121 T^{10} - 14770930034 T^{11} - 97381156095 T^{12} - 462309453256 T^{13} - 532081823789 T^{14} + 20751731115570 T^{15} + 322981226869109 T^{16} + 3255881578117756 T^{17} + 25880487254632935 T^{18} + 163718045299677974 T^{19} + 713342911662882601 T^{20} )$$
$67$ $$1 + 16 T + 189 T^{2} + 1952 T^{3} + 18569 T^{4} + 166320 T^{5} + 1416997 T^{6} + 11528512 T^{7} + 89517393 T^{8} + 659867984 T^{9} + 4560222413 T^{10} + 44211154928 T^{11} + 401843577177 T^{12} + 3467349854656 T^{13} + 28554078003637 T^{14} + 224552807796240 T^{15} + 1679721698496161 T^{16} + 11830509053590496 T^{17} + 76746791058205149 T^{18} + 435304550340719152 T^{19} + 1822837804551761449 T^{20}$$
$71$ $$( 1 + 71 T^{2} + 5041 T^{4} + 357911 T^{6} + 25411681 T^{8} + 1804229351 T^{10} + 128100283921 T^{12} + 9095120158391 T^{14} + 645753531245761 T^{16} + 45848500718449031 T^{18} + 3255243551009881201 T^{20} )^{2}$$
$73$ $$( 1 - 10 T + 73 T^{2} )^{10}( 1 + 10 T + 27 T^{2} - 460 T^{3} - 6571 T^{4} - 32130 T^{5} + 158383 T^{6} + 3929320 T^{7} + 27731241 T^{8} - 9527950 T^{9} - 2119660093 T^{10} - 695540350 T^{11} + 147779783289 T^{12} + 1528572278440 T^{13} + 4497798604303 T^{14} - 66607790283090 T^{15} - 994417200945019 T^{16} - 5081803318784620 T^{17} + 21774422481140187 T^{18} + 588715867082679130 T^{19} + 4297625829703557649 T^{20} )$$
$79$ $$( 1 - 4 T + 79 T^{2} )^{10}( 1 - 4 T - 63 T^{2} + 568 T^{3} + 2705 T^{4} - 55692 T^{5} + 9073 T^{6} + 4363376 T^{7} - 18170271 T^{8} - 272025620 T^{9} + 2523553889 T^{10} - 21490023980 T^{11} - 113400661311 T^{12} + 2151314539664 T^{13} + 353394084913 T^{14} - 171367424973108 T^{15} + 657551567184305 T^{16} + 10907820304138312 T^{17} - 95577855024113343 T^{18} - 479406383930473276 T^{19} + 9468276082626847201 T^{20} )$$
$83$ $$( 1 + 83 T^{2} + 6889 T^{4} + 571787 T^{6} + 47458321 T^{8} + 3939040643 T^{10} + 326940373369 T^{12} + 27136050989627 T^{14} + 2252292232139041 T^{16} + 186940255267540403 T^{18} + 15516041187205853449 T^{20} )^{2}$$
$89$ $$( 1 + 89 T^{2} + 7921 T^{4} + 704969 T^{6} + 62742241 T^{8} + 5584059449 T^{10} + 496981290961 T^{12} + 44231334895529 T^{14} + 3936588805702081 T^{16} + 350356403707485209 T^{18} + 31181719929966183601 T^{20} )^{2}$$
$97$ $$( 1 - 14 T + 99 T^{2} - 28 T^{3} - 9211 T^{4} + 131670 T^{5} - 949913 T^{6} + 526792 T^{7} + 84766473 T^{8} - 1237829446 T^{9} + 9107264363 T^{10} - 120069456262 T^{11} + 797567744457 T^{12} + 480788835016 T^{13} - 84095114902553 T^{14} + 1130695091639190 T^{15} - 7672505137401019 T^{16} - 2262351965387164 T^{17} + 775905925843319139 T^{18} - 10643234821163913038 T^{19} + 73742412689492826049 T^{20} )( 1 + 14 T + 99 T^{2} + 28 T^{3} - 9211 T^{4} - 131670 T^{5} - 949913 T^{6} - 526792 T^{7} + 84766473 T^{8} + 1237829446 T^{9} + 9107264363 T^{10} + 120069456262 T^{11} + 797567744457 T^{12} - 480788835016 T^{13} - 84095114902553 T^{14} - 1130695091639190 T^{15} - 7672505137401019 T^{16} + 2262351965387164 T^{17} + 775905925843319139 T^{18} + 10643234821163913038 T^{19} + 73742412689492826049 T^{20} )$$