# Properties

 Label 2592.2.s.j Level $2592$ Weight $2$ Character orbit 2592.s Analytic conductor $20.697$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2592 = 2^{5} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2592.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$20.6972242039$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 12 x^{14} + 49 x^{12} - 12 x^{10} - 600 x^{8} + 108 x^{6} + 4057 x^{4} + 18252 x^{2} + 28561$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{5} - \beta_{9} + \beta_{12} + \beta_{14} + \beta_{15} ) q^{5} + ( -\beta_{2} - \beta_{6} + 2 \beta_{8} + \beta_{10} ) q^{7} +O(q^{10})$$ $$q + ( -\beta_{5} - \beta_{9} + \beta_{12} + \beta_{14} + \beta_{15} ) q^{5} + ( -\beta_{2} - \beta_{6} + 2 \beta_{8} + \beta_{10} ) q^{7} + ( 2 \beta_{4} - \beta_{11} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{11} + ( 1 + \beta_{2} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{10} ) q^{13} + ( -3 \beta_{4} + \beta_{5} + 2 \beta_{11} - 3 \beta_{12} - \beta_{13} - 3 \beta_{14} + \beta_{15} ) q^{17} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} - 3 \beta_{8} ) q^{19} + ( \beta_{4} + \beta_{5} - \beta_{9} - \beta_{14} + \beta_{15} ) q^{23} + ( 3 + 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{6} - \beta_{7} - 3 \beta_{8} ) q^{25} + ( \beta_{4} - \beta_{9} + 2 \beta_{12} + 2 \beta_{14} - 2 \beta_{15} ) q^{29} + ( 6 - \beta_{1} - \beta_{6} - 3 \beta_{8} - \beta_{10} ) q^{31} + ( -5 \beta_{4} - \beta_{5} - 2 \beta_{9} + 3 \beta_{12} + 7 \beta_{14} + 4 \beta_{15} ) q^{35} + ( 4 - \beta_{2} - \beta_{3} - \beta_{6} + \beta_{8} ) q^{37} + ( \beta_{4} + \beta_{5} + \beta_{9} + \beta_{11} - 4 \beta_{12} - 2 \beta_{13} ) q^{41} + ( 4 + \beta_{2} + 4 \beta_{3} + \beta_{6} + 2 \beta_{8} - \beta_{10} ) q^{43} + ( -2 \beta_{5} - \beta_{9} - \beta_{11} - \beta_{12} + \beta_{13} - 3 \beta_{14} - 2 \beta_{15} ) q^{47} + ( 3 \beta_{1} - 6 \beta_{3} - \beta_{6} + 4 \beta_{8} + \beta_{10} ) q^{49} + ( -2 \beta_{4} - \beta_{5} - 2 \beta_{11} - \beta_{12} + \beta_{13} - 2 \beta_{14} ) q^{53} + ( -1 - 8 \beta_{1} + 2 \beta_{2} + 10 \beta_{3} - \beta_{6} - \beta_{7} + 4 \beta_{8} - \beta_{10} ) q^{55} + ( 4 \beta_{4} - \beta_{5} + \beta_{9} - \beta_{11} + 5 \beta_{12} + 3 \beta_{14} - 3 \beta_{15} ) q^{59} + ( 3 + 8 \beta_{1} + \beta_{2} - 4 \beta_{3} - \beta_{6} + 2 \beta_{7} - 3 \beta_{8} + \beta_{10} ) q^{61} + ( 3 \beta_{4} + 5 \beta_{12} + 2 \beta_{14} - 5 \beta_{15} ) q^{65} + ( 3 - 2 \beta_{1} + \beta_{2} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{10} ) q^{67} + ( \beta_{4} + \beta_{5} + 2 \beta_{9} + \beta_{12} - 3 \beta_{14} - 4 \beta_{15} ) q^{71} + ( 6 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{7} - \beta_{8} - \beta_{10} ) q^{73} + ( 4 \beta_{4} + \beta_{5} + \beta_{9} - \beta_{11} - 3 \beta_{12} + 2 \beta_{13} + 3 \beta_{14} + 3 \beta_{15} ) q^{77} + ( 2 - \beta_{1} + 2 \beta_{2} + 7 \beta_{3} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{10} ) q^{79} + ( 4 \beta_{4} + 4 \beta_{5} + 2 \beta_{9} + 4 \beta_{11} - 10 \beta_{12} - 4 \beta_{13} - 6 \beta_{15} ) q^{83} + ( -3 + 8 \beta_{1} - 3 \beta_{2} - 16 \beta_{3} - 2 \beta_{6} + 3 \beta_{7} + 5 \beta_{8} + 2 \beta_{10} ) q^{85} + ( -\beta_{4} - 2 \beta_{5} - 4 \beta_{11} + 4 \beta_{12} + 2 \beta_{13} - \beta_{14} + 3 \beta_{15} ) q^{89} + ( 3 - 8 \beta_{1} + 8 \beta_{3} - \beta_{6} + \beta_{7} - 6 \beta_{8} + \beta_{10} ) q^{91} + ( -\beta_{4} - 2 \beta_{5} + 2 \beta_{9} - \beta_{11} + \beta_{12} - 12 \beta_{14} + 12 \beta_{15} ) q^{95} + ( 2 + 5 \beta_{1} + 3 \beta_{2} - \beta_{3} - 3 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 12 q^{7} + O(q^{10})$$ $$16 q + 12 q^{7} + 4 q^{13} + 24 q^{25} + 72 q^{31} + 72 q^{37} + 84 q^{43} + 24 q^{49} + 28 q^{61} + 36 q^{67} + 96 q^{73} + 12 q^{79} + 12 q^{85} + 16 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 12 x^{14} + 49 x^{12} - 12 x^{10} - 600 x^{8} + 108 x^{6} + 4057 x^{4} + 18252 x^{2} + 28561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$4506447 \nu^{14} - 121241175 \nu^{12} + 1034556480 \nu^{10} - 3947923892 \nu^{8} + 2931273884 \nu^{6} + 20300847776 \nu^{4} + 37148084919 \nu^{2} - 42326241815$$$$)/ 228043751728$$ $$\beta_{2}$$ $$=$$ $$($$$$-206016 \nu^{14} - 375731 \nu^{12} + 17700672 \nu^{10} - 197309312 \nu^{8} + 824440588 \nu^{6} - 1211039720 \nu^{4} + 1959061112 \nu^{2} + 8568605045$$$$)/ 8770913528$$ $$\beta_{3}$$ $$=$$ $$($$$$-13282279 \nu^{14} + 200884791 \nu^{12} - 1177246560 \nu^{10} + 3219861076 \nu^{8} - 159249356 \nu^{6} - 4656520480 \nu^{4} + 8443725137 \nu^{2} - 296484914921$$$$)/ 228043751728$$ $$\beta_{4}$$ $$=$$ $$($$$$19681995 \nu^{15} - 459692017 \nu^{13} + 3195349744 \nu^{11} - 7267451924 \nu^{9} - 31251487660 \nu^{7} + 263429498832 \nu^{5} - 261204590093 \nu^{3} + 67134153359 \nu$$$$)/ 2964568772464$$ $$\beta_{5}$$ $$=$$ $$($$$$5080251 \nu^{15} + 21986582 \nu^{13} + 260696896 \nu^{11} - 2899209176 \nu^{9} + 7686905160 \nu^{7} + 39732514304 \nu^{5} - 234381538917 \nu^{3} - 1205684410658 \nu$$$$)/ 741142193116$$ $$\beta_{6}$$ $$=$$ $$($$$$-4209 \nu^{14} + 71399 \nu^{12} - 338828 \nu^{10} + 257052 \nu^{8} + 4901436 \nu^{6} - 13795796 \nu^{4} - 37041417 \nu^{2} - 83054881$$$$)/43420364$$ $$\beta_{7}$$ $$=$$ $$($$$$11703162 \nu^{14} - 120180421 \nu^{12} + 499243320 \nu^{10} - 345490376 \nu^{8} - 16177284 \nu^{6} - 27206713808 \nu^{4} + 59498003698 \nu^{2} + 202336233411$$$$)/ 114021875864$$ $$\beta_{8}$$ $$=$$ $$($$$$30695 \nu^{14} - 419209 \nu^{12} + 2520928 \nu^{10} - 7255428 \nu^{8} + 1602740 \nu^{6} + 16353072 \nu^{4} - 2947761 \nu^{2} + 379906423$$$$)/ 289763344$$ $$\beta_{9}$$ $$=$$ $$($$$$-16691371 \nu^{15} + 405805691 \nu^{13} - 5421908520 \nu^{11} + 32843046644 \nu^{9} - 97384894396 \nu^{7} + 44135143800 \nu^{5} + 312178275453 \nu^{3} + 175474228267 \nu$$$$)/ 1482284386232$$ $$\beta_{10}$$ $$=$$ $$($$$$25286987 \nu^{14} - 353399906 \nu^{12} + 1901413416 \nu^{10} - 4117794124 \nu^{8} - 7018524912 \nu^{6} - 521098768 \nu^{4} + 141521319795 \nu^{2} + 313392480206$$$$)/ 114021875864$$ $$\beta_{11}$$ $$=$$ $$($$$$-5281529 \nu^{15} - 16310755 \nu^{13} + 607738864 \nu^{11} - 4165926004 \nu^{9} + 9763253772 \nu^{7} + 21841421760 \nu^{5} - 59135450497 \nu^{3} - 198374633795 \nu$$$$)/ 228043751728$$ $$\beta_{12}$$ $$=$$ $$($$$$-599257 \nu^{15} + 8739462 \nu^{13} - 46755552 \nu^{11} + 102847788 \nu^{9} + 81288264 \nu^{7} + 258200712 \nu^{5} - 5035326929 \nu^{3} - 9359839554 \nu$$$$)/ 14676083032$$ $$\beta_{13}$$ $$=$$ $$($$$$144082388 \nu^{15} - 2115255563 \nu^{13} + 13246128560 \nu^{11} - 45286056288 \nu^{9} + 60022314908 \nu^{7} - 64125150616 \nu^{5} - 1751936724 \nu^{3} + 1840873477261 \nu$$$$)/ 1482284386232$$ $$\beta_{14}$$ $$=$$ $$($$$$319559619 \nu^{15} - 4593759155 \nu^{13} + 25201681824 \nu^{11} - 52599749252 \nu^{9} - 109014107700 \nu^{7} + 256060089488 \nu^{5} + 1338327702155 \nu^{3} + 3620395773293 \nu$$$$)/ 2964568772464$$ $$\beta_{15}$$ $$=$$ $$($$$$1582155 \nu^{15} - 21956880 \nu^{13} + 119050416 \nu^{11} - 229166428 \nu^{9} - 584143488 \nu^{7} + 1504637640 \nu^{5} + 4359965067 \nu^{3} + 14779694904 \nu$$$$)/ 14676083032$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{15} + \beta_{14} + \beta_{12} - \beta_{11} - \beta_{9} - \beta_{5} + 2 \beta_{4}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{8} + \beta_{7} - \beta_{6} + 7 \beta_{3} + \beta_{2} + \beta_{1} + 2$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$-3 \beta_{15} + \beta_{14} - \beta_{13} - 6 \beta_{12} - \beta_{11} - \beta_{9} + \beta_{5} + 4 \beta_{4}$$ $$\nu^{4}$$ $$=$$ $$($$$$-9 \beta_{10} + 21 \beta_{8} + 4 \beta_{7} - 9 \beta_{6} + 24 \beta_{3} + 4 \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-10 \beta_{15} - 27 \beta_{14} + 3 \beta_{13} - 67 \beta_{12} - 13 \beta_{11} - 3 \beta_{9} + 10 \beta_{5} + 84 \beta_{4}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-31 \beta_{10} + 25 \beta_{8} + 31 \beta_{7} - 20 \beta_{6} + 16 \beta_{3} + 11 \beta_{2} + 15 \beta_{1} - 28$$ $$\nu^{7}$$ $$=$$ $$($$$$247 \beta_{15} - 453 \beta_{14} + 16 \beta_{13} - 453 \beta_{12} - 13 \beta_{11} - 3 \beta_{9} + 13 \beta_{5} + 260 \beta_{4}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$-260 \beta_{10} - 172 \beta_{8} + 363 \beta_{7} - 103 \beta_{6} - 289 \beta_{3} - 103 \beta_{2} + 475 \beta_{1} - 88$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$1274 \beta_{15} - 1413 \beta_{14} - 141 \beta_{13} - 753 \beta_{12} + 93 \beta_{11} - 93 \beta_{9} - 141 \beta_{5}$$ $$\nu^{10}$$ $$=$$ $$($$$$-473 \beta_{10} - 1595 \beta_{8} + 1320 \beta_{7} + 473 \beta_{6} - 1320 \beta_{3} - 1320 \beta_{2} + 3863 \beta_{1} + 2244$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$12650 \beta_{15} - 9635 \beta_{14} - 3389 \beta_{13} - 947 \beta_{12} + 947 \beta_{11} - 3389 \beta_{9} - 2442 \beta_{5} - 5104 \beta_{4}$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$896 \beta_{10} + 896 \beta_{8} + 896 \beta_{7} + 2552 \beta_{6} + 6156 \beta_{3} - 3448 \beta_{2} + 7052 \beta_{1} + 12335$$ $$\nu^{13}$$ $$=$$ $$($$$$23781 \beta_{15} - 2821 \beta_{14} - 19208 \beta_{13} + 16971 \beta_{12} - 7075 \beta_{11} - 26283 \beta_{9} - 7075 \beta_{5} - 16706 \beta_{4}$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$($$$$9896 \beta_{10} + 120086 \beta_{8} - 3405 \beta_{7} + 13301 \beta_{6} + 194461 \beta_{3} - 13301 \beta_{2} - 13301 \beta_{1} + 110190$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$-53817 \beta_{15} + 60043 \beta_{14} - 21919 \beta_{13} + 4418 \beta_{12} - 60043 \beta_{11} - 60043 \beta_{9} + 21919 \beta_{5} + 63796 \beta_{4}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$1217$$ $$2431$$ $$\chi(n)$$ $$1$$ $$\beta_{8}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
863.1
 −2.13219 + 1.08896i 0.115299 − 1.50155i −2.39101 + 0.123030i −0.850627 − 1.24273i 0.850627 + 1.24273i 2.39101 − 0.123030i −0.115299 + 1.50155i 2.13219 − 1.08896i −2.13219 − 1.08896i 0.115299 + 1.50155i −2.39101 − 0.123030i −0.850627 + 1.24273i 0.850627 − 1.24273i 2.39101 + 0.123030i −0.115299 − 1.50155i 2.13219 + 1.08896i
0 0 0 −3.47996 2.00916i 0 2.66739 1.54002i 0 0 0
863.2 0 0 0 −2.35218 1.35803i 0 3.67803 2.12351i 0 0 0
863.3 0 0 0 −2.25522 1.30205i 0 −0.301361 + 0.173991i 0 0 0
863.4 0 0 0 −1.12743 0.650924i 0 −3.04406 + 1.75749i 0 0 0
863.5 0 0 0 1.12743 + 0.650924i 0 −3.04406 + 1.75749i 0 0 0
863.6 0 0 0 2.25522 + 1.30205i 0 −0.301361 + 0.173991i 0 0 0
863.7 0 0 0 2.35218 + 1.35803i 0 3.67803 2.12351i 0 0 0
863.8 0 0 0 3.47996 + 2.00916i 0 2.66739 1.54002i 0 0 0
1727.1 0 0 0 −3.47996 + 2.00916i 0 2.66739 + 1.54002i 0 0 0
1727.2 0 0 0 −2.35218 + 1.35803i 0 3.67803 + 2.12351i 0 0 0
1727.3 0 0 0 −2.25522 + 1.30205i 0 −0.301361 0.173991i 0 0 0
1727.4 0 0 0 −1.12743 + 0.650924i 0 −3.04406 1.75749i 0 0 0
1727.5 0 0 0 1.12743 0.650924i 0 −3.04406 1.75749i 0 0 0
1727.6 0 0 0 2.25522 1.30205i 0 −0.301361 0.173991i 0 0 0
1727.7 0 0 0 2.35218 1.35803i 0 3.67803 + 2.12351i 0 0 0
1727.8 0 0 0 3.47996 2.00916i 0 2.66739 + 1.54002i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1727.8 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
36.f odd 6 1 inner
36.h even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2592.2.s.j 16
3.b odd 2 1 inner 2592.2.s.j 16
4.b odd 2 1 2592.2.s.i 16
9.c even 3 1 2592.2.c.b 16
9.c even 3 1 2592.2.s.i 16
9.d odd 6 1 2592.2.c.b 16
9.d odd 6 1 2592.2.s.i 16
12.b even 2 1 2592.2.s.i 16
36.f odd 6 1 2592.2.c.b 16
36.f odd 6 1 inner 2592.2.s.j 16
36.h even 6 1 2592.2.c.b 16
36.h even 6 1 inner 2592.2.s.j 16
72.j odd 6 1 5184.2.c.l 16
72.l even 6 1 5184.2.c.l 16
72.n even 6 1 5184.2.c.l 16
72.p odd 6 1 5184.2.c.l 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2592.2.c.b 16 9.c even 3 1
2592.2.c.b 16 9.d odd 6 1
2592.2.c.b 16 36.f odd 6 1
2592.2.c.b 16 36.h even 6 1
2592.2.s.i 16 4.b odd 2 1
2592.2.s.i 16 9.c even 3 1
2592.2.s.i 16 9.d odd 6 1
2592.2.s.i 16 12.b even 2 1
2592.2.s.j 16 1.a even 1 1 trivial
2592.2.s.j 16 3.b odd 2 1 inner
2592.2.s.j 16 36.f odd 6 1 inner
2592.2.s.j 16 36.h even 6 1 inner
5184.2.c.l 16 72.j odd 6 1
5184.2.c.l 16 72.l even 6 1
5184.2.c.l 16 72.n even 6 1
5184.2.c.l 16 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2592, [\chi])$$:

 $$T_{5}^{16} - \cdots$$ $$T_{7}^{8} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$1874161 - 1752320 T^{2} + 1186630 T^{4} - 334784 T^{6} + 66571 T^{8} - 8000 T^{10} + 694 T^{12} - 32 T^{14} + T^{16}$$
$7$ $$( 256 + 1152 T + 1504 T^{2} - 1008 T^{3} + 36 T^{4} + 84 T^{5} - 2 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$11$ $$65536 + 180224 T^{2} + 412672 T^{4} + 205568 T^{6} + 73744 T^{8} + 12848 T^{10} + 1612 T^{12} + 44 T^{14} + T^{16}$$
$13$ $$( 1369 - 74 T + 670 T^{2} + 184 T^{3} + 283 T^{4} + 40 T^{5} + 22 T^{6} - 2 T^{7} + T^{8} )^{2}$$
$17$ $$( 9 + 13824 T^{2} + 2298 T^{4} + 96 T^{6} + T^{8} )^{2}$$
$19$ $$( 389376 + 77184 T^{2} + 4836 T^{4} + 120 T^{6} + T^{8} )^{2}$$
$23$ $$1871773696 + 994033664 T^{2} + 389278720 T^{4} + 63577856 T^{6} + 7557136 T^{8} + 325712 T^{10} + 10252 T^{12} + 116 T^{14} + T^{16}$$
$29$ $$1 - 56 T^{2} + 2542 T^{4} - 33152 T^{6} + 349699 T^{8} - 33152 T^{10} + 2542 T^{12} - 56 T^{14} + T^{16}$$
$31$ $$( 262144 + 98304 T - 51200 T^{2} - 23808 T^{3} + 18192 T^{4} - 4464 T^{5} + 556 T^{6} - 36 T^{7} + T^{8} )^{2}$$
$37$ $$( 141 - 210 T + 102 T^{2} - 18 T^{3} + T^{4} )^{4}$$
$41$ $$6505390336 - 2952654848 T^{2} + 974290048 T^{4} - 145405952 T^{6} + 15808816 T^{8} - 507392 T^{10} + 11848 T^{12} - 128 T^{14} + T^{16}$$
$43$ $$( 565504 + 379008 T - 49184 T^{2} - 89712 T^{3} + 39492 T^{4} - 7476 T^{5} + 766 T^{6} - 42 T^{7} + T^{8} )^{2}$$
$47$ $$268435456 + 184549376 T^{2} + 86769664 T^{4} + 21807104 T^{6} + 3993856 T^{8} + 408320 T^{10} + 28528 T^{12} + 176 T^{14} + T^{16}$$
$53$ $$( 144 + 1728 T^{2} + 2136 T^{4} + 144 T^{6} + T^{8} )^{2}$$
$59$ $$16777216 + 46137344 T^{2} + 104267776 T^{4} + 60735488 T^{6} + 28483840 T^{8} + 948992 T^{10} + 25456 T^{12} + 176 T^{14} + T^{16}$$
$61$ $$( 10182481 - 3707942 T + 1177930 T^{2} - 152096 T^{3} + 22375 T^{4} - 1568 T^{5} + 250 T^{6} - 14 T^{7} + T^{8} )^{2}$$
$67$ $$( 19642624 + 5212032 T - 26528 T^{2} - 129360 T^{3} + 612 T^{4} + 1980 T^{5} - 2 T^{6} - 18 T^{7} + T^{8} )^{2}$$
$71$ $$( 565504 - 134720 T^{2} + 8100 T^{4} - 164 T^{6} + T^{8} )^{2}$$
$73$ $$( -1263 - 120 T + 162 T^{2} - 24 T^{3} + T^{4} )^{4}$$
$79$ $$( 565504 - 776064 T + 437728 T^{2} - 113520 T^{3} + 10788 T^{4} + 660 T^{5} - 98 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$83$ $$905540854733602816 + 22084651103289344 T^{2} + 345852378873856 T^{4} + 3269794463744 T^{6} + 22626586624 T^{8} + 105909248 T^{10} + 362944 T^{12} + 752 T^{14} + T^{16}$$
$89$ $$( 1418481 + 1617192 T^{2} + 42114 T^{4} + 360 T^{6} + T^{8} )^{2}$$
$97$ $$( 177209344 + 8519680 T + 3764224 T^{2} + 51712 T^{3} + 55312 T^{4} + 736 T^{5} + 316 T^{6} - 8 T^{7} + T^{8} )^{2}$$