# Properties

 Label 1400.2.bh.j Level $1400$ Weight $2$ Character orbit 1400.bh Analytic conductor $11.179$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1400 = 2^{3} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1400.bh (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.1790562830$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 15 x^{14} + 170 x^{12} - 789 x^{10} + 2754 x^{8} - 960 x^{6} + 269 x^{4} - 18 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{9} ) q^{3} + ( \beta_{6} - \beta_{7} ) q^{7} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{10} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{9} ) q^{3} + ( \beta_{6} - \beta_{7} ) q^{7} + ( 1 - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{10} ) q^{9} + ( 1 + \beta_{2} + \beta_{5} - \beta_{10} - \beta_{11} ) q^{11} + ( \beta_{7} - \beta_{9} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{13} + ( -\beta_{1} + \beta_{7} - 2 \beta_{9} ) q^{17} + ( -3 - 3 \beta_{5} + 2 \beta_{8} - \beta_{10} + 2 \beta_{11} ) q^{19} + ( 3 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{8} + \beta_{11} ) q^{21} + ( 2 \beta_{6} + 2 \beta_{12} - 3 \beta_{14} ) q^{23} + ( 2 \beta_{1} - 2 \beta_{6} - \beta_{7} - 2 \beta_{9} - 5 \beta_{12} - 2 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{27} + ( -2 - \beta_{2} + 2 \beta_{8} ) q^{29} + ( -1 - \beta_{2} - 4 \beta_{5} + \beta_{10} ) q^{31} + ( 4 \beta_{6} + 2 \beta_{12} - 2 \beta_{14} + \beta_{15} ) q^{33} + ( -2 \beta_{6} - \beta_{12} + \beta_{14} + \beta_{15} ) q^{37} + ( -1 - \beta_{2} - 4 \beta_{4} + 3 \beta_{5} + \beta_{10} ) q^{39} + ( -3 + 2 \beta_{3} + 3 \beta_{8} ) q^{41} + ( \beta_{1} - \beta_{6} + 2 \beta_{7} + 2 \beta_{9} + 5 \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{43} + ( -3 \beta_{6} - 4 \beta_{12} + \beta_{15} ) q^{47} + ( 4 + \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{10} ) q^{49} + ( -\beta_{3} - \beta_{4} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{51} + ( \beta_{7} + 2 \beta_{9} ) q^{53} + ( -5 \beta_{1} + 5 \beta_{6} - \beta_{7} + 6 \beta_{9} + 6 \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{57} + ( -3 \beta_{4} + \beta_{5} ) q^{59} + ( 5 - \beta_{3} - \beta_{4} + 5 \beta_{5} + 3 \beta_{8} + 3 \beta_{11} ) q^{61} + ( 4 \beta_{1} + \beta_{7} - 5 \beta_{9} - 2 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{63} + ( 4 \beta_{1} - 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{13} ) q^{67} + ( 4 + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{8} ) q^{69} + ( -5 - 4 \beta_{3} + 2 \beta_{8} ) q^{71} + ( 2 \beta_{1} + 2 \beta_{7} - 2 \beta_{9} + 2 \beta_{13} ) q^{73} + ( -4 \beta_{1} + 3 \beta_{6} + \beta_{7} + 4 \beta_{12} + 3 \beta_{14} ) q^{77} + ( -1 - \beta_{5} + 3 \beta_{8} + 3 \beta_{11} ) q^{79} + ( -1 - \beta_{2} - 8 \beta_{4} + 6 \beta_{5} + \beta_{10} + 3 \beta_{11} ) q^{81} + ( 2 \beta_{7} + 5 \beta_{9} + 8 \beta_{12} + \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{83} + ( -3 \beta_{1} - \beta_{7} + 4 \beta_{9} + \beta_{13} ) q^{87} + ( -5 + 3 \beta_{3} + 3 \beta_{4} - 5 \beta_{5} - 2 \beta_{8} + \beta_{10} - 2 \beta_{11} ) q^{89} + ( -2 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{8} + 2 \beta_{10} - \beta_{11} ) q^{91} + ( \beta_{12} + \beta_{14} - \beta_{15} ) q^{93} + ( 2 \beta_{1} - 2 \beta_{6} + \beta_{7} + 2 \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} ) q^{97} + ( 15 + 2 \beta_{2} - 5 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 10q^{9} + O(q^{10})$$ $$16q + 10q^{9} - 20q^{19} + 50q^{21} - 8q^{29} + 28q^{31} - 20q^{39} - 16q^{41} + 26q^{49} - 10q^{51} - 2q^{59} + 50q^{61} + 64q^{69} - 80q^{71} + 4q^{79} - 48q^{81} - 38q^{89} + 34q^{91} + 204q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 15 x^{14} + 170 x^{12} - 789 x^{10} + 2754 x^{8} - 960 x^{6} + 269 x^{4} - 18 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$4102340 \nu^{14} - 60216631 \nu^{12} + 677333628 \nu^{10} - 3009625800 \nu^{8} + 10229021874 \nu^{6} - 299619996 \nu^{4} + 20064172 \nu^{2} - 1313504207$$$$)/ 359036523$$ $$\beta_{3}$$ $$=$$ $$($$$$-5600705 \nu^{14} + 82189681 \nu^{12} - 924727311 \nu^{10} + 4108880850 \nu^{8} - 13986921834 \nu^{6} + 409055127 \nu^{4} - 27392539 \nu^{2} - 98128630$$$$)/ 359036523$$ $$\beta_{4}$$ $$=$$ $$($$$$-5297 \nu^{14} + 80335 \nu^{12} - 913449 \nu^{10} + 4324629 \nu^{8} - 15233538 \nu^{6} + 7255677 \nu^{4} - 1489165 \nu^{2} + 99650$$$$)/259983$$ $$\beta_{5}$$ $$=$$ $$($$$$2723695 \nu^{14} - 40688940 \nu^{12} + 460586700 \nu^{10} - 2121507168 \nu^{8} + 7378916580 \nu^{6} - 2197202760 \nu^{4} + 720514496 \nu^{2} - 48212247$$$$)/39892947$$ $$\beta_{6}$$ $$=$$ $$($$$$-2723695 \nu^{15} + 40688940 \nu^{13} - 460586700 \nu^{11} + 2121507168 \nu^{9} - 7378916580 \nu^{7} + 2197202760 \nu^{5} - 720514496 \nu^{3} + 48212247 \nu$$$$)/39892947$$ $$\beta_{7}$$ $$=$$ $$($$$$35385680 \nu^{15} - 519187807 \nu^{13} + 5842497456 \nu^{11} - 25960221600 \nu^{9} + 88439791707 \nu^{7} - 2584441392 \nu^{5} + 173068144 \nu^{3} + 3763758820 \nu$$$$)/ 359036523$$ $$\beta_{8}$$ $$=$$ $$($$$$-39488020 \nu^{14} + 579404438 \nu^{12} - 6519831084 \nu^{10} + 28969847400 \nu^{8} - 98668813581 \nu^{6} + 2884061388 \nu^{4} - 193132316 \nu^{2} - 1014108521$$$$)/ 359036523$$ $$\beta_{9}$$ $$=$$ $$($$$$-39488020 \nu^{15} + 579404438 \nu^{13} - 6519831084 \nu^{11} + 28969847400 \nu^{9} - 98668813581 \nu^{7} + 2884061388 \nu^{5} - 193132316 \nu^{3} - 1373145044 \nu$$$$)/ 359036523$$ $$\beta_{10}$$ $$=$$ $$($$$$-60623903 \nu^{14} + 905469064 \nu^{12} - 10249640520 \nu^{10} + 47199500037 \nu^{8} - 164206309848 \nu^{6} + 48895329456 \nu^{4} - 17010925465 \nu^{2} + 185133080$$$$)/ 359036523$$ $$\beta_{11}$$ $$=$$ $$($$$$-73615375 \nu^{14} + 1111329695 \nu^{12} - 12618776481 \nu^{10} + 59254651869 \nu^{8} - 207944878650 \nu^{6} + 88415581794 \nu^{4} - 20321026133 \nu^{2} + 1359797656$$$$)/ 359036523$$ $$\beta_{12}$$ $$=$$ $$($$$$-98128630 \nu^{15} + 1477530155 \nu^{13} - 16764056781 \nu^{11} + 78348216381 \nu^{9} - 274355127870 \nu^{7} + 108190406634 \nu^{5} - 26805656597 \nu^{3} + 1793707879 \nu$$$$)/ 359036523$$ $$\beta_{13}$$ $$=$$ $$($$$$112863355 \nu^{15} - 1656023633 \nu^{13} + 18634765941 \nu^{11} - 82800661350 \nu^{9} + 282019518909 \nu^{7} - 8243129037 \nu^{5} + 552004409 \nu^{3} + 3662269979 \nu$$$$)/ 359036523$$ $$\beta_{14}$$ $$=$$ $$-\nu^{15} + 15 \nu^{13} - 170 \nu^{11} + 789 \nu^{9} - 2754 \nu^{7} + 960 \nu^{5} - 269 \nu^{3} + 18 \nu$$ $$\beta_{15}$$ $$=$$ $$($$$$360686108 \nu^{15} - 5432977525 \nu^{13} + 61649473755 \nu^{11} - 288326988363 \nu^{9} + 1009972746282 \nu^{7} - 402966711759 \nu^{5} + 98681459059 \nu^{3} - 6603304643 \nu$$$$)/ 359036523$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{10} + 3 \beta_{5} + \beta_{4} + \beta_{3} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{15} + \beta_{14} + \beta_{13} + 2 \beta_{12} + 2 \beta_{9} - \beta_{7} - 8 \beta_{6} + 8 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{11} + 8 \beta_{10} + 22 \beta_{5} + 13 \beta_{4} - 8 \beta_{2} - 8$$ $$\nu^{5}$$ $$=$$ $$13 \beta_{15} + 8 \beta_{14} + 35 \beta_{12} - 66 \beta_{6}$$ $$\nu^{6}$$ $$=$$ $$13 \beta_{8} - 140 \beta_{3} - 66 \beta_{2} - 243$$ $$\nu^{7}$$ $$=$$ $$-140 \beta_{13} - 341 \beta_{9} + 66 \beta_{7} - 568 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$140 \beta_{11} - 568 \beta_{10} + 140 \beta_{8} - 1498 \beta_{5} - 1403 \beta_{4} - 1403 \beta_{3} - 1498$$ $$\nu^{9}$$ $$=$$ $$-1403 \beta_{15} - 568 \beta_{14} - 1403 \beta_{13} - 4336 \beta_{12} - 3501 \beta_{9} + 568 \beta_{7} + 5033 \beta_{6} - 5033 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$1403 \beta_{11} - 5033 \beta_{10} - 13128 \beta_{5} - 13578 \beta_{4} + 5033 \beta_{2} + 5033$$ $$\nu^{11}$$ $$=$$ $$-13578 \beta_{15} - 5033 \beta_{14} - 42843 \beta_{12} + 45435 \beta_{6}$$ $$\nu^{12}$$ $$=$$ $$-13578 \beta_{8} + 129012 \beta_{3} + 45435 \beta_{2} + 163129$$ $$\nu^{13}$$ $$=$$ $$129012 \beta_{13} + 328023 \beta_{9} - 45435 \beta_{7} + 414868 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$-129012 \beta_{11} + 414868 \beta_{10} - 129012 \beta_{8} + 1070157 \beta_{5} + 1213504 \beta_{4} + 1213504 \beta_{3} + 1070157$$ $$\nu^{15}$$ $$=$$ $$1213504 \beta_{15} + 414868 \beta_{14} + 1213504 \beta_{13} + 3895268 \beta_{12} + 3096632 \beta_{9} - 414868 \beta_{7} - 3814121 \beta_{6} + 3814121 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$$.

 $$n$$ $$351$$ $$701$$ $$801$$ $$1177$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1 - \beta_{5}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
249.1
 −2.00531 − 1.15777i −2.63854 − 1.52336i −0.229810 − 0.132681i 0.462601 + 0.267083i −0.462601 − 0.267083i 0.229810 + 0.132681i 2.63854 + 1.52336i 2.00531 + 1.15777i −2.00531 + 1.15777i −2.63854 + 1.52336i −0.229810 + 0.132681i 0.462601 − 0.267083i −0.462601 + 0.267083i 0.229810 − 0.132681i 2.63854 − 1.52336i 2.00531 − 1.15777i
0 −2.87134 1.65777i 0 0 0 −2.49733 + 0.873699i 0 3.99638 + 6.92193i 0
249.2 0 −1.77252 1.02336i 0 0 0 −1.48827 + 2.18747i 0 0.594550 + 1.02979i 0
249.3 0 −1.09584 0.632681i 0 0 0 2.16772 + 1.51690i 0 −0.699429 1.21145i 0
249.4 0 −0.403424 0.232917i 0 0 0 −2.02469 1.70312i 0 −1.39150 2.41015i 0
249.5 0 0.403424 + 0.232917i 0 0 0 2.02469 + 1.70312i 0 −1.39150 2.41015i 0
249.6 0 1.09584 + 0.632681i 0 0 0 −2.16772 1.51690i 0 −0.699429 1.21145i 0
249.7 0 1.77252 + 1.02336i 0 0 0 1.48827 2.18747i 0 0.594550 + 1.02979i 0
249.8 0 2.87134 + 1.65777i 0 0 0 2.49733 0.873699i 0 3.99638 + 6.92193i 0
849.1 0 −2.87134 + 1.65777i 0 0 0 −2.49733 0.873699i 0 3.99638 6.92193i 0
849.2 0 −1.77252 + 1.02336i 0 0 0 −1.48827 2.18747i 0 0.594550 1.02979i 0
849.3 0 −1.09584 + 0.632681i 0 0 0 2.16772 1.51690i 0 −0.699429 + 1.21145i 0
849.4 0 −0.403424 + 0.232917i 0 0 0 −2.02469 + 1.70312i 0 −1.39150 + 2.41015i 0
849.5 0 0.403424 0.232917i 0 0 0 2.02469 1.70312i 0 −1.39150 + 2.41015i 0
849.6 0 1.09584 0.632681i 0 0 0 −2.16772 + 1.51690i 0 −0.699429 + 1.21145i 0
849.7 0 1.77252 1.02336i 0 0 0 1.48827 + 2.18747i 0 0.594550 1.02979i 0
849.8 0 2.87134 1.65777i 0 0 0 2.49733 + 0.873699i 0 3.99638 6.92193i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 849.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.2.bh.j 16
5.b even 2 1 inner 1400.2.bh.j 16
5.c odd 4 1 1400.2.q.l 8
5.c odd 4 1 1400.2.q.m yes 8
7.c even 3 1 inner 1400.2.bh.j 16
35.j even 6 1 inner 1400.2.bh.j 16
35.k even 12 1 9800.2.a.ck 4
35.k even 12 1 9800.2.a.cu 4
35.l odd 12 1 1400.2.q.l 8
35.l odd 12 1 1400.2.q.m yes 8
35.l odd 12 1 9800.2.a.cj 4
35.l odd 12 1 9800.2.a.ct 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.q.l 8 5.c odd 4 1
1400.2.q.l 8 35.l odd 12 1
1400.2.q.m yes 8 5.c odd 4 1
1400.2.q.m yes 8 35.l odd 12 1
1400.2.bh.j 16 1.a even 1 1 trivial
1400.2.bh.j 16 5.b even 2 1 inner
1400.2.bh.j 16 7.c even 3 1 inner
1400.2.bh.j 16 35.j even 6 1 inner
9800.2.a.cj 4 35.l odd 12 1
9800.2.a.ck 4 35.k even 12 1
9800.2.a.ct 4 35.l odd 12 1
9800.2.a.cu 4 35.k even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1400, [\chi])$$:

 $$T_{3}^{16} - \cdots$$ $$T_{11}^{8} + 37 T_{11}^{6} - 98 T_{11}^{5} + 1235 T_{11}^{4} - 1813 T_{11}^{3} + 7359 T_{11}^{2} + 6566 T_{11} + 17956$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$256 - 1424 T^{2} + 6737 T^{4} - 6042 T^{6} + 3947 T^{8} - 1080 T^{10} + 215 T^{12} - 17 T^{14} + T^{16}$$
$5$ $$T^{16}$$
$7$ $$5764801 - 1529437 T^{2} + 463393 T^{4} - 77077 T^{6} + 13465 T^{8} - 1573 T^{10} + 193 T^{12} - 13 T^{14} + T^{16}$$
$11$ $$( 17956 + 6566 T + 7359 T^{2} - 1813 T^{3} + 1235 T^{4} - 98 T^{5} + 37 T^{6} + T^{8} )^{2}$$
$13$ $$( 7056 + 7161 T^{2} + 1201 T^{4} + 62 T^{6} + T^{8} )^{2}$$
$17$ $$625 - 6350 T^{2} + 56741 T^{4} - 77244 T^{6} + 87806 T^{8} - 10377 T^{10} + 914 T^{12} - 35 T^{14} + T^{16}$$
$19$ $$( 678976 + 295816 T + 118169 T^{2} + 21147 T^{3} + 4583 T^{4} + 588 T^{5} + 113 T^{6} + 10 T^{7} + T^{8} )^{2}$$
$23$ $$824843587681 - 144273540695 T^{2} + 17835732302 T^{4} - 1007197641 T^{6} + 40366310 T^{8} - 969516 T^{10} + 16817 T^{12} - 158 T^{14} + T^{16}$$
$29$ $$( -222 - 219 T - 49 T^{2} + 2 T^{3} + T^{4} )^{4}$$
$31$ $$( 625 - 75 T + 1234 T^{2} - 553 T^{3} + 2468 T^{4} - 692 T^{5} + 147 T^{6} - 14 T^{7} + T^{8} )^{2}$$
$37$ $$71639296 - 69548688 T^{2} + 49126817 T^{4} - 16332021 T^{6} + 3973935 T^{8} - 179136 T^{10} + 5927 T^{12} - 90 T^{14} + T^{16}$$
$41$ $$( 3319 - 125 T - 127 T^{2} + 4 T^{3} + T^{4} )^{4}$$
$43$ $$( 3825936 + 401289 T^{2} + 14734 T^{4} + 217 T^{6} + T^{8} )^{2}$$
$47$ $$82632190550625 - 6107594914350 T^{2} + 283707055521 T^{4} - 8342728236 T^{6} + 181518598 T^{8} - 2770801 T^{10} + 31278 T^{12} - 223 T^{14} + T^{16}$$
$53$ $$1048576 - 2073600 T^{2} + 3571217 T^{4} - 960909 T^{6} + 181215 T^{8} - 17664 T^{10} + 1247 T^{12} - 42 T^{14} + T^{16}$$
$59$ $$( 4900 - 5390 T + 10549 T^{2} + 4942 T^{3} + 4363 T^{4} + 88 T^{5} + 67 T^{6} + T^{7} + T^{8} )^{2}$$
$61$ $$( 4536900 - 121410 T + 352569 T^{2} - 97152 T^{3} + 30451 T^{4} - 4214 T^{5} + 461 T^{6} - 25 T^{7} + T^{8} )^{2}$$
$67$ $$4694952902656 - 1139485704192 T^{2} + 201223442432 T^{4} - 16723989504 T^{6} + 1017327360 T^{8} - 11464704 T^{10} + 94832 T^{12} - 360 T^{14} + T^{16}$$
$71$ $$( -1035 - 732 T + 34 T^{2} + 20 T^{3} + T^{4} )^{4}$$
$73$ $$1761205026816 - 758296608768 T^{2} + 276462305280 T^{4} - 20551827456 T^{6} + 1207103488 T^{8} - 12880128 T^{10} + 100688 T^{12} - 372 T^{14} + T^{16}$$
$79$ $$( 22201 - 39485 T + 59050 T^{2} - 20471 T^{3} + 6304 T^{4} - 380 T^{5} + 79 T^{6} - 2 T^{7} + T^{8} )^{2}$$
$83$ $$( 7986276 + 1057149 T^{2} + 30097 T^{4} + 302 T^{6} + T^{8} )^{2}$$
$89$ $$( 2064969 + 681138 T + 300837 T^{2} + 29484 T^{3} + 13252 T^{4} + 1955 T^{5} + 308 T^{6} + 19 T^{7} + T^{8} )^{2}$$
$97$ $$( 729 + 3483 T^{2} + 4167 T^{4} + 150 T^{6} + T^{8} )^{2}$$