# Properties

 Label 3584.2.b.g Level $3584$ Weight $2$ Character orbit 3584.b Analytic conductor $28.618$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3584 = 2^{9} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3584.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$28.6183840844$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 4x^{9} + 12x^{8} - 12x^{7} + 3x^{6} - 48x^{5} + 158x^{4} - 80x^{3} - 66x^{2} + 72x + 36$$ x^10 - 4*x^9 + 12*x^8 - 12*x^7 + 3*x^6 - 48*x^5 + 158*x^4 - 80*x^3 - 66*x^2 + 72*x + 36 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{9}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} - \beta_{6} q^{5} - q^{7} + ( - \beta_{5} - 2) q^{9}+O(q^{10})$$ q - b1 * q^3 - b6 * q^5 - q^7 + (-b5 - 2) * q^9 $$q - \beta_1 q^{3} - \beta_{6} q^{5} - q^{7} + ( - \beta_{5} - 2) q^{9} + \beta_{8} q^{11} - \beta_{9} q^{13} + (\beta_{5} + \beta_{4} - \beta_{2} + 1) q^{15} + (\beta_{3} + 2) q^{17} + ( - \beta_{9} + \beta_{7} - \beta_{6} - \beta_1) q^{19} + \beta_1 q^{21} + ( - \beta_{2} - 1) q^{23} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 2) q^{25} + ( - \beta_{9} + \beta_{7} - \beta_{6} + 2 \beta_1) q^{27} + (\beta_{9} - \beta_{8} + 2 \beta_{6} + \beta_1) q^{29} + ( - \beta_{5} - \beta_{4} + \beta_{3}) q^{31} + ( - \beta_{5} - \beta_{4} + 2 \beta_{2} + 2) q^{33} + \beta_{6} q^{35} + (\beta_{9} - \beta_{8} - \beta_{7} - \beta_1) q^{37} + (2 \beta_{3} + \beta_{2} + 1) q^{39} + (\beta_{4} + \beta_{2} - 2) q^{41} + ( - \beta_{8} + \beta_{7} + 2 \beta_{6} + 2 \beta_1) q^{43} + (\beta_{9} + 2 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_1) q^{45} + ( - \beta_{4} + \beta_{2} - 2) q^{47} + q^{49} + (2 \beta_{9} + \beta_{7} + 2 \beta_{6} - 2 \beta_1) q^{51} + ( - \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{6} - 3 \beta_1) q^{53} + (3 \beta_{5} - \beta_{3} - \beta_{2} + 2) q^{55} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - 3) q^{57} + (\beta_{9} + \beta_{7} + \beta_{6} + \beta_1) q^{59} + (2 \beta_{7} - \beta_{6}) q^{61} + (\beta_{5} + 2) q^{63} + ( - \beta_{5} + 2 \beta_{4} + 3) q^{65} + (\beta_{8} + \beta_{7} - 2 \beta_1) q^{67} + ( - \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + \beta_{6} + 2 \beta_1) q^{69} + ( - \beta_{5} + \beta_{4}) q^{71} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 4) q^{73} + (\beta_{9} - 2 \beta_{8} + 3 \beta_{7} - 3 \beta_{6} + 3 \beta_1) q^{75} - \beta_{8} q^{77} + (\beta_{5} - \beta_{4} + 4) q^{79} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 6) q^{81} + (2 \beta_{7} - \beta_1) q^{83} + (3 \beta_{7} + 2 \beta_1) q^{85} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2}) q^{87} + (\beta_{5} + \beta_{4} - 2) q^{89} + \beta_{9} q^{91} + (\beta_{7} - 2 \beta_{6} + 2 \beta_1) q^{93} + ( - 2 \beta_{5} + 2 \beta_{4} + \beta_{2} - 3) q^{95} + ( - 3 \beta_{5} + \beta_{2} + 4) q^{97} + ( - \beta_{8} + 4 \beta_{7} - 6 \beta_{6} - 2 \beta_1) q^{99}+O(q^{100})$$ q - b1 * q^3 - b6 * q^5 - q^7 + (-b5 - 2) * q^9 + b8 * q^11 - b9 * q^13 + (b5 + b4 - b2 + 1) * q^15 + (b3 + 2) * q^17 + (-b9 + b7 - b6 - b1) * q^19 + b1 * q^21 + (-b2 - 1) * q^23 + (-b5 - b4 + b3 + b2 - 2) * q^25 + (-b9 + b7 - b6 + 2*b1) * q^27 + (b9 - b8 + 2*b6 + b1) * q^29 + (-b5 - b4 + b3) * q^31 + (-b5 - b4 + 2*b2 + 2) * q^33 + b6 * q^35 + (b9 - b8 - b7 - b1) * q^37 + (2*b3 + b2 + 1) * q^39 + (b4 + b2 - 2) * q^41 + (-b8 + b7 + 2*b6 + 2*b1) * q^43 + (b9 + 2*b8 - 2*b7 + 2*b6 - 2*b1) * q^45 + (-b4 + b2 - 2) * q^47 + q^49 + (2*b9 + b7 + 2*b6 - 2*b1) * q^51 + (-b9 + b8 + b7 - 2*b6 - 3*b1) * q^53 + (3*b5 - b3 - b2 + 2) * q^55 + (b5 + b4 + b3 - b2 - 3) * q^57 + (b9 + b7 + b6 + b1) * q^59 + (2*b7 - b6) * q^61 + (b5 + 2) * q^63 + (-b5 + 2*b4 + 3) * q^65 + (b8 + b7 - 2*b1) * q^67 + (-b9 + 2*b8 - 2*b7 + b6 + 2*b1) * q^69 + (-b5 + b4) * q^71 + (-b4 + b3 + b2 - 4) * q^73 + (b9 - 2*b8 + 3*b7 - 3*b6 + 3*b1) * q^75 - b8 * q^77 + (b5 - b4 + 4) * q^79 + (b5 + b4 + b3 - b2 + 6) * q^81 + (2*b7 - b1) * q^83 + (3*b7 + 2*b1) * q^85 + (-b4 - 2*b3 - b2) * q^87 + (b5 + b4 - 2) * q^89 + b9 * q^91 + (b7 - 2*b6 + 2*b1) * q^93 + (-2*b5 + 2*b4 + b2 - 3) * q^95 + (-3*b5 + b2 + 4) * q^97 + (-b8 + 4*b7 - 6*b6 - 2*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 10 q^{7} - 22 q^{9}+O(q^{10})$$ 10 * q - 10 * q^7 - 22 * q^9 $$10 q - 10 q^{7} - 22 q^{9} + 16 q^{15} + 16 q^{17} - 8 q^{23} - 30 q^{25} - 8 q^{31} + 12 q^{33} - 20 q^{41} - 24 q^{47} + 10 q^{49} + 32 q^{55} - 28 q^{57} + 22 q^{63} + 32 q^{65} - 48 q^{73} + 40 q^{79} + 62 q^{81} + 8 q^{87} - 16 q^{89} - 32 q^{95} + 32 q^{97}+O(q^{100})$$ 10 * q - 10 * q^7 - 22 * q^9 + 16 * q^15 + 16 * q^17 - 8 * q^23 - 30 * q^25 - 8 * q^31 + 12 * q^33 - 20 * q^41 - 24 * q^47 + 10 * q^49 + 32 * q^55 - 28 * q^57 + 22 * q^63 + 32 * q^65 - 48 * q^73 + 40 * q^79 + 62 * q^81 + 8 * q^87 - 16 * q^89 - 32 * q^95 + 32 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 4x^{9} + 12x^{8} - 12x^{7} + 3x^{6} - 48x^{5} + 158x^{4} - 80x^{3} - 66x^{2} + 72x + 36$$ :

 $$\beta_{1}$$ $$=$$ $$( 1883 \nu^{9} + 39443 \nu^{8} - 140969 \nu^{7} + 463067 \nu^{6} - 313772 \nu^{5} + 12764 \nu^{4} - 2064154 \nu^{3} + 6984030 \nu^{2} - 1109304 \nu - 2273868 ) / 1451844$$ (1883*v^9 + 39443*v^8 - 140969*v^7 + 463067*v^6 - 313772*v^5 + 12764*v^4 - 2064154*v^3 + 6984030*v^2 - 1109304*v - 2273868) / 1451844 $$\beta_{2}$$ $$=$$ $$( 15829 \nu^{9} - 84722 \nu^{8} + 209765 \nu^{7} - 289322 \nu^{6} - 201334 \nu^{5} - 877304 \nu^{4} + 3566650 \nu^{3} - 1103322 \nu^{2} - 3911724 \nu - 9846 ) / 725922$$ (15829*v^9 - 84722*v^8 + 209765*v^7 - 289322*v^6 - 201334*v^5 - 877304*v^4 + 3566650*v^3 - 1103322*v^2 - 3911724*v - 9846) / 725922 $$\beta_{3}$$ $$=$$ $$( - 19133 \nu^{9} + 66337 \nu^{8} - 198991 \nu^{7} + 125815 \nu^{6} - 18106 \nu^{5} + 934966 \nu^{4} - 2799902 \nu^{3} + 582306 \nu^{2} + 2932884 \nu - 1858968 ) / 725922$$ (-19133*v^9 + 66337*v^8 - 198991*v^7 + 125815*v^6 - 18106*v^5 + 934966*v^4 - 2799902*v^3 + 582306*v^2 + 2932884*v - 1858968) / 725922 $$\beta_{4}$$ $$=$$ $$( 22726 \nu^{9} - 88931 \nu^{8} + 257948 \nu^{7} - 199391 \nu^{6} - 159244 \nu^{5} - 620576 \nu^{4} + 2918896 \nu^{3} - 850908 \nu^{2} - 3111984 \nu + 4740426 ) / 725922$$ (22726*v^9 - 88931*v^8 + 257948*v^7 - 199391*v^6 - 159244*v^5 - 620576*v^4 + 2918896*v^3 - 850908*v^2 - 3111984*v + 4740426) / 725922 $$\beta_{5}$$ $$=$$ $$( - 31664 \nu^{9} + 124663 \nu^{8} - 353158 \nu^{7} + 285913 \nu^{6} + 124556 \nu^{5} + 1332430 \nu^{4} - 4770464 \nu^{3} + 1214952 \nu^{2} + 5074416 \nu - 1480806 ) / 725922$$ (-31664*v^9 + 124663*v^8 - 353158*v^7 + 285913*v^6 + 124556*v^5 + 1332430*v^4 - 4770464*v^3 + 1214952*v^2 + 5074416*v - 1480806) / 725922 $$\beta_{6}$$ $$=$$ $$( 77231 \nu^{9} - 432283 \nu^{8} + 1411729 \nu^{7} - 2393731 \nu^{6} + 1741774 \nu^{5} - 4281784 \nu^{4} + 18554846 \nu^{3} - 24722286 \nu^{2} + \cdots + 9111492 ) / 1451844$$ (77231*v^9 - 432283*v^8 + 1411729*v^7 - 2393731*v^6 + 1741774*v^5 - 4281784*v^4 + 18554846*v^3 - 24722286*v^2 + 3643452*v + 9111492) / 1451844 $$\beta_{7}$$ $$=$$ $$( 3820 \nu^{9} - 15476 \nu^{8} + 48281 \nu^{7} - 55454 \nu^{6} + 33773 \nu^{5} - 207770 \nu^{4} + 604930 \nu^{3} - 416874 \nu^{2} + 19098 \nu + 165024 ) / 40329$$ (3820*v^9 - 15476*v^8 + 48281*v^7 - 55454*v^6 + 33773*v^5 - 207770*v^4 + 604930*v^3 - 416874*v^2 + 19098*v + 165024) / 40329 $$\beta_{8}$$ $$=$$ $$( - 76448 \nu^{9} + 287347 \nu^{8} - 878263 \nu^{7} + 850897 \nu^{6} - 413401 \nu^{5} + 4084954 \nu^{4} - 11128874 \nu^{3} + 5808030 \nu^{2} + \cdots - 2399184 ) / 725922$$ (-76448*v^9 + 287347*v^8 - 878263*v^7 + 850897*v^6 - 413401*v^5 + 4084954*v^4 - 11128874*v^3 + 5808030*v^2 + 129366*v - 2399184) / 725922 $$\beta_{9}$$ $$=$$ $$( - 209987 \nu^{9} + 975355 \nu^{8} - 3098881 \nu^{7} + 4281475 \nu^{6} - 2816962 \nu^{5} + 11110924 \nu^{4} - 41125610 \nu^{3} + 40233750 \nu^{2} + \cdots - 15620868 ) / 1451844$$ (-209987*v^9 + 975355*v^8 - 3098881*v^7 + 4281475*v^6 - 2816962*v^5 + 11110924*v^4 - 41125610*v^3 + 40233750*v^2 - 5172948*v - 15620868) / 1451844
 $$\nu$$ $$=$$ $$( -\beta_{9} + \beta_{8} + \beta_{7} - 2\beta_{6} + \beta_{3} - \beta _1 + 2 ) / 4$$ (-b9 + b8 + b7 - 2*b6 + b3 - b1 + 2) / 4 $$\nu^{2}$$ $$=$$ $$( -\beta_{9} + 3\beta_{8} + 3\beta_{7} - 2\beta_{6} + 2\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 - 4 ) / 4$$ (-b9 + 3*b8 + 3*b7 - 2*b6 + 2*b5 + b4 - b3 + b2 + b1 - 4) / 4 $$\nu^{3}$$ $$=$$ $$( - \beta_{9} - 3 \beta_{8} - 6 \beta_{7} + 2 \beta_{6} + 8 \beta_{5} + 3 \beta_{4} - 7 \beta_{3} + 3 \beta_{2} + 5 \beta _1 - 22 ) / 4$$ (-b9 - 3*b8 - 6*b7 + 2*b6 + 8*b5 + 3*b4 - 7*b3 + 3*b2 + 5*b1 - 22) / 4 $$\nu^{4}$$ $$=$$ $$( 11\beta_{9} - 31\beta_{8} - 40\beta_{7} + 38\beta_{6} + 3\beta_{4} + \beta_{3} - \beta_{2} + 37\beta _1 - 18 ) / 4$$ (11*b9 - 31*b8 - 40*b7 + 38*b6 + 3*b4 + b3 - b2 + 37*b1 - 18) / 4 $$\nu^{5}$$ $$=$$ $$( 19 \beta_{9} - 41 \beta_{8} - 52 \beta_{7} + 60 \beta_{6} - 71 \beta_{5} - 39 \beta_{4} + 48 \beta_{3} - 24 \beta_{2} + 59 \beta _1 + 230 ) / 4$$ (19*b9 - 41*b8 - 52*b7 + 60*b6 - 71*b5 - 39*b4 + 48*b3 - 24*b2 + 59*b1 + 230) / 4 $$\nu^{6}$$ $$=$$ $$( - 99 \beta_{9} + 203 \beta_{8} + 220 \beta_{7} - 258 \beta_{6} - 186 \beta_{5} - 95 \beta_{4} + 133 \beta_{3} - 65 \beta_{2} - 209 \beta _1 + 578 ) / 4$$ (-99*b9 + 203*b8 + 220*b7 - 258*b6 - 186*b5 - 95*b4 + 133*b3 - 65*b2 - 209*b1 + 578) / 4 $$\nu^{7}$$ $$=$$ $$( - 377 \beta_{9} + 915 \beta_{8} + 1048 \beta_{7} - 1064 \beta_{6} + 215 \beta_{5} + 131 \beta_{4} - 178 \beta_{3} + 40 \beta_{2} - 865 \beta _1 - 870 ) / 4$$ (-377*b9 + 915*b8 + 1048*b7 - 1064*b6 + 215*b5 + 131*b4 - 178*b3 + 40*b2 - 865*b1 - 870) / 4 $$\nu^{8}$$ $$=$$ $$( - 103 \beta_{9} + 291 \beta_{8} + 372 \beta_{7} - 350 \beta_{6} + 2288 \beta_{5} + 1165 \beta_{4} - 1849 \beta_{3} + 649 \beta_{2} - 329 \beta _1 - 7654 ) / 4$$ (-103*b9 + 291*b8 + 372*b7 - 350*b6 + 2288*b5 + 1165*b4 - 1849*b3 + 649*b2 - 329*b1 - 7654) / 4 $$\nu^{9}$$ $$=$$ $$( 3395 \beta_{9} - 7965 \beta_{8} - 8946 \beta_{7} + 9296 \beta_{6} + 3431 \beta_{5} + 1827 \beta_{4} - 2686 \beta_{3} + 972 \beta_{2} + 7379 \beta _1 - 11770 ) / 4$$ (3395*b9 - 7965*b8 - 8946*b7 + 9296*b6 + 3431*b5 + 1827*b4 - 2686*b3 + 972*b2 + 7379*b1 - 11770) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times$$.

 $$n$$ $$1023$$ $$1025$$ $$1541$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1793.1
 0.875338 + 0.673961i 1.68909 + 0.861941i −1.29337 − 1.34353i −0.484731 − 0.120245i 1.21367 − 2.82548i 1.21367 + 2.82548i −0.484731 + 0.120245i −1.29337 + 1.34353i 1.68909 − 0.861941i 0.875338 − 0.673961i
0 3.17593i 0 4.23755i 0 −1.00000 0 −7.08655 0
1793.2 0 2.99347i 0 0.369776i 0 −1.00000 0 −5.96088 0
1793.3 0 2.47533i 0 2.59708i 0 −1.00000 0 −3.12724 0
1793.4 0 0.783186i 0 3.56843i 0 −1.00000 0 2.38662 0
1793.5 0 0.460386i 0 1.55819i 0 −1.00000 0 2.78804 0
1793.6 0 0.460386i 0 1.55819i 0 −1.00000 0 2.78804 0
1793.7 0 0.783186i 0 3.56843i 0 −1.00000 0 2.38662 0
1793.8 0 2.47533i 0 2.59708i 0 −1.00000 0 −3.12724 0
1793.9 0 2.99347i 0 0.369776i 0 −1.00000 0 −5.96088 0
1793.10 0 3.17593i 0 4.23755i 0 −1.00000 0 −7.08655 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1793.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3584.2.b.g 10
4.b odd 2 1 3584.2.b.h 10
8.b even 2 1 inner 3584.2.b.g 10
8.d odd 2 1 3584.2.b.h 10
16.e even 4 2 3584.2.a.p yes 10
16.f odd 4 2 3584.2.a.o 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3584.2.a.o 10 16.f odd 4 2
3584.2.a.p yes 10 16.e even 4 2
3584.2.b.g 10 1.a even 1 1 trivial
3584.2.b.g 10 8.b even 2 1 inner
3584.2.b.h 10 4.b odd 2 1
3584.2.b.h 10 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3584, [\chi])$$:

 $$T_{3}^{10} + 26T_{3}^{8} + 228T_{3}^{6} + 728T_{3}^{4} + 484T_{3}^{2} + 72$$ T3^10 + 26*T3^8 + 228*T3^6 + 728*T3^4 + 484*T3^2 + 72 $$T_{23}^{5} + 4T_{23}^{4} - 60T_{23}^{3} - 112T_{23}^{2} + 644T_{23} + 1296$$ T23^5 + 4*T23^4 - 60*T23^3 - 112*T23^2 + 644*T23 + 1296

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$T^{10} + 26 T^{8} + 228 T^{6} + \cdots + 72$$
$5$ $$T^{10} + 40 T^{8} + 532 T^{6} + \cdots + 512$$
$7$ $$(T + 1)^{10}$$
$11$ $$T^{10} + 98 T^{8} + 3648 T^{6} + \cdots + 1492992$$
$13$ $$T^{10} + 104 T^{8} + 3540 T^{6} + \cdots + 4608$$
$17$ $$(T^{5} - 8 T^{4} - 24 T^{3} + 256 T^{2} + \cdots - 1152)^{2}$$
$19$ $$T^{10} + 122 T^{8} + 5028 T^{6} + \cdots + 5832$$
$23$ $$(T^{5} + 4 T^{4} - 60 T^{3} - 112 T^{2} + \cdots + 1296)^{2}$$
$29$ $$T^{10} + 184 T^{8} + 11296 T^{6} + \cdots + 18432$$
$31$ $$(T^{5} + 4 T^{4} - 96 T^{3} - 640 T^{2} + \cdots - 384)^{2}$$
$37$ $$T^{10} + 216 T^{8} + 15648 T^{6} + \cdots + 1492992$$
$41$ $$(T^{5} + 10 T^{4} - 120 T^{3} + \cdots + 28128)^{2}$$
$43$ $$T^{10} + 274 T^{8} + 25984 T^{6} + \cdots + 18432$$
$47$ $$(T^{5} + 12 T^{4} - 48 T^{3} - 768 T^{2} + \cdots + 10368)^{2}$$
$53$ $$T^{10} + 280 T^{8} + 22240 T^{6} + \cdots + 5752832$$
$59$ $$T^{10} + 202 T^{8} + 6628 T^{6} + \cdots + 52488$$
$61$ $$T^{10} + 200 T^{8} + 13140 T^{6} + \cdots + 7558272$$
$67$ $$T^{10} + 194 T^{8} + 11904 T^{6} + \cdots + 2654208$$
$71$ $$(T^{5} - 120 T^{3} - 416 T^{2} + \cdots + 1152)^{2}$$
$73$ $$(T^{5} + 24 T^{4} + 104 T^{3} + \cdots - 10496)^{2}$$
$79$ $$(T^{5} - 20 T^{4} + 40 T^{3} + 1216 T^{2} + \cdots + 11584)^{2}$$
$83$ $$T^{10} + 170 T^{8} + 10020 T^{6} + \cdots + 7235208$$
$89$ $$(T^{5} + 8 T^{4} - 72 T^{3} - 288 T^{2} + \cdots - 2304)^{2}$$
$97$ $$(T^{5} - 16 T^{4} - 264 T^{3} + 4288 T^{2} + \cdots - 1536)^{2}$$