# Properties

 Label 975.2.bb.i Level $975$ Weight $2$ Character orbit 975.bb Analytic conductor $7.785$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$975 = 3 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 975.bb (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.78541419707$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.1731891456.1 Defining polynomial: $$x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256$$ x^8 - 9*x^6 + 65*x^4 - 144*x^2 + 256 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{5} - \beta_{4} + 3 \beta_{2} + 2) q^{4} + ( - \beta_{5} + \beta_{4} - \beta_{2}) q^{6} + (\beta_{7} - \beta_{6} + \beta_{3} + \beta_1) q^{7} + ( - 4 \beta_{7} + \beta_{6}) q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10})$$ q + b1 * q^2 - b3 * q^3 + (b5 - b4 + 3*b2 + 2) * q^4 + (-b5 + b4 - b2) * q^6 + (b7 - b6 + b3 + b1) * q^7 + (-4*b7 + b6) * q^8 + (b2 + 1) * q^9 $$q + \beta_1 q^{2} - \beta_{3} q^{3} + (\beta_{5} - \beta_{4} + 3 \beta_{2} + 2) q^{4} + ( - \beta_{5} + \beta_{4} - \beta_{2}) q^{6} + (\beta_{7} - \beta_{6} + \beta_{3} + \beta_1) q^{7} + ( - 4 \beta_{7} + \beta_{6}) q^{8} + (\beta_{2} + 1) q^{9} - 2 \beta_{2} q^{11} + (2 \beta_{7} - \beta_{6}) q^{12} + ( - \beta_{7} - \beta_{6} - \beta_{3} + 2 \beta_1) q^{13} + ( - 2 \beta_{4} + 4) q^{14} + (3 \beta_{5} + 3 \beta_{2}) q^{16} + (\beta_{6} - \beta_1) q^{17} + \beta_{6} q^{18} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2} + 4) q^{19} + (\beta_{4} - 1) q^{21} + ( - 2 \beta_{6} + 2 \beta_1) q^{22} - 2 \beta_{3} q^{23} + ( - \beta_{5} - 5 \beta_{2}) q^{24} + (\beta_{5} - \beta_{4} + 5 \beta_{2} + 8) q^{26} + \beta_{7} q^{27} + (6 \beta_{3} + 4 \beta_1) q^{28} + (3 \beta_{5} + \beta_{2}) q^{29} + (\beta_{4} + 1) q^{31} + ( - 4 \beta_{7} + \beta_{6} - 4 \beta_{3} - \beta_1) q^{32} + ( - 2 \beta_{7} - 2 \beta_{3}) q^{33} + (\beta_{4} - 4) q^{34} + (\beta_{5} + 3 \beta_{2}) q^{36} + (6 \beta_{3} - \beta_1) q^{37} + (8 \beta_{7} + 2 \beta_{6}) q^{38} + ( - \beta_{5} + 2 \beta_{4} - \beta_{2} + 1) q^{39} + (\beta_{5} + \beta_{2}) q^{41} + ( - 4 \beta_{3} - 2 \beta_1) q^{42} + (3 \beta_{7} + \beta_{6} + 3 \beta_{3} - \beta_1) q^{43} + ( - 2 \beta_{4} + 4) q^{44} + ( - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{2}) q^{46} + (2 \beta_{7} + 4 \beta_{6}) q^{47} + ( - 3 \beta_{6} + 3 \beta_1) q^{48} + ( - 3 \beta_{5} - \beta_{2}) q^{49} - \beta_{4} q^{51} + ( - 4 \beta_{7} + 3 \beta_{6} + 2 \beta_{3} + 2 \beta_1) q^{52} + ( - 4 \beta_{7} + 3 \beta_{6}) q^{53} + ( - \beta_{5} - \beta_{2}) q^{54} + (6 \beta_{5} - 6 \beta_{4} + 14 \beta_{2} + 8) q^{56} + (4 \beta_{7} + 2 \beta_{6}) q^{57} + ( - 12 \beta_{7} + \beta_{6} - 12 \beta_{3} - \beta_1) q^{58} + ( - 2 \beta_{5} + 2 \beta_{4} - 8 \beta_{2} - 6) q^{59} + ( - 2 \beta_{5} + 2 \beta_{4} - 9 \beta_{2} - 7) q^{61} - 4 \beta_{3} q^{62} + (\beta_{3} + \beta_1) q^{63} + ( - \beta_{4} - 4) q^{64} + 2 \beta_{4} q^{66} + (3 \beta_{3} - \beta_1) q^{67} + ( - 4 \beta_{3} - 3 \beta_1) q^{68} + (2 \beta_{2} + 2) q^{69} + ( - 14 \beta_{2} - 14) q^{71} + ( - 4 \beta_{7} + \beta_{6} - 4 \beta_{3} - \beta_1) q^{72} + (7 \beta_{7} + 2 \beta_{6}) q^{73} + (5 \beta_{5} - 5 \beta_{4} + \beta_{2} - 4) q^{74} + ( - 2 \beta_{5} - 2 \beta_{2}) q^{76} + (2 \beta_{7} - 2 \beta_{6}) q^{77} + (4 \beta_{7} - \beta_{6} - 4 \beta_{3}) q^{78} + (\beta_{4} - 7) q^{79} + \beta_{2} q^{81} + ( - 4 \beta_{7} + \beta_{6} - 4 \beta_{3} - \beta_1) q^{82} + (4 \beta_{7} - 2 \beta_{6}) q^{83} + ( - 4 \beta_{5} + 4 \beta_{4} - 10 \beta_{2} - 6) q^{84} + ( - 2 \beta_{4} - 4) q^{86} + ( - 2 \beta_{7} - 3 \beta_{6} - 2 \beta_{3} + 3 \beta_1) q^{87} + (8 \beta_{3} + 2 \beta_1) q^{88} + ( - 2 \beta_{5} - 10 \beta_{2}) q^{89} + ( - \beta_{5} - 2 \beta_{4} - 4 \beta_{2} + 4) q^{91} + (4 \beta_{7} - 2 \beta_{6}) q^{92} + ( - \beta_{3} + \beta_1) q^{93} + (2 \beta_{5} + 18 \beta_{2}) q^{94} + ( - \beta_{4} + 4) q^{96} + (7 \beta_{7} + \beta_{6} + 7 \beta_{3} - \beta_1) q^{97} + (12 \beta_{7} - \beta_{6} + 12 \beta_{3} + \beta_1) q^{98} + 2 q^{99}+O(q^{100})$$ q + b1 * q^2 - b3 * q^3 + (b5 - b4 + 3*b2 + 2) * q^4 + (-b5 + b4 - b2) * q^6 + (b7 - b6 + b3 + b1) * q^7 + (-4*b7 + b6) * q^8 + (b2 + 1) * q^9 - 2*b2 * q^11 + (2*b7 - b6) * q^12 + (-b7 - b6 - b3 + 2*b1) * q^13 + (-2*b4 + 4) * q^14 + (3*b5 + 3*b2) * q^16 + (b6 - b1) * q^17 + b6 * q^18 + (-2*b5 + 2*b4 + 2*b2 + 4) * q^19 + (b4 - 1) * q^21 + (-2*b6 + 2*b1) * q^22 - 2*b3 * q^23 + (-b5 - 5*b2) * q^24 + (b5 - b4 + 5*b2 + 8) * q^26 + b7 * q^27 + (6*b3 + 4*b1) * q^28 + (3*b5 + b2) * q^29 + (b4 + 1) * q^31 + (-4*b7 + b6 - 4*b3 - b1) * q^32 + (-2*b7 - 2*b3) * q^33 + (b4 - 4) * q^34 + (b5 + 3*b2) * q^36 + (6*b3 - b1) * q^37 + (8*b7 + 2*b6) * q^38 + (-b5 + 2*b4 - b2 + 1) * q^39 + (b5 + b2) * q^41 + (-4*b3 - 2*b1) * q^42 + (3*b7 + b6 + 3*b3 - b1) * q^43 + (-2*b4 + 4) * q^44 + (-2*b5 + 2*b4 - 2*b2) * q^46 + (2*b7 + 4*b6) * q^47 + (-3*b6 + 3*b1) * q^48 + (-3*b5 - b2) * q^49 - b4 * q^51 + (-4*b7 + 3*b6 + 2*b3 + 2*b1) * q^52 + (-4*b7 + 3*b6) * q^53 + (-b5 - b2) * q^54 + (6*b5 - 6*b4 + 14*b2 + 8) * q^56 + (4*b7 + 2*b6) * q^57 + (-12*b7 + b6 - 12*b3 - b1) * q^58 + (-2*b5 + 2*b4 - 8*b2 - 6) * q^59 + (-2*b5 + 2*b4 - 9*b2 - 7) * q^61 - 4*b3 * q^62 + (b3 + b1) * q^63 + (-b4 - 4) * q^64 + 2*b4 * q^66 + (3*b3 - b1) * q^67 + (-4*b3 - 3*b1) * q^68 + (2*b2 + 2) * q^69 + (-14*b2 - 14) * q^71 + (-4*b7 + b6 - 4*b3 - b1) * q^72 + (7*b7 + 2*b6) * q^73 + (5*b5 - 5*b4 + b2 - 4) * q^74 + (-2*b5 - 2*b2) * q^76 + (2*b7 - 2*b6) * q^77 + (4*b7 - b6 - 4*b3) * q^78 + (b4 - 7) * q^79 + b2 * q^81 + (-4*b7 + b6 - 4*b3 - b1) * q^82 + (4*b7 - 2*b6) * q^83 + (-4*b5 + 4*b4 - 10*b2 - 6) * q^84 + (-2*b4 - 4) * q^86 + (-2*b7 - 3*b6 - 2*b3 + 3*b1) * q^87 + (8*b3 + 2*b1) * q^88 + (-2*b5 - 10*b2) * q^89 + (-b5 - 2*b4 - 4*b2 + 4) * q^91 + (4*b7 - 2*b6) * q^92 + (-b3 + b1) * q^93 + (2*b5 + 18*b2) * q^94 + (-b4 + 4) * q^96 + (7*b7 + b6 + 7*b3 - b1) * q^97 + (12*b7 - b6 + 12*b3 + b1) * q^98 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 10 q^{4} - 2 q^{6} + 4 q^{9}+O(q^{10})$$ 8 * q + 10 * q^4 - 2 * q^6 + 4 * q^9 $$8 q + 10 q^{4} - 2 q^{6} + 4 q^{9} + 8 q^{11} + 40 q^{14} - 6 q^{16} + 12 q^{19} - 12 q^{21} + 18 q^{24} + 50 q^{26} + 2 q^{29} + 4 q^{31} - 36 q^{34} - 10 q^{36} + 2 q^{39} - 2 q^{41} + 40 q^{44} - 4 q^{46} - 2 q^{49} + 4 q^{51} + 2 q^{54} + 44 q^{56} - 28 q^{59} - 32 q^{61} - 28 q^{64} - 8 q^{66} + 8 q^{69} - 56 q^{71} - 6 q^{74} + 4 q^{76} - 60 q^{79} - 4 q^{81} - 32 q^{84} - 24 q^{86} + 36 q^{89} + 54 q^{91} - 68 q^{94} + 36 q^{96} + 16 q^{99}+O(q^{100})$$ 8 * q + 10 * q^4 - 2 * q^6 + 4 * q^9 + 8 * q^11 + 40 * q^14 - 6 * q^16 + 12 * q^19 - 12 * q^21 + 18 * q^24 + 50 * q^26 + 2 * q^29 + 4 * q^31 - 36 * q^34 - 10 * q^36 + 2 * q^39 - 2 * q^41 + 40 * q^44 - 4 * q^46 - 2 * q^49 + 4 * q^51 + 2 * q^54 + 44 * q^56 - 28 * q^59 - 32 * q^61 - 28 * q^64 - 8 * q^66 + 8 * q^69 - 56 * q^71 - 6 * q^74 + 4 * q^76 - 60 * q^79 - 4 * q^81 - 32 * q^84 - 24 * q^86 + 36 * q^89 + 54 * q^91 - 68 * q^94 + 36 * q^96 + 16 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 9\nu^{6} - 65\nu^{4} + 585\nu^{2} - 1296 ) / 1040$$ (9*v^6 - 65*v^4 + 585*v^2 - 1296) / 1040 $$\beta_{3}$$ $$=$$ $$( -\nu^{7} - 181\nu ) / 260$$ (-v^7 - 181*v) / 260 $$\beta_{4}$$ $$=$$ $$( \nu^{6} + 116 ) / 65$$ (v^6 + 116) / 65 $$\beta_{5}$$ $$=$$ $$( -29\nu^{6} + 325\nu^{4} - 1885\nu^{2} + 4176 ) / 1040$$ (-29*v^6 + 325*v^4 - 1885*v^2 + 4176) / 1040 $$\beta_{6}$$ $$=$$ $$( 9\nu^{7} - 65\nu^{5} + 585\nu^{3} - 256\nu ) / 1040$$ (9*v^7 - 65*v^5 + 585*v^3 - 256*v) / 1040 $$\beta_{7}$$ $$=$$ $$( 9\nu^{7} - 65\nu^{5} + 377\nu^{3} - 256\nu ) / 832$$ (9*v^7 - 65*v^5 + 377*v^3 - 256*v) / 832
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} - \beta_{4} + 5\beta_{2} + 4$$ b5 - b4 + 5*b2 + 4 $$\nu^{3}$$ $$=$$ $$-4\beta_{7} + 5\beta_{6}$$ -4*b7 + 5*b6 $$\nu^{4}$$ $$=$$ $$9\beta_{5} + 29\beta_{2}$$ 9*b5 + 29*b2 $$\nu^{5}$$ $$=$$ $$-36\beta_{7} + 29\beta_{6} - 36\beta_{3} - 29\beta_1$$ -36*b7 + 29*b6 - 36*b3 - 29*b1 $$\nu^{6}$$ $$=$$ $$65\beta_{4} - 116$$ 65*b4 - 116 $$\nu^{7}$$ $$=$$ $$-260\beta_{3} - 181\beta_1$$ -260*b3 - 181*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$326$$ $$352$$ $$\chi(n)$$ $$-1 - \beta_{2}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
724.1
 −2.21837 − 1.28078i −1.35234 − 0.780776i 1.35234 + 0.780776i 2.21837 + 1.28078i −2.21837 + 1.28078i −1.35234 + 0.780776i 1.35234 − 0.780776i 2.21837 − 1.28078i
−2.21837 1.28078i 0.866025 + 0.500000i 2.28078 + 3.95042i 0 −1.28078 2.21837i −3.08440 + 1.78078i 6.56155i 0.500000 + 0.866025i 0
724.2 −1.35234 0.780776i −0.866025 0.500000i 0.219224 + 0.379706i 0 0.780776 + 1.35234i −0.486319 + 0.280776i 2.43845i 0.500000 + 0.866025i 0
724.3 1.35234 + 0.780776i 0.866025 + 0.500000i 0.219224 + 0.379706i 0 0.780776 + 1.35234i 0.486319 0.280776i 2.43845i 0.500000 + 0.866025i 0
724.4 2.21837 + 1.28078i −0.866025 0.500000i 2.28078 + 3.95042i 0 −1.28078 2.21837i 3.08440 1.78078i 6.56155i 0.500000 + 0.866025i 0
874.1 −2.21837 + 1.28078i 0.866025 0.500000i 2.28078 3.95042i 0 −1.28078 + 2.21837i −3.08440 1.78078i 6.56155i 0.500000 0.866025i 0
874.2 −1.35234 + 0.780776i −0.866025 + 0.500000i 0.219224 0.379706i 0 0.780776 1.35234i −0.486319 0.280776i 2.43845i 0.500000 0.866025i 0
874.3 1.35234 0.780776i 0.866025 0.500000i 0.219224 0.379706i 0 0.780776 1.35234i 0.486319 + 0.280776i 2.43845i 0.500000 0.866025i 0
874.4 2.21837 1.28078i −0.866025 + 0.500000i 2.28078 3.95042i 0 −1.28078 + 2.21837i 3.08440 + 1.78078i 6.56155i 0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 874.4 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
13.c even 3 1 inner
65.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bb.i 8
5.b even 2 1 inner 975.2.bb.i 8
5.c odd 4 1 39.2.e.b 4
5.c odd 4 1 975.2.i.k 4
13.c even 3 1 inner 975.2.bb.i 8
15.e even 4 1 117.2.g.c 4
20.e even 4 1 624.2.q.h 4
60.l odd 4 1 1872.2.t.r 4
65.f even 4 1 507.2.j.g 8
65.h odd 4 1 507.2.e.g 4
65.k even 4 1 507.2.j.g 8
65.n even 6 1 inner 975.2.bb.i 8
65.o even 12 1 507.2.b.d 4
65.o even 12 1 507.2.j.g 8
65.q odd 12 1 39.2.e.b 4
65.q odd 12 1 507.2.a.g 2
65.q odd 12 1 975.2.i.k 4
65.r odd 12 1 507.2.a.d 2
65.r odd 12 1 507.2.e.g 4
65.t even 12 1 507.2.b.d 4
65.t even 12 1 507.2.j.g 8
195.bc odd 12 1 1521.2.b.h 4
195.bf even 12 1 1521.2.a.m 2
195.bl even 12 1 117.2.g.c 4
195.bl even 12 1 1521.2.a.g 2
195.bn odd 12 1 1521.2.b.h 4
260.bg even 12 1 8112.2.a.bo 2
260.bj even 12 1 624.2.q.h 4
260.bj even 12 1 8112.2.a.bk 2
780.cj odd 12 1 1872.2.t.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.b 4 5.c odd 4 1
39.2.e.b 4 65.q odd 12 1
117.2.g.c 4 15.e even 4 1
117.2.g.c 4 195.bl even 12 1
507.2.a.d 2 65.r odd 12 1
507.2.a.g 2 65.q odd 12 1
507.2.b.d 4 65.o even 12 1
507.2.b.d 4 65.t even 12 1
507.2.e.g 4 65.h odd 4 1
507.2.e.g 4 65.r odd 12 1
507.2.j.g 8 65.f even 4 1
507.2.j.g 8 65.k even 4 1
507.2.j.g 8 65.o even 12 1
507.2.j.g 8 65.t even 12 1
624.2.q.h 4 20.e even 4 1
624.2.q.h 4 260.bj even 12 1
975.2.i.k 4 5.c odd 4 1
975.2.i.k 4 65.q odd 12 1
975.2.bb.i 8 1.a even 1 1 trivial
975.2.bb.i 8 5.b even 2 1 inner
975.2.bb.i 8 13.c even 3 1 inner
975.2.bb.i 8 65.n even 6 1 inner
1521.2.a.g 2 195.bl even 12 1
1521.2.a.m 2 195.bf even 12 1
1521.2.b.h 4 195.bc odd 12 1
1521.2.b.h 4 195.bn odd 12 1
1872.2.t.r 4 60.l odd 4 1
1872.2.t.r 4 780.cj odd 12 1
8112.2.a.bk 2 260.bj even 12 1
8112.2.a.bo 2 260.bg even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(975, [\chi])$$:

 $$T_{2}^{8} - 9T_{2}^{6} + 65T_{2}^{4} - 144T_{2}^{2} + 256$$ T2^8 - 9*T2^6 + 65*T2^4 - 144*T2^2 + 256 $$T_{7}^{8} - 13T_{7}^{6} + 165T_{7}^{4} - 52T_{7}^{2} + 16$$ T7^8 - 13*T7^6 + 165*T7^4 - 52*T7^2 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 9 T^{6} + 65 T^{4} - 144 T^{2} + \cdots + 256$$
$3$ $$(T^{4} - T^{2} + 1)^{2}$$
$5$ $$T^{8}$$
$7$ $$T^{8} - 13 T^{6} + 165 T^{4} + \cdots + 16$$
$11$ $$(T^{2} - 2 T + 4)^{4}$$
$13$ $$(T^{4} - 25 T^{2} + 169)^{2}$$
$17$ $$T^{8} - 9 T^{6} + 65 T^{4} - 144 T^{2} + \cdots + 256$$
$19$ $$(T^{4} - 6 T^{3} + 44 T^{2} + 48 T + 64)^{2}$$
$23$ $$(T^{4} - 4 T^{2} + 16)^{2}$$
$29$ $$(T^{4} - T^{3} + 39 T^{2} + 38 T + 1444)^{2}$$
$31$ $$(T^{2} - T - 4)^{4}$$
$37$ $$T^{8} - 69 T^{6} + 4085 T^{4} + \cdots + 456976$$
$41$ $$(T^{4} + T^{3} + 5 T^{2} - 4 T + 16)^{2}$$
$43$ $$T^{8} - 21 T^{6} + 437 T^{4} + \cdots + 16$$
$47$ $$(T^{2} + 68)^{4}$$
$53$ $$(T^{4} + 137 T^{2} + 64)^{2}$$
$59$ $$(T^{4} + 14 T^{3} + 164 T^{2} + 448 T + 1024)^{2}$$
$61$ $$(T^{4} + 16 T^{3} + 209 T^{2} + 752 T + 2209)^{2}$$
$67$ $$T^{8} - 21 T^{6} + 437 T^{4} + \cdots + 16$$
$71$ $$(T^{2} + 14 T + 196)^{4}$$
$73$ $$(T^{4} + 106 T^{2} + 361)^{2}$$
$79$ $$(T^{2} + 15 T + 52)^{4}$$
$83$ $$(T^{4} + 84 T^{2} + 64)^{2}$$
$89$ $$(T^{4} - 18 T^{3} + 260 T^{2} - 1152 T + 4096)^{2}$$
$97$ $$T^{8} - 93 T^{6} + 7205 T^{4} + \cdots + 2085136$$