# Properties

 Label 39.2.e.b Level $39$ Weight $2$ Character orbit 39.e Analytic conductor $0.311$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$39 = 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 39.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.311416567883$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{17})$$ Defining polynomial: $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ x^4 - x^3 + 5*x^2 + 4*x + 16 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - \beta_{2} q^{3} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{4} + ( - \beta_{3} - 2) q^{5} + (\beta_{3} + \beta_1) q^{6} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{7} + ( - \beta_{3} + 4) q^{8} + (\beta_{2} - 1) q^{9}+O(q^{10})$$ q - b1 * q^2 - b2 * q^3 + (b3 + 2*b2 + b1 - 2) * q^4 + (-b3 - 2) * q^5 + (b3 + b1) * q^6 + (-b3 - b2 - b1 + 1) * q^7 + (-b3 + 4) * q^8 + (b2 - 1) * q^9 $$q - \beta_1 q^{2} - \beta_{2} q^{3} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{4} + ( - \beta_{3} - 2) q^{5} + (\beta_{3} + \beta_1) q^{6} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{7} + ( - \beta_{3} + 4) q^{8} + (\beta_{2} - 1) q^{9} + ( - 4 \beta_{2} + \beta_1) q^{10} + 2 \beta_{2} q^{11} + ( - \beta_{3} + 2) q^{12} + (\beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{13} + (2 \beta_{3} - 4) q^{14} + (2 \beta_{2} - \beta_1) q^{15} - 3 \beta_1 q^{16} + (\beta_{3} + \beta_1) q^{17} - \beta_{3} q^{18} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 4) q^{19} + (\beta_{3} + \beta_1) q^{20} + (\beta_{3} - 1) q^{21} + ( - 2 \beta_{3} - 2 \beta_1) q^{22} - 2 \beta_{2} q^{23} + ( - 4 \beta_{2} - \beta_1) q^{24} + (3 \beta_{3} + 3) q^{25} + ( - \beta_{3} - 4 \beta_{2} - \beta_1 + 8) q^{26} + q^{27} + (6 \beta_{2} + 4 \beta_1) q^{28} + ( - 2 \beta_{2} + 3 \beta_1) q^{29} + ( - \beta_{3} + 4 \beta_{2} - \beta_1 - 4) q^{30} + (\beta_{3} + 1) q^{31} + (\beta_{3} + 4 \beta_{2} + \beta_1 - 4) q^{32} + ( - 2 \beta_{2} + 2) q^{33} + ( - \beta_{3} + 4) q^{34} + ( - 2 \beta_{2} + 2) q^{35} + ( - 2 \beta_{2} - \beta_1) q^{36} + ( - 6 \beta_{2} + \beta_1) q^{37} + ( - 2 \beta_{3} - 8) q^{38} + ( - 2 \beta_{3} - \beta_1 - 1) q^{39} + ( - 3 \beta_{3} - 4) q^{40} - \beta_1 q^{41} + (4 \beta_{2} + 2 \beta_1) q^{42} + ( - \beta_{3} + 3 \beta_{2} - \beta_1 - 3) q^{43} + (2 \beta_{3} - 4) q^{44} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{45} + (2 \beta_{3} + 2 \beta_1) q^{46} + (4 \beta_{3} + 2) q^{47} + (3 \beta_{3} + 3 \beta_1) q^{48} + (2 \beta_{2} - 3 \beta_1) q^{49} + 12 \beta_{2} q^{50} - \beta_{3} q^{51} + (3 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 4) q^{52} + ( - 3 \beta_{3} + 4) q^{53} - \beta_1 q^{54} + ( - 4 \beta_{2} + 2 \beta_1) q^{55} + ( - 6 \beta_{3} - 8 \beta_{2} - 6 \beta_1 + 8) q^{56} + (2 \beta_{3} + 4) q^{57} + ( - \beta_{3} - 12 \beta_{2} - \beta_1 + 12) q^{58} + ( - 2 \beta_{3} - 6 \beta_{2} - 2 \beta_1 + 6) q^{59} - \beta_{3} q^{60} + (2 \beta_{3} + 7 \beta_{2} + 2 \beta_1 - 7) q^{61} + 4 \beta_{2} q^{62} + (\beta_{2} + \beta_1) q^{63} + (\beta_{3} + 4) q^{64} + ( - 2 \beta_{3} + 10 \beta_{2} - 3 \beta_1 - 6) q^{65} + 2 \beta_{3} q^{66} + ( - 3 \beta_{2} + \beta_1) q^{67} + ( - 4 \beta_{2} - 3 \beta_1) q^{68} + (2 \beta_{2} - 2) q^{69} + 2 \beta_{3} q^{70} + (14 \beta_{2} - 14) q^{71} + (\beta_{3} + 4 \beta_{2} + \beta_1 - 4) q^{72} + ( - 2 \beta_{3} - 7) q^{73} + (5 \beta_{3} - 4 \beta_{2} + 5 \beta_1 + 4) q^{74} + ( - 3 \beta_{2} + 3 \beta_1) q^{75} + 2 \beta_1 q^{76} + ( - 2 \beta_{3} + 2) q^{77} + (\beta_{3} - 4 \beta_{2} - 4) q^{78} + ( - \beta_{3} + 7) q^{79} + ( - 12 \beta_{2} + 3 \beta_1) q^{80} - \beta_{2} q^{81} + (\beta_{3} + 4 \beta_{2} + \beta_1 - 4) q^{82} + (2 \beta_{3} - 4) q^{83} + ( - 4 \beta_{3} - 6 \beta_{2} - 4 \beta_1 + 6) q^{84} + ( - \beta_{3} + 4 \beta_{2} - \beta_1 - 4) q^{85} + ( - 2 \beta_{3} - 4) q^{86} + ( - 3 \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 2) q^{87} + (8 \beta_{2} + 2 \beta_1) q^{88} + ( - 8 \beta_{2} - 2 \beta_1) q^{89} + (\beta_{3} + 4) q^{90} + ( - 2 \beta_{3} + 3 \beta_{2} + \beta_1 + 4) q^{91} + ( - 2 \beta_{3} + 4) q^{92} + ( - \beta_{2} + \beta_1) q^{93} + (16 \beta_{2} + 2 \beta_1) q^{94} + (6 \beta_{3} - 16 \beta_{2} + 6 \beta_1 + 16) q^{95} + ( - \beta_{3} + 4) q^{96} + (\beta_{3} - 7 \beta_{2} + \beta_1 + 7) q^{97} + (\beta_{3} + 12 \beta_{2} + \beta_1 - 12) q^{98} - 2 q^{99}+O(q^{100})$$ q - b1 * q^2 - b2 * q^3 + (b3 + 2*b2 + b1 - 2) * q^4 + (-b3 - 2) * q^5 + (b3 + b1) * q^6 + (-b3 - b2 - b1 + 1) * q^7 + (-b3 + 4) * q^8 + (b2 - 1) * q^9 + (-4*b2 + b1) * q^10 + 2*b2 * q^11 + (-b3 + 2) * q^12 + (b3 - b2 + 2*b1 + 1) * q^13 + (2*b3 - 4) * q^14 + (2*b2 - b1) * q^15 - 3*b1 * q^16 + (b3 + b1) * q^17 - b3 * q^18 + (-2*b3 + 4*b2 - 2*b1 - 4) * q^19 + (b3 + b1) * q^20 + (b3 - 1) * q^21 + (-2*b3 - 2*b1) * q^22 - 2*b2 * q^23 + (-4*b2 - b1) * q^24 + (3*b3 + 3) * q^25 + (-b3 - 4*b2 - b1 + 8) * q^26 + q^27 + (6*b2 + 4*b1) * q^28 + (-2*b2 + 3*b1) * q^29 + (-b3 + 4*b2 - b1 - 4) * q^30 + (b3 + 1) * q^31 + (b3 + 4*b2 + b1 - 4) * q^32 + (-2*b2 + 2) * q^33 + (-b3 + 4) * q^34 + (-2*b2 + 2) * q^35 + (-2*b2 - b1) * q^36 + (-6*b2 + b1) * q^37 + (-2*b3 - 8) * q^38 + (-2*b3 - b1 - 1) * q^39 + (-3*b3 - 4) * q^40 - b1 * q^41 + (4*b2 + 2*b1) * q^42 + (-b3 + 3*b2 - b1 - 3) * q^43 + (2*b3 - 4) * q^44 + (b3 - 2*b2 + b1 + 2) * q^45 + (2*b3 + 2*b1) * q^46 + (4*b3 + 2) * q^47 + (3*b3 + 3*b1) * q^48 + (2*b2 - 3*b1) * q^49 + 12*b2 * q^50 - b3 * q^51 + (3*b3 - 2*b2 - 2*b1 - 4) * q^52 + (-3*b3 + 4) * q^53 - b1 * q^54 + (-4*b2 + 2*b1) * q^55 + (-6*b3 - 8*b2 - 6*b1 + 8) * q^56 + (2*b3 + 4) * q^57 + (-b3 - 12*b2 - b1 + 12) * q^58 + (-2*b3 - 6*b2 - 2*b1 + 6) * q^59 - b3 * q^60 + (2*b3 + 7*b2 + 2*b1 - 7) * q^61 + 4*b2 * q^62 + (b2 + b1) * q^63 + (b3 + 4) * q^64 + (-2*b3 + 10*b2 - 3*b1 - 6) * q^65 + 2*b3 * q^66 + (-3*b2 + b1) * q^67 + (-4*b2 - 3*b1) * q^68 + (2*b2 - 2) * q^69 + 2*b3 * q^70 + (14*b2 - 14) * q^71 + (b3 + 4*b2 + b1 - 4) * q^72 + (-2*b3 - 7) * q^73 + (5*b3 - 4*b2 + 5*b1 + 4) * q^74 + (-3*b2 + 3*b1) * q^75 + 2*b1 * q^76 + (-2*b3 + 2) * q^77 + (b3 - 4*b2 - 4) * q^78 + (-b3 + 7) * q^79 + (-12*b2 + 3*b1) * q^80 - b2 * q^81 + (b3 + 4*b2 + b1 - 4) * q^82 + (2*b3 - 4) * q^83 + (-4*b3 - 6*b2 - 4*b1 + 6) * q^84 + (-b3 + 4*b2 - b1 - 4) * q^85 + (-2*b3 - 4) * q^86 + (-3*b3 + 2*b2 - 3*b1 - 2) * q^87 + (8*b2 + 2*b1) * q^88 + (-8*b2 - 2*b1) * q^89 + (b3 + 4) * q^90 + (-2*b3 + 3*b2 + b1 + 4) * q^91 + (-2*b3 + 4) * q^92 + (-b2 + b1) * q^93 + (16*b2 + 2*b1) * q^94 + (6*b3 - 16*b2 + 6*b1 + 16) * q^95 + (-b3 + 4) * q^96 + (b3 - 7*b2 + b1 + 7) * q^97 + (b3 + 12*b2 + b1 - 12) * q^98 - 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{2} - 2 q^{3} - 5 q^{4} - 6 q^{5} - q^{6} + 3 q^{7} + 18 q^{8} - 2 q^{9}+O(q^{10})$$ 4 * q - q^2 - 2 * q^3 - 5 * q^4 - 6 * q^5 - q^6 + 3 * q^7 + 18 * q^8 - 2 * q^9 $$4 q - q^{2} - 2 q^{3} - 5 q^{4} - 6 q^{5} - q^{6} + 3 q^{7} + 18 q^{8} - 2 q^{9} - 7 q^{10} + 4 q^{11} + 10 q^{12} + 2 q^{13} - 20 q^{14} + 3 q^{15} - 3 q^{16} - q^{17} + 2 q^{18} - 6 q^{19} - q^{20} - 6 q^{21} + 2 q^{22} - 4 q^{23} - 9 q^{24} + 6 q^{25} + 25 q^{26} + 4 q^{27} + 16 q^{28} - q^{29} - 7 q^{30} + 2 q^{31} - 9 q^{32} + 4 q^{33} + 18 q^{34} + 4 q^{35} - 5 q^{36} - 11 q^{37} - 28 q^{38} - q^{39} - 10 q^{40} - q^{41} + 10 q^{42} - 5 q^{43} - 20 q^{44} + 3 q^{45} - 2 q^{46} - 3 q^{48} + q^{49} + 24 q^{50} + 2 q^{51} - 28 q^{52} + 22 q^{53} - q^{54} - 6 q^{55} + 22 q^{56} + 12 q^{57} + 25 q^{58} + 14 q^{59} + 2 q^{60} - 16 q^{61} + 8 q^{62} + 3 q^{63} + 14 q^{64} - 3 q^{65} - 4 q^{66} - 5 q^{67} - 11 q^{68} - 4 q^{69} - 4 q^{70} - 28 q^{71} - 9 q^{72} - 24 q^{73} + 3 q^{74} - 3 q^{75} + 2 q^{76} + 12 q^{77} - 26 q^{78} + 30 q^{79} - 21 q^{80} - 2 q^{81} - 9 q^{82} - 20 q^{83} + 16 q^{84} - 7 q^{85} - 12 q^{86} - q^{87} + 18 q^{88} - 18 q^{89} + 14 q^{90} + 27 q^{91} + 20 q^{92} - q^{93} + 34 q^{94} + 26 q^{95} + 18 q^{96} + 13 q^{97} - 25 q^{98} - 8 q^{99}+O(q^{100})$$ 4 * q - q^2 - 2 * q^3 - 5 * q^4 - 6 * q^5 - q^6 + 3 * q^7 + 18 * q^8 - 2 * q^9 - 7 * q^10 + 4 * q^11 + 10 * q^12 + 2 * q^13 - 20 * q^14 + 3 * q^15 - 3 * q^16 - q^17 + 2 * q^18 - 6 * q^19 - q^20 - 6 * q^21 + 2 * q^22 - 4 * q^23 - 9 * q^24 + 6 * q^25 + 25 * q^26 + 4 * q^27 + 16 * q^28 - q^29 - 7 * q^30 + 2 * q^31 - 9 * q^32 + 4 * q^33 + 18 * q^34 + 4 * q^35 - 5 * q^36 - 11 * q^37 - 28 * q^38 - q^39 - 10 * q^40 - q^41 + 10 * q^42 - 5 * q^43 - 20 * q^44 + 3 * q^45 - 2 * q^46 - 3 * q^48 + q^49 + 24 * q^50 + 2 * q^51 - 28 * q^52 + 22 * q^53 - q^54 - 6 * q^55 + 22 * q^56 + 12 * q^57 + 25 * q^58 + 14 * q^59 + 2 * q^60 - 16 * q^61 + 8 * q^62 + 3 * q^63 + 14 * q^64 - 3 * q^65 - 4 * q^66 - 5 * q^67 - 11 * q^68 - 4 * q^69 - 4 * q^70 - 28 * q^71 - 9 * q^72 - 24 * q^73 + 3 * q^74 - 3 * q^75 + 2 * q^76 + 12 * q^77 - 26 * q^78 + 30 * q^79 - 21 * q^80 - 2 * q^81 - 9 * q^82 - 20 * q^83 + 16 * q^84 - 7 * q^85 - 12 * q^86 - q^87 + 18 * q^88 - 18 * q^89 + 14 * q^90 + 27 * q^91 + 20 * q^92 - q^93 + 34 * q^94 + 26 * q^95 + 18 * q^96 + 13 * q^97 - 25 * q^98 - 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20$$ (-v^3 + 5*v^2 - 5*v + 16) / 20 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 4 ) / 5$$ (v^3 + 4) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4\beta_{2} + \beta _1 - 4$$ b3 + 4*b2 + b1 - 4 $$\nu^{3}$$ $$=$$ $$5\beta_{3} - 4$$ 5*b3 - 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/39\mathbb{Z}\right)^\times$$.

 $$n$$ $$14$$ $$28$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1
 1.28078 + 2.21837i −0.780776 − 1.35234i 1.28078 − 2.21837i −0.780776 + 1.35234i
−1.28078 2.21837i −0.500000 0.866025i −2.28078 + 3.95042i 0.561553 −1.28078 + 2.21837i 1.78078 3.08440i 6.56155 −0.500000 + 0.866025i −0.719224 1.24573i
16.2 0.780776 + 1.35234i −0.500000 0.866025i −0.219224 + 0.379706i −3.56155 0.780776 1.35234i −0.280776 + 0.486319i 2.43845 −0.500000 + 0.866025i −2.78078 4.81645i
22.1 −1.28078 + 2.21837i −0.500000 + 0.866025i −2.28078 3.95042i 0.561553 −1.28078 2.21837i 1.78078 + 3.08440i 6.56155 −0.500000 0.866025i −0.719224 + 1.24573i
22.2 0.780776 1.35234i −0.500000 + 0.866025i −0.219224 0.379706i −3.56155 0.780776 + 1.35234i −0.280776 0.486319i 2.43845 −0.500000 0.866025i −2.78078 + 4.81645i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.2.e.b 4
3.b odd 2 1 117.2.g.c 4
4.b odd 2 1 624.2.q.h 4
5.b even 2 1 975.2.i.k 4
5.c odd 4 2 975.2.bb.i 8
12.b even 2 1 1872.2.t.r 4
13.b even 2 1 507.2.e.g 4
13.c even 3 1 inner 39.2.e.b 4
13.c even 3 1 507.2.a.g 2
13.d odd 4 2 507.2.j.g 8
13.e even 6 1 507.2.a.d 2
13.e even 6 1 507.2.e.g 4
13.f odd 12 2 507.2.b.d 4
13.f odd 12 2 507.2.j.g 8
39.h odd 6 1 1521.2.a.m 2
39.i odd 6 1 117.2.g.c 4
39.i odd 6 1 1521.2.a.g 2
39.k even 12 2 1521.2.b.h 4
52.i odd 6 1 8112.2.a.bo 2
52.j odd 6 1 624.2.q.h 4
52.j odd 6 1 8112.2.a.bk 2
65.n even 6 1 975.2.i.k 4
65.q odd 12 2 975.2.bb.i 8
156.p even 6 1 1872.2.t.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.b 4 1.a even 1 1 trivial
39.2.e.b 4 13.c even 3 1 inner
117.2.g.c 4 3.b odd 2 1
117.2.g.c 4 39.i odd 6 1
507.2.a.d 2 13.e even 6 1
507.2.a.g 2 13.c even 3 1
507.2.b.d 4 13.f odd 12 2
507.2.e.g 4 13.b even 2 1
507.2.e.g 4 13.e even 6 1
507.2.j.g 8 13.d odd 4 2
507.2.j.g 8 13.f odd 12 2
624.2.q.h 4 4.b odd 2 1
624.2.q.h 4 52.j odd 6 1
975.2.i.k 4 5.b even 2 1
975.2.i.k 4 65.n even 6 1
975.2.bb.i 8 5.c odd 4 2
975.2.bb.i 8 65.q odd 12 2
1521.2.a.g 2 39.i odd 6 1
1521.2.a.m 2 39.h odd 6 1
1521.2.b.h 4 39.k even 12 2
1872.2.t.r 4 12.b even 2 1
1872.2.t.r 4 156.p even 6 1
8112.2.a.bk 2 52.j odd 6 1
8112.2.a.bo 2 52.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + T_{2}^{3} + 5T_{2}^{2} - 4T_{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(39, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + T^{3} + 5 T^{2} - 4 T + 16$$
$3$ $$(T^{2} + T + 1)^{2}$$
$5$ $$(T^{2} + 3 T - 2)^{2}$$
$7$ $$T^{4} - 3 T^{3} + 11 T^{2} + 6 T + 4$$
$11$ $$(T^{2} - 2 T + 4)^{2}$$
$13$ $$(T^{2} - T + 13)^{2}$$
$17$ $$T^{4} + T^{3} + 5 T^{2} - 4 T + 16$$
$19$ $$T^{4} + 6 T^{3} + 44 T^{2} - 48 T + 64$$
$23$ $$(T^{2} + 2 T + 4)^{2}$$
$29$ $$T^{4} + T^{3} + 39 T^{2} - 38 T + 1444$$
$31$ $$(T^{2} - T - 4)^{2}$$
$37$ $$T^{4} + 11 T^{3} + 95 T^{2} + \cdots + 676$$
$41$ $$T^{4} + T^{3} + 5 T^{2} - 4 T + 16$$
$43$ $$T^{4} + 5 T^{3} + 23 T^{2} + 10 T + 4$$
$47$ $$(T^{2} - 68)^{2}$$
$53$ $$(T^{2} - 11 T - 8)^{2}$$
$59$ $$T^{4} - 14 T^{3} + 164 T^{2} + \cdots + 1024$$
$61$ $$T^{4} + 16 T^{3} + 209 T^{2} + \cdots + 2209$$
$67$ $$T^{4} + 5 T^{3} + 23 T^{2} + 10 T + 4$$
$71$ $$(T^{2} + 14 T + 196)^{2}$$
$73$ $$(T^{2} + 12 T + 19)^{2}$$
$79$ $$(T^{2} - 15 T + 52)^{2}$$
$83$ $$(T^{2} + 10 T + 8)^{2}$$
$89$ $$T^{4} + 18 T^{3} + 260 T^{2} + \cdots + 4096$$
$97$ $$T^{4} - 13 T^{3} + 131 T^{2} + \cdots + 1444$$