# Properties

 Label 420.2.bh.b Level $420$ Weight $2$ Character orbit 420.bh Analytic conductor $3.354$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$420 = 2^{2} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 420.bh (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.35371688489$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{6})$$ Coefficient field: 10.0.29471584693248.1 Defining polynomial: $$x^{10} - 2x^{9} + 2x^{8} - 4x^{7} + 13x^{6} - 36x^{5} + 39x^{4} - 36x^{3} + 54x^{2} - 162x + 243$$ x^10 - 2*x^9 + 2*x^8 - 4*x^7 + 13*x^6 - 36*x^5 + 39*x^4 - 36*x^3 + 54*x^2 - 162*x + 243 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{3} + (\beta_{7} + 1) q^{5} + (\beta_{9} - \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{7} + (\beta_{9} - \beta_{7} + \beta_{3} - \beta_{2}) q^{9}+O(q^{10})$$ q + b6 * q^3 + (b7 + 1) * q^5 + (b9 - b7 + b5 - b4 + b3 + b2 - 1) * q^7 + (b9 - b7 + b3 - b2) * q^9 $$q + \beta_{6} q^{3} + (\beta_{7} + 1) q^{5} + (\beta_{9} - \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{7} + (\beta_{9} - \beta_{7} + \beta_{3} - \beta_{2}) q^{9} + (\beta_{8} + \beta_{7} + \beta_{6} - \beta_{3} - \beta_1 + 1) q^{11} + (\beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{2} + 1) q^{13} + \beta_1 q^{15} + ( - 2 \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} - 2 \beta_1) q^{17} + ( - \beta_{8} + \beta_{6}) q^{19} + (2 \beta_{8} + 2 \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + 3) q^{21} + (2 \beta_{9} + \beta_{7} - \beta_{4} + 2 \beta_{2} - 3) q^{23} + \beta_{7} q^{25} + ( - 2 \beta_{9} - \beta_{8} + \beta_{6} - \beta_{5} - 2 \beta_{2}) q^{27} + ( - 2 \beta_{9} - \beta_{8} + 4 \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + \cdots + 2) q^{29}+ \cdots + ( - \beta_{9} + 3 \beta_{8} - 2 \beta_{7} - \beta_{6} - 3 \beta_{5} - \beta_{3} - 2 \beta_{2} + \cdots + 3) q^{99}+O(q^{100})$$ q + b6 * q^3 + (b7 + 1) * q^5 + (b9 - b7 + b5 - b4 + b3 + b2 - 1) * q^7 + (b9 - b7 + b3 - b2) * q^9 + (b8 + b7 + b6 - b3 - b1 + 1) * q^11 + (b8 + 2*b7 + b6 - b5 - 2*b4 + b2 + 1) * q^13 + b1 * q^15 + (-2*b8 + b7 + b6 + b5 + b4 - b2 - 2*b1) * q^17 + (-b8 + b6) * q^19 + (2*b8 + 2*b7 - b5 - b4 + b3 + 3) * q^21 + (2*b9 + b7 - b4 + 2*b2 - 3) * q^23 + b7 * q^25 + (-2*b9 - b8 + b6 - b5 - 2*b2) * q^27 + (-2*b9 - b8 + 4*b7 + b6 - b5 + 2*b4 - 2*b3 - b2 - 2*b1 + 2) * q^29 + (-2*b8 - 2*b6 + b4 + b3 + b2 + 2*b1 + 1) * q^31 + (b9 - 2*b8 + b7 + b5 + 2*b3 + b1) * q^33 + (b9 + b8 - b7 + b2 - 1) * q^35 + (b9 + b8 - b7 + b6 - b5 + 2*b3 - b1) * q^37 + (2*b9 - 3*b7 - b6 + 3*b5 - 2*b4 + 3*b3 + 2*b1 - 3) * q^39 + (3*b8 - b6 - b5 - b2 - 2*b1 + 2) * q^41 + (b9 - b3 - 2*b2 - 3) * q^43 + (b9 - b4) * q^45 + (-2*b9 - 2*b5 - 4*b3 - 2*b1 - 2) * q^47 + (-3*b9 + b8 + b7 + 2*b6 - b5 + b4 - 2*b3 - 3*b1) * q^49 + (-2*b9 - b8 - 3*b7 - b6 - b5 + 3*b4 - 2*b2 + b1 + 3) * q^51 + (2*b7 + 2*b6 + 2*b5 + 2*b4 - 2*b3 + 2*b2 + 2) * q^53 + (-b9 + b8 + 2*b7 + b6 - b5 - b3 + 1) * q^55 + (b9 - 4*b7 + b3 - b2) * q^57 + (2*b9 - b8 - b7 - b6 + 2*b5 - 4*b4 + b3 + 4*b2 - b1 - 1) * q^59 + (b8 - 3*b7 - b6 + 3) * q^61 + (-b9 + b8 + 3*b7 + b6 + b5 - b2 + 2*b1 - 3) * q^63 + (b7 - b5 - b4 + 2*b2 + b1 - 1) * q^65 + (b8 - 3*b7 - 4*b6 + 3*b5 + b4 - b2 + b1) * q^67 + (2*b9 + b8 - 3*b7 - 4*b6 - 2*b5 - 3*b4 + 3*b3 + 2*b2 + 3*b1) * q^69 + (-3*b9 - b8 + 2*b7 - b6 + b5 - 3*b3 + 1) * q^71 + (-4*b8 - b7 - b6 + 3*b5 - 2*b4 + b3 - 2*b2 + 4*b1 - 1) * q^73 + (-b6 + b1) * q^75 + (b8 - 4*b7 - 2*b6 + 2*b3 + b2 + 3*b1 + 1) * q^77 + (-2*b9 + b8 + 2*b7 + b6 + 4*b4 - 4*b3) * q^79 + (-b9 - 2*b8 - 5*b7 + 4*b5 + 2*b4 - b3 - 3*b2 - 2*b1 - 3) * q^81 + (-4*b6 - 4*b5 + 5*b2 + 4*b1 - 1) * q^83 + (-b8 + 2*b6 + 2*b5 - b2 - b1 - 1) * q^85 + (-2*b8 - 5*b7 - 2*b6 + b5 + 4*b4 - b3 + 2*b1 - 3) * q^87 + (-6*b7 - b5 + 3*b4 - b1 - 6) * q^89 + (b9 + 3*b8 - 4*b7 - 4*b6 - b5 + 2*b4 + b3 - b2 - b1 - 2) * q^91 + (5*b8 - 2*b7 + b6 - 4*b5 - 4*b3 + 2*b2) * q^93 + (-b8 + b5 + b1) * q^95 + (-b9 - 2*b8 + b6 - b5 + 2*b4 - b3 - b2 - 3*b1) * q^97 + (-b9 + 3*b8 - 2*b7 - b6 - 3*b5 - b3 - 2*b2 + 2*b1 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + q^{3} + 5 q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10})$$ 10 * q + q^3 + 5 * q^5 - 5 * q^7 + 3 * q^9 $$10 q + q^{3} + 5 q^{5} - 5 q^{7} + 3 q^{9} + 6 q^{11} + 2 q^{15} - 6 q^{17} + 3 q^{19} + 12 q^{21} - 24 q^{23} - 5 q^{25} - 8 q^{27} + 15 q^{31} - 4 q^{33} - q^{35} - q^{37} - 21 q^{39} + 8 q^{41} - 26 q^{43} + 3 q^{45} - 14 q^{47} - 13 q^{49} + 40 q^{51} + 24 q^{53} + 18 q^{57} + 42 q^{61} - 49 q^{63} - 9 q^{65} + 7 q^{67} + 14 q^{69} - 3 q^{73} + q^{75} + 26 q^{77} + q^{79} - 13 q^{81} + 8 q^{83} - 12 q^{85} + 8 q^{87} - 28 q^{89} - 11 q^{91} + 25 q^{93} + 3 q^{95} + 36 q^{99}+O(q^{100})$$ 10 * q + q^3 + 5 * q^5 - 5 * q^7 + 3 * q^9 + 6 * q^11 + 2 * q^15 - 6 * q^17 + 3 * q^19 + 12 * q^21 - 24 * q^23 - 5 * q^25 - 8 * q^27 + 15 * q^31 - 4 * q^33 - q^35 - q^37 - 21 * q^39 + 8 * q^41 - 26 * q^43 + 3 * q^45 - 14 * q^47 - 13 * q^49 + 40 * q^51 + 24 * q^53 + 18 * q^57 + 42 * q^61 - 49 * q^63 - 9 * q^65 + 7 * q^67 + 14 * q^69 - 3 * q^73 + q^75 + 26 * q^77 + q^79 - 13 * q^81 + 8 * q^83 - 12 * q^85 + 8 * q^87 - 28 * q^89 - 11 * q^91 + 25 * q^93 + 3 * q^95 + 36 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2x^{9} + 2x^{8} - 4x^{7} + 13x^{6} - 36x^{5} + 39x^{4} - 36x^{3} + 54x^{2} - 162x + 243$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{7} + 2\nu^{6} - 2\nu^{5} + 4\nu^{4} - 4\nu^{3} + 18\nu^{2} - 21\nu + 18 ) / 18$$ (-v^7 + 2*v^6 - 2*v^5 + 4*v^4 - 4*v^3 + 18*v^2 - 21*v + 18) / 18 $$\beta_{3}$$ $$=$$ $$( \nu^{9} - \nu^{8} - 9\nu^{7} + 7\nu^{6} - 18\nu^{5} + 22\nu^{4} - 51\nu^{3} + 21\nu^{2} - 171\nu + 135 ) / 216$$ (v^9 - v^8 - 9*v^7 + 7*v^6 - 18*v^5 + 22*v^4 - 51*v^3 + 21*v^2 - 171*v + 135) / 216 $$\beta_{4}$$ $$=$$ $$( 5\nu^{9} - 4\nu^{8} + 7\nu^{7} + 28\nu^{6} + 32\nu^{5} - 30\nu^{4} + 123\nu^{3} - 216\nu^{2} - 243\nu - 486 ) / 648$$ (5*v^9 - 4*v^8 + 7*v^7 + 28*v^6 + 32*v^5 - 30*v^4 + 123*v^3 - 216*v^2 - 243*v - 486) / 648 $$\beta_{5}$$ $$=$$ $$( 3\nu^{9} + 2\nu^{8} - \nu^{7} + 4\nu^{6} + 16\nu^{5} - 58\nu^{4} + 9\nu^{3} + 6\nu^{2} + 9\nu - 486 ) / 216$$ (3*v^9 + 2*v^8 - v^7 + 4*v^6 + 16*v^5 - 58*v^4 + 9*v^3 + 6*v^2 + 9*v - 486) / 216 $$\beta_{6}$$ $$=$$ $$( -\nu^{9} - \nu^{7} + 6\nu^{6} - 20\nu^{5} + 22\nu^{4} - 15\nu^{3} + 48\nu^{2} - 63\nu + 216 ) / 72$$ (-v^9 - v^7 + 6*v^6 - 20*v^5 + 22*v^4 - 15*v^3 + 48*v^2 - 63*v + 216) / 72 $$\beta_{7}$$ $$=$$ $$( 8\nu^{9} - 7\nu^{8} + 16\nu^{7} - 23\nu^{6} + 50\nu^{5} - 108\nu^{4} + 114\nu^{3} - 153\nu^{2} - 729 ) / 648$$ (8*v^9 - 7*v^8 + 16*v^7 - 23*v^6 + 50*v^5 - 108*v^4 + 114*v^3 - 153*v^2 - 729) / 648 $$\beta_{8}$$ $$=$$ $$( \nu^{9} - 2\nu^{8} + 2\nu^{7} - 4\nu^{6} + 13\nu^{5} - 36\nu^{4} + 39\nu^{3} - 36\nu^{2} + 54\nu - 162 ) / 81$$ (v^9 - 2*v^8 + 2*v^7 - 4*v^6 + 13*v^5 - 36*v^4 + 39*v^3 - 36*v^2 + 54*v - 162) / 81 $$\beta_{9}$$ $$=$$ $$( - 13 \nu^{9} + 5 \nu^{8} + 25 \nu^{7} - 35 \nu^{6} - 94 \nu^{5} + 186 \nu^{4} + 231 \nu^{3} - 297 \nu^{2} + 243 \nu + 1701 ) / 648$$ (-13*v^9 + 5*v^8 + 25*v^7 - 35*v^6 - 94*v^5 + 186*v^4 + 231*v^3 - 297*v^2 + 243*v + 1701) / 648
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{7} - \beta_{4} - \beta_{3} + \beta_{2}$$ b7 - b4 - b3 + b2 $$\nu^{3}$$ $$=$$ $$2\beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} + 2\beta_{2}$$ 2*b9 + b8 - b6 + b5 + 2*b2 $$\nu^{4}$$ $$=$$ $$\beta_{9} - 2\beta_{8} + 3\beta_{7} - 2\beta_{6} - 2\beta_{5} + \beta_{4} + 2\beta_{2} + 2\beta _1 - 3$$ b9 - 2*b8 + 3*b7 - 2*b6 - 2*b5 + b4 + 2*b2 + 2*b1 - 3 $$\nu^{5}$$ $$=$$ $$-2\beta_{9} - \beta_{8} - 2\beta_{7} - 4\beta_{6} - 4\beta_{5} + 2\beta_{4} - 4\beta_{3} + 2\beta_{2} - 2\beta _1 + 6$$ -2*b9 - b8 - 2*b7 - 4*b6 - 4*b5 + 2*b4 - 4*b3 + 2*b2 - 2*b1 + 6 $$\nu^{6}$$ $$=$$ $$-4\beta_{9} - 8\beta_{7} + 4\beta_{6} + 12\beta_{4} - 4\beta_{3} - 2\beta_{2} + 4\beta _1 + 3$$ -4*b9 - 8*b7 + 4*b6 + 12*b4 - 4*b3 - 2*b2 + 4*b1 + 3 $$\nu^{7}$$ $$=$$ $$-8\beta_{9} - 10\beta_{8} + 18\beta_{7} + 12\beta_{6} - 4\beta_{5} + 6\beta_{4} - 18\beta_{3} - 8\beta_{2} - \beta_1$$ -8*b9 - 10*b8 + 18*b7 + 12*b6 - 4*b5 + 6*b4 - 18*b3 - 8*b2 - b1 $$\nu^{8}$$ $$=$$ $$16 \beta_{9} - 16 \beta_{8} - 7 \beta_{7} - 10 \beta_{6} + 38 \beta_{5} - 5 \beta_{4} - 5 \beta_{3} + 3 \beta_{2} - 8 \beta _1 + 30$$ 16*b9 - 16*b8 - 7*b7 - 10*b6 + 38*b5 - 5*b4 - 5*b3 + 3*b2 - 8*b1 + 30 $$\nu^{9}$$ $$=$$ $$16\beta_{9} - 29\beta_{8} + 88\beta_{7} - 9\beta_{6} + 25\beta_{5} + 26\beta_{3} + 18\beta_{2} + 46\beta _1 + 48$$ 16*b9 - 29*b8 + 88*b7 - 9*b6 + 25*b5 + 26*b3 + 18*b2 + 46*b1 + 48

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$$.

 $$n$$ $$211$$ $$241$$ $$281$$ $$337$$ $$\chi(n)$$ $$1$$ $$1 + \beta_{7}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
101.1
 −1.31611 − 1.12599i 0.527154 − 1.64988i −1.08831 + 1.34743i 1.72689 + 0.133595i 1.15038 + 1.29484i −1.31611 + 1.12599i 0.527154 + 1.64988i −1.08831 − 1.34743i 1.72689 − 0.133595i 1.15038 − 1.29484i
0 −1.63319 + 0.576792i 0 0.500000 + 0.866025i 0 −1.73439 1.99797i 0 2.33462 1.88402i 0
101.2 0 −1.16526 1.28147i 0 0.500000 + 0.866025i 0 −1.80025 + 1.93884i 0 −0.284326 + 2.98650i 0
101.3 0 0.622752 + 1.61622i 0 0.500000 + 0.866025i 0 2.57325 0.615143i 0 −2.22436 + 2.01301i 0
101.4 0 0.979142 1.42873i 0 0.500000 + 0.866025i 0 −0.456468 2.60608i 0 −1.08256 2.79787i 0
101.5 0 1.69656 0.348838i 0 0.500000 + 0.866025i 0 −1.08214 + 2.41433i 0 2.75662 1.18365i 0
341.1 0 −1.63319 0.576792i 0 0.500000 0.866025i 0 −1.73439 + 1.99797i 0 2.33462 + 1.88402i 0
341.2 0 −1.16526 + 1.28147i 0 0.500000 0.866025i 0 −1.80025 1.93884i 0 −0.284326 2.98650i 0
341.3 0 0.622752 1.61622i 0 0.500000 0.866025i 0 2.57325 + 0.615143i 0 −2.22436 2.01301i 0
341.4 0 0.979142 + 1.42873i 0 0.500000 0.866025i 0 −0.456468 + 2.60608i 0 −1.08256 + 2.79787i 0
341.5 0 1.69656 + 0.348838i 0 0.500000 0.866025i 0 −1.08214 2.41433i 0 2.75662 + 1.18365i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 341.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 420.2.bh.b yes 10
3.b odd 2 1 420.2.bh.a 10
5.b even 2 1 2100.2.bi.j 10
5.c odd 4 2 2100.2.bo.g 20
7.c even 3 1 2940.2.d.a 10
7.d odd 6 1 420.2.bh.a 10
7.d odd 6 1 2940.2.d.b 10
15.d odd 2 1 2100.2.bi.k 10
15.e even 4 2 2100.2.bo.h 20
21.g even 6 1 inner 420.2.bh.b yes 10
21.g even 6 1 2940.2.d.a 10
21.h odd 6 1 2940.2.d.b 10
35.i odd 6 1 2100.2.bi.k 10
35.k even 12 2 2100.2.bo.h 20
105.p even 6 1 2100.2.bi.j 10
105.w odd 12 2 2100.2.bo.g 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.bh.a 10 3.b odd 2 1
420.2.bh.a 10 7.d odd 6 1
420.2.bh.b yes 10 1.a even 1 1 trivial
420.2.bh.b yes 10 21.g even 6 1 inner
2100.2.bi.j 10 5.b even 2 1
2100.2.bi.j 10 105.p even 6 1
2100.2.bi.k 10 15.d odd 2 1
2100.2.bi.k 10 35.i odd 6 1
2100.2.bo.g 20 5.c odd 4 2
2100.2.bo.g 20 105.w odd 12 2
2100.2.bo.h 20 15.e even 4 2
2100.2.bo.h 20 35.k even 12 2
2940.2.d.a 10 7.c even 3 1
2940.2.d.a 10 21.g even 6 1
2940.2.d.b 10 7.d odd 6 1
2940.2.d.b 10 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{10} - 6 T_{11}^{9} - 4 T_{11}^{8} + 96 T_{11}^{7} + 84 T_{11}^{6} - 2280 T_{11}^{5} + 8156 T_{11}^{4} - 14352 T_{11}^{3} + 14128 T_{11}^{2} - 7488 T_{11} + 1728$$ acting on $$S_{2}^{\mathrm{new}}(420, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$T^{10} - T^{9} - T^{8} + 4 T^{7} + T^{6} + \cdots + 243$$
$5$ $$(T^{2} - T + 1)^{5}$$
$7$ $$T^{10} + 5 T^{9} + 19 T^{8} + \cdots + 16807$$
$11$ $$T^{10} - 6 T^{9} - 4 T^{8} + 96 T^{7} + \cdots + 1728$$
$13$ $$T^{10} + 81 T^{8} + 2183 T^{6} + \cdots + 3888$$
$17$ $$T^{10} + 6 T^{9} + 70 T^{8} + \cdots + 69696$$
$19$ $$T^{10} - 3 T^{9} - 12 T^{8} + 45 T^{7} + \cdots + 192$$
$23$ $$T^{10} + 24 T^{9} + 223 T^{8} + \cdots + 2462508$$
$29$ $$T^{10} + 142 T^{8} + 6537 T^{6} + \cdots + 8748$$
$31$ $$T^{10} - 15 T^{9} + 12 T^{8} + \cdots + 1978032$$
$37$ $$T^{10} + T^{9} + 68 T^{8} - 65 T^{7} + \cdots + 12544$$
$41$ $$(T^{5} - 4 T^{4} - 115 T^{3} + 64 T^{2} + \cdots - 1338)^{2}$$
$43$ $$(T^{5} + 13 T^{4} - 42 T^{3} - 568 T^{2} + \cdots + 1559)^{2}$$
$47$ $$T^{10} + 14 T^{9} + \cdots + 295289856$$
$53$ $$T^{10} - 24 T^{9} + \cdots + 113246208$$
$59$ $$T^{10} + 160 T^{8} + \cdots + 125081856$$
$61$ $$T^{10} - 42 T^{9} + 807 T^{8} + \cdots + 1338672$$
$67$ $$T^{10} - 7 T^{9} + 241 T^{8} + \cdots + 45684081$$
$71$ $$T^{10} + 288 T^{8} + \cdots + 88259328$$
$73$ $$T^{10} + 3 T^{9} - 250 T^{8} + \cdots + 1572528$$
$79$ $$T^{10} - T^{9} + 194 T^{8} + \cdots + 138485824$$
$83$ $$(T^{5} - 4 T^{4} - 379 T^{3} + 1024 T^{2} + \cdots + 28794)^{2}$$
$89$ $$T^{10} + 28 T^{9} + 563 T^{8} + \cdots + 272484$$
$97$ $$T^{10} + 212 T^{8} + 11040 T^{6} + \cdots + 1051392$$