Properties

 Label 99.3.b.a Level $99$ Weight $3$ Character orbit 99.b Analytic conductor $2.698$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$99 = 3^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 99.b (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$2.69755461717$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.65306824704.6 Defining polynomial: $$x^{8} - 4x^{7} - 2x^{6} + 20x^{5} + x^{4} - 40x^{3} + 36x^{2} - 12x + 27$$ x^8 - 4*x^7 - 2*x^6 + 20*x^5 + x^4 - 40*x^3 + 36*x^2 - 12*x + 27 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{6}\cdot 3^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + ( - \beta_{3} - 2) q^{4} + ( - \beta_{2} + \beta_1) q^{5} + ( - \beta_{7} + 2) q^{7} + (\beta_{6} + 3 \beta_{5} + 4 \beta_{2} - \beta_1) q^{8}+O(q^{10})$$ q - b2 * q^2 + (-b3 - 2) * q^4 + (-b2 + b1) * q^5 + (-b7 + 2) * q^7 + (b6 + 3*b5 + 4*b2 - b1) * q^8 $$q - \beta_{2} q^{2} + ( - \beta_{3} - 2) q^{4} + ( - \beta_{2} + \beta_1) q^{5} + ( - \beta_{7} + 2) q^{7} + (\beta_{6} + 3 \beta_{5} + 4 \beta_{2} - \beta_1) q^{8} + ( - \beta_{7} - 2 \beta_{3} - 6) q^{10} + \beta_{5} q^{11} + (2 \beta_{7} + \beta_{4} - \beta_{3} - 1) q^{13} + ( - 2 \beta_{6} - 3 \beta_{2} - \beta_1) q^{14} + (4 \beta_{4} + 4 \beta_{3} + 13) q^{16} + ( - \beta_{6} - 3 \beta_{5} - 5 \beta_{2} - \beta_1) q^{17} + ( - \beta_{7} - 3 \beta_{4} + \beta_{3} + 5) q^{19} + (6 \beta_{5} + 13 \beta_{2} + \beta_1) q^{20} + (\beta_{4} + \beta_{3}) q^{22} + ( - \beta_{6} - 6 \beta_{5} + \beta_1) q^{23} + ( - 2 \beta_{4} + \beta_{3} - 14) q^{25} + (3 \beta_{6} + 8 \beta_{2} + \beta_1) q^{26} + ( - \beta_{7} - 2 \beta_{4} - 2 \beta_{3} - 4) q^{28} + (6 \beta_{6} - 6 \beta_{2} - 2 \beta_1) q^{29} + (4 \beta_{7} - 4 \beta_{4} + 3 \beta_{3} - 7) q^{31} + ( - 8 \beta_{6} - 12 \beta_{5} - 25 \beta_{2}) q^{32} + (2 \beta_{7} - 4 \beta_{4} - 7 \beta_{3} - 27) q^{34} + (3 \beta_{6} - 9 \beta_{5} + 5 \beta_{2} + 3 \beta_1) q^{35} + (6 \beta_{7} - 2 \beta_{4} + \beta_{3} + 17) q^{37} + (3 \beta_{6} + 6 \beta_{5} - 9 \beta_{2}) q^{38} + ( - 5 \beta_{7} + 6 \beta_{4} + 10 \beta_{3} + 54) q^{40} + (5 \beta_{6} - 3 \beta_{5} + 13 \beta_{2} + \beta_1) q^{41} + (\beta_{7} + 5 \beta_{4} + 7 \beta_{3} - 13) q^{43} + ( - 3 \beta_{6} - 2 \beta_{5} - 7 \beta_{2} + \beta_1) q^{44} + ( - 7 \beta_{4} - 7 \beta_{3} + 3) q^{46} + ( - 11 \beta_{6} + 6 \beta_{5} + 2 \beta_{2} + \beta_1) q^{47} + ( - 8 \beta_{7} - 2 \beta_{4} - 3 \beta_{3} - 12) q^{49} + (3 \beta_{6} + 3 \beta_{5} + 10 \beta_{2} + \beta_1) q^{50} + (4 \beta_{7} + 7 \beta_{4} + 3 \beta_{3} + 35) q^{52} + ( - 8 \beta_{6} + \beta_{2} - 5 \beta_1) q^{53} + ( - 3 \beta_{7} + 2 \beta_{3}) q^{55} + ( - 4 \beta_{6} + 12 \beta_{5} + 5 \beta_{2} - 7 \beta_1) q^{56} + ( - 4 \beta_{7} + 6 \beta_{4} - 4 \beta_{3} - 54) q^{58} + (8 \beta_{6} - 6 \beta_{5} + 12 \beta_{2} - 2 \beta_1) q^{59} + ( - 2 \beta_{7} + 9 \beta_{4} + 3 \beta_{3} - 1) q^{61} + (13 \beta_{6} + 3 \beta_{5} - 3 \beta_{2} + 7 \beta_1) q^{62} + (8 \beta_{7} - 4 \beta_{4} - 21 \beta_{3} - 74) q^{64} + ( - 14 \beta_{6} + 24 \beta_{5} - 6 \beta_{2} + 2 \beta_1) q^{65} + (6 \beta_{7} + 2 \beta_{4} + 4 \beta_{3} + 14) q^{67} + (15 \beta_{6} + 21 \beta_{5} + 55 \beta_{2} - 9 \beta_1) q^{68} + ( - 6 \beta_{7} - 6 \beta_{4} - 7 \beta_{3} + 21) q^{70} + ( - \beta_{6} - 6 \beta_{5} - 4 \beta_{2} - 11 \beta_1) q^{71} + ( - 6 \beta_{7} + 10 \beta_{4} - 6 \beta_{3} + 56) q^{73} + (15 \beta_{6} + 3 \beta_{5} - 15 \beta_{2} + 7 \beta_1) q^{74} + ( - 7 \beta_{7} - 3 \beta_{4} + \beta_{3} - 43) q^{76} + ( - 2 \beta_{6} + 2 \beta_{5} - \beta_{2} - 3 \beta_1) q^{77} + ( - \beta_{7} - 4 \beta_{4} + 6 \beta_{3} + 56) q^{79} + ( - 32 \beta_{6} - 24 \beta_{5} - 73 \beta_{2} + 9 \beta_1) q^{80} + ( - 6 \beta_{7} + 2 \beta_{4} + 9 \beta_{3} + 63) q^{82} + ( - 7 \beta_{6} + 9 \beta_{5} - 25 \beta_{2} + 9 \beta_1) q^{83} + (2 \beta_{7} - 2 \beta_{4} - 18 \beta_{3} + 6) q^{85} + ( - 15 \beta_{6} - 36 \beta_{5} - 33 \beta_{2} + 8 \beta_1) q^{86} + (2 \beta_{7} - \beta_{4} - 6 \beta_{3} - 33) q^{88} + (9 \beta_{6} + 12 \beta_{5} + 27 \beta_{2} + 14 \beta_1) q^{89} + (12 \beta_{7} + 6 \beta_{3} - 68) q^{91} + (17 \beta_{6} + 18 \beta_{5} + 46 \beta_{2} - 3 \beta_1) q^{92} + (10 \beta_{7} - 5 \beta_{4} + 7 \beta_{3} + 45) q^{94} + (27 \beta_{6} - 15 \beta_{5} + 3 \beta_{2} - 3 \beta_1) q^{95} + ( - 16 \beta_{7} - 2 \beta_{4} + \beta_{3} - 19) q^{97} + ( - 9 \beta_{6} + 15 \beta_{5} + 24 \beta_{2} - 11 \beta_1) q^{98}+O(q^{100})$$ q - b2 * q^2 + (-b3 - 2) * q^4 + (-b2 + b1) * q^5 + (-b7 + 2) * q^7 + (b6 + 3*b5 + 4*b2 - b1) * q^8 + (-b7 - 2*b3 - 6) * q^10 + b5 * q^11 + (2*b7 + b4 - b3 - 1) * q^13 + (-2*b6 - 3*b2 - b1) * q^14 + (4*b4 + 4*b3 + 13) * q^16 + (-b6 - 3*b5 - 5*b2 - b1) * q^17 + (-b7 - 3*b4 + b3 + 5) * q^19 + (6*b5 + 13*b2 + b1) * q^20 + (b4 + b3) * q^22 + (-b6 - 6*b5 + b1) * q^23 + (-2*b4 + b3 - 14) * q^25 + (3*b6 + 8*b2 + b1) * q^26 + (-b7 - 2*b4 - 2*b3 - 4) * q^28 + (6*b6 - 6*b2 - 2*b1) * q^29 + (4*b7 - 4*b4 + 3*b3 - 7) * q^31 + (-8*b6 - 12*b5 - 25*b2) * q^32 + (2*b7 - 4*b4 - 7*b3 - 27) * q^34 + (3*b6 - 9*b5 + 5*b2 + 3*b1) * q^35 + (6*b7 - 2*b4 + b3 + 17) * q^37 + (3*b6 + 6*b5 - 9*b2) * q^38 + (-5*b7 + 6*b4 + 10*b3 + 54) * q^40 + (5*b6 - 3*b5 + 13*b2 + b1) * q^41 + (b7 + 5*b4 + 7*b3 - 13) * q^43 + (-3*b6 - 2*b5 - 7*b2 + b1) * q^44 + (-7*b4 - 7*b3 + 3) * q^46 + (-11*b6 + 6*b5 + 2*b2 + b1) * q^47 + (-8*b7 - 2*b4 - 3*b3 - 12) * q^49 + (3*b6 + 3*b5 + 10*b2 + b1) * q^50 + (4*b7 + 7*b4 + 3*b3 + 35) * q^52 + (-8*b6 + b2 - 5*b1) * q^53 + (-3*b7 + 2*b3) * q^55 + (-4*b6 + 12*b5 + 5*b2 - 7*b1) * q^56 + (-4*b7 + 6*b4 - 4*b3 - 54) * q^58 + (8*b6 - 6*b5 + 12*b2 - 2*b1) * q^59 + (-2*b7 + 9*b4 + 3*b3 - 1) * q^61 + (13*b6 + 3*b5 - 3*b2 + 7*b1) * q^62 + (8*b7 - 4*b4 - 21*b3 - 74) * q^64 + (-14*b6 + 24*b5 - 6*b2 + 2*b1) * q^65 + (6*b7 + 2*b4 + 4*b3 + 14) * q^67 + (15*b6 + 21*b5 + 55*b2 - 9*b1) * q^68 + (-6*b7 - 6*b4 - 7*b3 + 21) * q^70 + (-b6 - 6*b5 - 4*b2 - 11*b1) * q^71 + (-6*b7 + 10*b4 - 6*b3 + 56) * q^73 + (15*b6 + 3*b5 - 15*b2 + 7*b1) * q^74 + (-7*b7 - 3*b4 + b3 - 43) * q^76 + (-2*b6 + 2*b5 - b2 - 3*b1) * q^77 + (-b7 - 4*b4 + 6*b3 + 56) * q^79 + (-32*b6 - 24*b5 - 73*b2 + 9*b1) * q^80 + (-6*b7 + 2*b4 + 9*b3 + 63) * q^82 + (-7*b6 + 9*b5 - 25*b2 + 9*b1) * q^83 + (2*b7 - 2*b4 - 18*b3 + 6) * q^85 + (-15*b6 - 36*b5 - 33*b2 + 8*b1) * q^86 + (2*b7 - b4 - 6*b3 - 33) * q^88 + (9*b6 + 12*b5 + 27*b2 + 14*b1) * q^89 + (12*b7 + 6*b3 - 68) * q^91 + (17*b6 + 18*b5 + 46*b2 - 3*b1) * q^92 + (10*b7 - 5*b4 + 7*b3 + 45) * q^94 + (27*b6 - 15*b5 + 3*b2 - 3*b1) * q^95 + (-16*b7 - 2*b4 + b3 - 19) * q^97 + (-9*b6 + 15*b5 + 24*b2 - 11*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 16 q^{4} + 16 q^{7}+O(q^{10})$$ 8 * q - 16 * q^4 + 16 * q^7 $$8 q - 16 q^{4} + 16 q^{7} - 48 q^{10} - 8 q^{13} + 104 q^{16} + 40 q^{19} - 112 q^{25} - 32 q^{28} - 56 q^{31} - 216 q^{34} + 136 q^{37} + 432 q^{40} - 104 q^{43} + 24 q^{46} - 96 q^{49} + 280 q^{52} - 432 q^{58} - 8 q^{61} - 592 q^{64} + 112 q^{67} + 168 q^{70} + 448 q^{73} - 344 q^{76} + 448 q^{79} + 504 q^{82} + 48 q^{85} - 264 q^{88} - 544 q^{91} + 360 q^{94} - 152 q^{97}+O(q^{100})$$ 8 * q - 16 * q^4 + 16 * q^7 - 48 * q^10 - 8 * q^13 + 104 * q^16 + 40 * q^19 - 112 * q^25 - 32 * q^28 - 56 * q^31 - 216 * q^34 + 136 * q^37 + 432 * q^40 - 104 * q^43 + 24 * q^46 - 96 * q^49 + 280 * q^52 - 432 * q^58 - 8 * q^61 - 592 * q^64 + 112 * q^67 + 168 * q^70 + 448 * q^73 - 344 * q^76 + 448 * q^79 + 504 * q^82 + 48 * q^85 - 264 * q^88 - 544 * q^91 + 360 * q^94 - 152 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} - 2x^{6} + 20x^{5} + x^{4} - 40x^{3} + 36x^{2} - 12x + 27$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{6} + 3\nu^{5} + 6\nu^{4} - 17\nu^{3} - 20\nu^{2} + 29\nu - 3 ) / 4$$ (-v^6 + 3*v^5 + 6*v^4 - 17*v^3 - 20*v^2 + 29*v - 3) / 4 $$\beta_{2}$$ $$=$$ $$( -7\nu^{7} + 11\nu^{6} + 60\nu^{5} - 83\nu^{4} - 170\nu^{3} - 21\nu^{2} + 21\nu + 54 ) / 324$$ (-7*v^7 + 11*v^6 + 60*v^5 - 83*v^4 - 170*v^3 - 21*v^2 + 21*v + 54) / 324 $$\beta_{3}$$ $$=$$ $$( -7\nu^{7} + 11\nu^{6} + 60\nu^{5} - 83\nu^{4} - 170\nu^{3} + 303\nu^{2} + 21\nu - 756 ) / 162$$ (-7*v^7 + 11*v^6 + 60*v^5 - 83*v^4 - 170*v^3 + 303*v^2 + 21*v - 756) / 162 $$\beta_{4}$$ $$=$$ $$( 19\nu^{7} - 53\nu^{6} - 186\nu^{5} + 503\nu^{4} + 716\nu^{3} - 1563\nu^{2} - 219\nu + 1080 ) / 324$$ (19*v^7 - 53*v^6 - 186*v^5 + 503*v^4 + 716*v^3 - 1563*v^2 - 219*v + 1080) / 324 $$\beta_{5}$$ $$=$$ $$( 8\nu^{7} - 28\nu^{6} - 30\nu^{5} + 145\nu^{4} + 40\nu^{3} - 219\nu^{2} + 300\nu - 108 ) / 81$$ (8*v^7 - 28*v^6 - 30*v^5 + 145*v^4 + 40*v^3 - 219*v^2 + 300*v - 108) / 81 $$\beta_{6}$$ $$=$$ $$( -35\nu^{7} + 136\nu^{6} + 57\nu^{5} - 577\nu^{4} - 121\nu^{3} + 1191\nu^{2} - 1596\nu + 513 ) / 324$$ (-35*v^7 + 136*v^6 + 57*v^5 - 577*v^4 - 121*v^3 + 1191*v^2 - 1596*v + 513) / 324 $$\beta_{7}$$ $$=$$ $$( 47\nu^{7} - 178\nu^{6} - 183\nu^{5} + 997\nu^{4} + 667\nu^{3} - 2127\nu^{2} - 546\nu - 27 ) / 324$$ (47*v^7 - 178*v^6 - 183*v^5 + 997*v^4 + 667*v^3 - 2127*v^2 - 546*v - 27) / 324
 $$\nu$$ $$=$$ $$( -\beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} + 3 ) / 6$$ (-b7 - b6 + b4 + b3 - b2 + 3) / 6 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 2\beta_{2} + 5 ) / 2$$ (b3 - 2*b2 + 5) / 2 $$\nu^{3}$$ $$=$$ $$( -\beta_{7} - 7\beta_{6} - 9\beta_{5} + 4\beta_{4} + 7\beta_{3} - 16\beta_{2} + 21 ) / 6$$ (-b7 - 7*b6 - 9*b5 + 4*b4 + 7*b3 - 16*b2 + 21) / 6 $$\nu^{4}$$ $$=$$ $$( -2\beta_{6} - 6\beta_{5} + 2\beta_{4} + 5\beta_{3} - 22\beta_{2} + 2\beta _1 + 17 ) / 2$$ (-2*b6 - 6*b5 + 2*b4 + 5*b3 - 22*b2 + 2*b1 + 17) / 2 $$\nu^{5}$$ $$=$$ $$( 2\beta_{7} - 49\beta_{6} - 90\beta_{5} + 7\beta_{4} + 25\beta_{3} - 184\beta_{2} + 15\beta _1 + 93 ) / 6$$ (2*b7 - 49*b6 - 90*b5 + 7*b4 + 25*b3 - 184*b2 + 15*b1 + 93) / 6 $$\nu^{6}$$ $$=$$ $$( -2\beta_{7} - 31\beta_{6} - 75\beta_{5} + 6\beta_{4} + 5\beta_{3} - 195\beta_{2} + 19\beta _1 - 1 ) / 2$$ (-2*b7 - 31*b6 - 75*b5 + 6*b4 + 5*b3 - 195*b2 + 19*b1 - 1) / 2 $$\nu^{7}$$ $$=$$ $$( 29\beta_{7} - 328\beta_{6} - 693\beta_{5} - 77\beta_{4} - 116\beta_{3} - 1588\beta_{2} + 147\beta _1 - 312 ) / 6$$ (29*b7 - 328*b6 - 693*b5 - 77*b4 - 116*b3 - 1588*b2 + 147*b1 - 312) / 6

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/99\mathbb{Z}\right)^\times$$.

 $$n$$ $$46$$ $$56$$ $$\chi(n)$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
89.1
 2.75726 − 0.707107i −1.75726 + 0.707107i 1.13623 − 0.707107i −0.136233 + 0.707107i −0.136233 − 0.707107i 1.13623 + 0.707107i −1.75726 − 0.707107i 2.75726 + 0.707107i
3.89935i 0 −11.2049 6.10332i 0 2.61086 28.0945i 0 −23.7990
89.2 2.48514i 0 −2.17590 4.68911i 0 −3.30128 4.53313i 0 −11.6531
89.3 1.60688i 0 1.41795 6.21249i 0 11.1468 8.70597i 0 9.98270
89.4 0.192662i 0 3.96288 7.62670i 0 −2.45638 1.53415i 0 1.46938
89.5 0.192662i 0 3.96288 7.62670i 0 −2.45638 1.53415i 0 1.46938
89.6 1.60688i 0 1.41795 6.21249i 0 11.1468 8.70597i 0 9.98270
89.7 2.48514i 0 −2.17590 4.68911i 0 −3.30128 4.53313i 0 −11.6531
89.8 3.89935i 0 −11.2049 6.10332i 0 2.61086 28.0945i 0 −23.7990
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 89.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

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

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.3.b.a 8
3.b odd 2 1 inner 99.3.b.a 8
4.b odd 2 1 1584.3.i.b 8
11.b odd 2 1 1089.3.b.g 8
12.b even 2 1 1584.3.i.b 8
33.d even 2 1 1089.3.b.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.3.b.a 8 1.a even 1 1 trivial
99.3.b.a 8 3.b odd 2 1 inner
1089.3.b.g 8 11.b odd 2 1
1089.3.b.g 8 33.d even 2 1
1584.3.i.b 8 4.b odd 2 1
1584.3.i.b 8 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(99, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 24 T^{6} + 150 T^{4} + 248 T^{2} + \cdots + 9$$
$3$ $$T^{8}$$
$5$ $$T^{8} + 156 T^{6} + 8796 T^{4} + \cdots + 1838736$$
$7$ $$(T^{4} - 8 T^{3} - 42 T^{2} + 56 T + 236)^{2}$$
$11$ $$(T^{2} + 11)^{4}$$
$13$ $$(T^{4} + 4 T^{3} - 390 T^{2} - 2416 T + 2828)^{2}$$
$17$ $$T^{8} + 1080 T^{6} + \cdots + 472801536$$
$19$ $$(T^{4} - 20 T^{3} - 576 T^{2} + \cdots - 37908)^{2}$$
$23$ $$T^{8} + 1788 T^{6} + \cdots + 11052737424$$
$29$ $$T^{8} + 4848 T^{6} + \cdots + 125180100864$$
$31$ $$(T^{4} + 28 T^{3} - 2412 T^{2} + \cdots + 244016)^{2}$$
$37$ $$(T^{4} - 68 T^{3} - 972 T^{2} + \cdots - 914256)^{2}$$
$41$ $$T^{8} + 4824 T^{6} + \cdots + 85142571264$$
$43$ $$(T^{4} + 52 T^{3} - 3936 T^{2} + \cdots - 3241332)^{2}$$
$47$ $$T^{8} + 11052 T^{6} + \cdots + 12929518743696$$
$53$ $$T^{8} + 8124 T^{6} + \cdots + 26983975824$$
$59$ $$T^{8} + 7872 T^{6} + \cdots + 1028100096$$
$61$ $$(T^{4} + 4 T^{3} - 6198 T^{2} + \cdots + 1712844)^{2}$$
$67$ $$(T^{4} - 56 T^{3} - 2520 T^{2} + \cdots - 2846576)^{2}$$
$71$ $$T^{8} + 17916 T^{6} + \cdots + 13312019262096$$
$73$ $$(T^{4} - 224 T^{3} + 7464 T^{2} + \cdots - 37468144)^{2}$$
$79$ $$(T^{4} - 224 T^{3} + 15318 T^{2} + \cdots - 4606964)^{2}$$
$83$ $$T^{8} + \cdots + 177188471398656$$
$89$ $$T^{8} + 49704 T^{6} + \cdots + 13\!\cdots\!84$$
$97$ $$(T^{4} + 76 T^{3} - 15060 T^{2} + \cdots - 9076976)^{2}$$