Properties

 Label 5184.2.d.r Level $5184$ Weight $2$ Character orbit 5184.d Analytic conductor $41.394$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$5184 = 2^{6} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5184.d (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$41.3944484078$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36$$ x^12 - 16*x^8 - 24*x^7 + 96*x^5 + 304*x^4 + 384*x^3 + 288*x^2 + 144*x + 36 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{12}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 576) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{8} q^{5} + \beta_{4} q^{7}+O(q^{10})$$ q + b8 * q^5 + b4 * q^7 $$q + \beta_{8} q^{5} + \beta_{4} q^{7} + (\beta_{9} + \beta_{7}) q^{11} + \beta_{10} q^{13} + \beta_1 q^{17} + \beta_{7} q^{19} + ( - \beta_{6} + \beta_{4} - \beta_{3}) q^{23} + 2 q^{25} + (\beta_{11} - \beta_{10} + \beta_{8}) q^{29} + ( - \beta_{6} - \beta_{4} + \beta_{3}) q^{31} + (\beta_{9} + \beta_{7} + \beta_{5}) q^{35} + ( - \beta_{10} - \beta_{8}) q^{37} + (\beta_1 + 3) q^{41} + ( - \beta_{9} + \beta_{7} - \beta_{5}) q^{43} + (2 \beta_{6} + \beta_{4} - \beta_{3}) q^{47} + ( - \beta_{2} + \beta_1 + 4) q^{49} + ( - \beta_{11} + \beta_{10} + 2 \beta_{8}) q^{53} + (\beta_{6} - \beta_{4}) q^{55} + ( - \beta_{9} + 2 \beta_{7} + \beta_{5}) q^{59} + (\beta_{11} - \beta_{10} - \beta_{8}) q^{61} + (\beta_{2} - \beta_1 + 1) q^{65} + ( - \beta_{9} + \beta_{7} + 2 \beta_{5}) q^{67} + ( - \beta_{6} - 2 \beta_{4} - 2 \beta_{3}) q^{71} + (\beta_{2} - \beta_1) q^{73} + (\beta_{11} + 2 \beta_{10} + 6 \beta_{8}) q^{77} + (\beta_{6} + \beta_{4} + \beta_{3}) q^{79} + ( - \beta_{9} + 2 \beta_{7} - \beta_{5}) q^{83} + ( - \beta_{11} + \beta_{10}) q^{85} + (\beta_{2} - \beta_1 + 4) q^{89} + (3 \beta_{9} + 3 \beta_{7}) q^{91} - \beta_{3} q^{95} + ( - 3 \beta_1 + 1) q^{97}+O(q^{100})$$ q + b8 * q^5 + b4 * q^7 + (b9 + b7) * q^11 + b10 * q^13 + b1 * q^17 + b7 * q^19 + (-b6 + b4 - b3) * q^23 + 2 * q^25 + (b11 - b10 + b8) * q^29 + (-b6 - b4 + b3) * q^31 + (b9 + b7 + b5) * q^35 + (-b10 - b8) * q^37 + (b1 + 3) * q^41 + (-b9 + b7 - b5) * q^43 + (2*b6 + b4 - b3) * q^47 + (-b2 + b1 + 4) * q^49 + (-b11 + b10 + 2*b8) * q^53 + (b6 - b4) * q^55 + (-b9 + 2*b7 + b5) * q^59 + (b11 - b10 - b8) * q^61 + (b2 - b1 + 1) * q^65 + (-b9 + b7 + 2*b5) * q^67 + (-b6 - 2*b4 - 2*b3) * q^71 + (b2 - b1) * q^73 + (b11 + 2*b10 + 6*b8) * q^77 + (b6 + b4 + b3) * q^79 + (-b9 + 2*b7 - b5) * q^83 + (-b11 + b10) * q^85 + (b2 - b1 + 4) * q^89 + (3*b9 + 3*b7) * q^91 - b3 * q^95 + (-3*b1 + 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q+O(q^{10})$$ 12 * q $$12 q + 24 q^{25} + 36 q^{41} + 48 q^{49} + 12 q^{65} + 48 q^{89} + 12 q^{97}+O(q^{100})$$ 12 * q + 24 * q^25 + 36 * q^41 + 48 * q^49 + 12 * q^65 + 48 * q^89 + 12 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36$$ :

 $$\beta_{1}$$ $$=$$ $$( - 275 \nu^{11} + 300 \nu^{10} - 650 \nu^{9} + 1148 \nu^{8} + 1747 \nu^{7} + 3300 \nu^{6} + 5062 \nu^{5} - 27262 \nu^{4} - 51252 \nu^{3} - 44964 \nu^{2} - 26082 \nu - 96192 ) / 51972$$ (-275*v^11 + 300*v^10 - 650*v^9 + 1148*v^8 + 1747*v^7 + 3300*v^6 + 5062*v^5 - 27262*v^4 - 51252*v^3 - 44964*v^2 - 26082*v - 96192) / 51972 $$\beta_{2}$$ $$=$$ $$( 1969 \nu^{11} - 2148 \nu^{10} + 4654 \nu^{9} - 11338 \nu^{8} - 17186 \nu^{7} - 23628 \nu^{6} - 17534 \nu^{5} + 254444 \nu^{4} + 441804 \nu^{3} + 378072 \nu^{2} + \cdots - 442176 ) / 51972$$ (1969*v^11 - 2148*v^10 + 4654*v^9 - 11338*v^8 - 17186*v^7 - 23628*v^6 - 17534*v^5 + 254444*v^4 + 441804*v^3 + 378072*v^2 + 214812*v - 442176) / 51972 $$\beta_{3}$$ $$=$$ $$( - 34 \nu^{11} - 4 \nu^{10} + 17 \nu^{9} - 12 \nu^{8} + 546 \nu^{7} + 896 \nu^{6} - 196 \nu^{5} - 3468 \nu^{4} - 10454 \nu^{3} - 13152 \nu^{2} - 7260 \nu - 2736 ) / 366$$ (-34*v^11 - 4*v^10 + 17*v^9 - 12*v^8 + 546*v^7 + 896*v^6 - 196*v^5 - 3468*v^4 - 10454*v^3 - 13152*v^2 - 7260*v - 2736) / 366 $$\beta_{4}$$ $$=$$ $$( - 11251 \nu^{11} - 3656 \nu^{10} + 8584 \nu^{9} - 6576 \nu^{8} + 184040 \nu^{7} + 325576 \nu^{6} - 51410 \nu^{5} - 1183104 \nu^{4} - 3657720 \nu^{3} + \cdots - 967896 ) / 103944$$ (-11251*v^11 - 3656*v^10 + 8584*v^9 - 6576*v^8 + 184040*v^7 + 325576*v^6 - 51410*v^5 - 1183104*v^4 - 3657720*v^3 - 4638708*v^2 - 2561328*v - 967896) / 103944 $$\beta_{5}$$ $$=$$ $$( 5765 \nu^{11} - 3533 \nu^{10} + 2602 \nu^{9} - 2490 \nu^{8} - 88123 \nu^{7} - 86504 \nu^{6} + 51058 \nu^{5} + 522216 \nu^{4} + 1425948 \nu^{3} + 1357440 \nu^{2} + \cdots + 355320 ) / 51972$$ (5765*v^11 - 3533*v^10 + 2602*v^9 - 2490*v^8 - 88123*v^7 - 86504*v^6 + 51058*v^5 + 522216*v^4 + 1425948*v^3 + 1357440*v^2 + 954990*v + 355320) / 51972 $$\beta_{6}$$ $$=$$ $$( 6604 \nu^{11} + 4175 \nu^{10} - 8182 \nu^{9} + 6486 \nu^{8} - 110771 \nu^{7} - 217840 \nu^{6} + 19196 \nu^{5} + 732168 \nu^{4} + 2347908 \nu^{3} + 3011652 \nu^{2} + \cdots + 631152 ) / 51972$$ (6604*v^11 + 4175*v^10 - 8182*v^9 + 6486*v^8 - 110771*v^7 - 217840*v^6 + 19196*v^5 + 732168*v^4 + 2347908*v^3 + 3011652*v^2 + 1663614*v + 631152) / 51972 $$\beta_{7}$$ $$=$$ $$( 1688 \nu^{11} - 1054 \nu^{10} + 840 \nu^{9} - 684 \nu^{8} - 26378 \nu^{7} - 24587 \nu^{6} + 12648 \nu^{5} + 151968 \nu^{4} + 420176 \nu^{3} + 400856 \nu^{2} + 282372 \nu + 105120 ) / 12993$$ (1688*v^11 - 1054*v^10 + 840*v^9 - 684*v^8 - 26378*v^7 - 24587*v^6 + 12648*v^5 + 151968*v^4 + 420176*v^3 + 400856*v^2 + 282372*v + 105120) / 12993 $$\beta_{8}$$ $$=$$ $$( - 118 \nu^{11} + 90 \nu^{10} - 53 \nu^{9} + 32 \nu^{8} + 1866 \nu^{7} + 1416 \nu^{6} - 1364 \nu^{5} - 10408 \nu^{4} - 27758 \nu^{3} - 22776 \nu^{2} - 12468 \nu - 4320 ) / 852$$ (-118*v^11 + 90*v^10 - 53*v^9 + 32*v^8 + 1866*v^7 + 1416*v^6 - 1364*v^5 - 10408*v^4 - 27758*v^3 - 22776*v^2 - 12468*v - 4320) / 852 $$\beta_{9}$$ $$=$$ $$( 26035 \nu^{11} - 16590 \nu^{10} + 14290 \nu^{9} - 9780 \nu^{8} - 416670 \nu^{7} - 347068 \nu^{6} + 155770 \nu^{5} + 2327880 \nu^{4} + 6525980 \nu^{3} + \cdots + 1639800 ) / 103944$$ (26035*v^11 - 16590*v^10 + 14290*v^9 - 9780*v^8 - 416670*v^7 - 347068*v^6 + 155770*v^5 + 2327880*v^4 + 6525980*v^3 + 6240620*v^2 + 4402140*v + 1639800) / 103944 $$\beta_{10}$$ $$=$$ $$( - 17920 \nu^{11} + 13314 \nu^{10} - 5985 \nu^{9} + 1146 \nu^{8} + 286651 \nu^{7} + 215040 \nu^{6} - 211740 \nu^{5} - 1622514 \nu^{4} - 4157302 \nu^{3} + \cdots - 639072 ) / 51972$$ (-17920*v^11 + 13314*v^10 - 5985*v^9 + 1146*v^8 + 286651*v^7 + 215040*v^6 - 211740*v^5 - 1622514*v^4 - 4157302*v^3 - 3407556*v^2 - 1863546*v - 639072) / 51972 $$\beta_{11}$$ $$=$$ $$( - 29743 \nu^{11} + 22494 \nu^{10} - 12243 \nu^{9} + 5262 \nu^{8} + 472510 \nu^{7} + 356916 \nu^{6} - 347886 \nu^{5} - 2656440 \nu^{4} - 6971578 \nu^{3} + \cdots - 1079712 ) / 51972$$ (-29743*v^11 + 22494*v^10 - 12243*v^9 + 5262*v^8 + 472510*v^7 + 356916*v^6 - 347886*v^5 - 2656440*v^4 - 6971578*v^3 - 5717928*v^2 - 3128892*v - 1079712) / 51972
 $$\nu$$ $$=$$ $$( \beta_{11} - \beta_{10} + \beta_{6} + 3\beta_{5} + 2\beta_{4} + 3\beta_1 ) / 12$$ (b11 - b10 + b6 + 3*b5 + 2*b4 + 3*b1) / 12 $$\nu^{2}$$ $$=$$ $$( 2\beta_{9} - 7\beta_{7} + 2\beta_{5} + 6\beta_{4} - 9\beta_{3} ) / 12$$ (2*b9 - 7*b7 + 2*b5 + 6*b4 - 9*b3) / 12 $$\nu^{3}$$ $$=$$ $$( -4\beta_{11} + 4\beta_{10} + 6\beta_{8} + 4\beta_{6} + 8\beta_{4} - 3\beta_{3} ) / 6$$ (-4*b11 + 4*b10 + 6*b8 + 4*b6 + 8*b4 - 3*b3) / 6 $$\nu^{4}$$ $$=$$ $$( -9\beta_{11} + 3\beta_{10} + 30\beta_{8} + 2\beta_{2} + 7\beta _1 + 32 ) / 6$$ (-9*b11 + 3*b10 + 30*b8 + 2*b2 + 7*b1 + 32) / 6 $$\nu^{5}$$ $$=$$ $$( - 19 \beta_{11} + 16 \beta_{10} + 6 \beta_{9} + 39 \beta_{8} - 57 \beta_{7} - 16 \beta_{6} + 54 \beta_{5} - 38 \beta_{4} + 21 \beta_{3} + 3 \beta_{2} + 54 \beta _1 + 120 ) / 12$$ (-19*b11 + 16*b10 + 6*b9 + 39*b8 - 57*b7 - 16*b6 + 54*b5 - 38*b4 + 21*b3 + 3*b2 + 54*b1 + 120) / 12 $$\nu^{6}$$ $$=$$ $$( 16\beta_{9} - 65\beta_{7} + 40\beta_{5} ) / 3$$ (16*b9 - 65*b7 + 40*b5) / 3 $$\nu^{7}$$ $$=$$ $$( - 47 \beta_{11} + 35 \beta_{10} + 24 \beta_{9} + 108 \beta_{8} - 156 \beta_{7} + 35 \beta_{6} + 129 \beta_{5} + 94 \beta_{4} - 60 \beta_{3} - 12 \beta_{2} - 129 \beta _1 - 336 ) / 6$$ (-47*b11 + 35*b10 + 24*b9 + 108*b8 - 156*b7 + 35*b6 + 129*b5 + 94*b4 - 60*b3 - 12*b2 - 129*b1 - 336) / 6 $$\nu^{8}$$ $$=$$ $$( -249\beta_{11} + 144\beta_{10} + 669\beta_{8} - 35\beta_{2} - 214\beta _1 - 704 ) / 6$$ (-249*b11 + 144*b10 + 669*b8 - 35*b2 - 214*b1 - 704) / 6 $$\nu^{9}$$ $$=$$ $$( -236\beta_{11} + 164\beta_{10} + 570\beta_{8} - 164\beta_{6} - 472\beta_{4} + 321\beta_{3} ) / 3$$ (-236*b11 + 164*b10 + 570*b8 - 164*b6 - 472*b4 + 321*b3) / 3 $$\nu^{10}$$ $$=$$ $$( 164\beta_{9} - 790\beta_{7} - 393\beta_{6} + 557\beta_{5} - 1278\beta_{4} + 954\beta_{3} ) / 3$$ (164*b9 - 790*b7 - 393*b6 + 557*b5 - 1278*b4 + 954*b3) / 3 $$\nu^{11}$$ $$=$$ $$( 1193 \beta_{11} - 800 \beta_{10} + 786 \beta_{9} - 2949 \beta_{8} - 4227 \beta_{7} - 800 \beta_{6} + 3186 \beta_{5} - 2386 \beta_{4} + 1671 \beta_{3} - 393 \beta_{2} - 3186 \beta _1 - 9240 ) / 6$$ (1193*b11 - 800*b10 + 786*b9 - 2949*b8 - 4227*b7 - 800*b6 + 3186*b5 - 2386*b4 + 1671*b3 - 393*b2 - 3186*b1 - 9240) / 6

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$1217$$ $$2431$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2593.1
 −0.673288 + 0.180407i 0.583700 + 2.17840i −1.50511 + 0.403293i −0.403293 − 1.50511i 2.17840 − 0.583700i −0.180407 − 0.673288i −0.673288 − 0.180407i 0.583700 − 2.17840i −1.50511 − 0.403293i −0.403293 + 1.50511i 2.17840 + 0.583700i −0.180407 + 0.673288i
0 0 0 1.73205i 0 −4.35461 0 0 0
2593.2 0 0 0 1.73205i 0 −3.61328 0 0 0
2593.3 0 0 0 1.73205i 0 −0.990721 0 0 0
2593.4 0 0 0 1.73205i 0 0.990721 0 0 0
2593.5 0 0 0 1.73205i 0 3.61328 0 0 0
2593.6 0 0 0 1.73205i 0 4.35461 0 0 0
2593.7 0 0 0 1.73205i 0 −4.35461 0 0 0
2593.8 0 0 0 1.73205i 0 −3.61328 0 0 0
2593.9 0 0 0 1.73205i 0 −0.990721 0 0 0
2593.10 0 0 0 1.73205i 0 0.990721 0 0 0
2593.11 0 0 0 1.73205i 0 3.61328 0 0 0
2593.12 0 0 0 1.73205i 0 4.35461 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2593.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

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

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5184.2.d.r 12
3.b odd 2 1 5184.2.d.q 12
4.b odd 2 1 inner 5184.2.d.r 12
8.b even 2 1 inner 5184.2.d.r 12
8.d odd 2 1 inner 5184.2.d.r 12
9.c even 3 1 1728.2.r.e 12
9.c even 3 1 1728.2.r.f 12
9.d odd 6 1 576.2.r.e 12
9.d odd 6 1 576.2.r.f yes 12
12.b even 2 1 5184.2.d.q 12
24.f even 2 1 5184.2.d.q 12
24.h odd 2 1 5184.2.d.q 12
36.f odd 6 1 1728.2.r.e 12
36.f odd 6 1 1728.2.r.f 12
36.h even 6 1 576.2.r.e 12
36.h even 6 1 576.2.r.f yes 12
72.j odd 6 1 576.2.r.e 12
72.j odd 6 1 576.2.r.f yes 12
72.l even 6 1 576.2.r.e 12
72.l even 6 1 576.2.r.f yes 12
72.n even 6 1 1728.2.r.e 12
72.n even 6 1 1728.2.r.f 12
72.p odd 6 1 1728.2.r.e 12
72.p odd 6 1 1728.2.r.f 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
576.2.r.e 12 9.d odd 6 1
576.2.r.e 12 36.h even 6 1
576.2.r.e 12 72.j odd 6 1
576.2.r.e 12 72.l even 6 1
576.2.r.f yes 12 9.d odd 6 1
576.2.r.f yes 12 36.h even 6 1
576.2.r.f yes 12 72.j odd 6 1
576.2.r.f yes 12 72.l even 6 1
1728.2.r.e 12 9.c even 3 1
1728.2.r.e 12 36.f odd 6 1
1728.2.r.e 12 72.n even 6 1
1728.2.r.e 12 72.p odd 6 1
1728.2.r.f 12 9.c even 3 1
1728.2.r.f 12 36.f odd 6 1
1728.2.r.f 12 72.n even 6 1
1728.2.r.f 12 72.p odd 6 1
5184.2.d.q 12 3.b odd 2 1
5184.2.d.q 12 12.b even 2 1
5184.2.d.q 12 24.f even 2 1
5184.2.d.q 12 24.h odd 2 1
5184.2.d.r 12 1.a even 1 1 trivial
5184.2.d.r 12 4.b odd 2 1 inner
5184.2.d.r 12 8.b even 2 1 inner
5184.2.d.r 12 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(5184, [\chi])$$:

 $$T_{5}^{2} + 3$$ T5^2 + 3 $$T_{7}^{6} - 33T_{7}^{4} + 279T_{7}^{2} - 243$$ T7^6 - 33*T7^4 + 279*T7^2 - 243 $$T_{17}^{3} - 24T_{17} - 36$$ T17^3 - 24*T17 - 36 $$T_{23}^{6} - 117T_{23}^{4} + 4347T_{23}^{2} - 49923$$ T23^6 - 117*T23^4 + 4347*T23^2 - 49923 $$T_{41}^{3} - 9T_{41}^{2} + 3T_{41} + 9$$ T41^3 - 9*T41^2 + 3*T41 + 9

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12}$$
$5$ $$(T^{2} + 3)^{6}$$
$7$ $$(T^{6} - 33 T^{4} + 279 T^{2} - 243)^{2}$$
$11$ $$(T^{6} + 39 T^{4} + 171 T^{2} + 81)^{2}$$
$13$ $$(T^{6} + 57 T^{4} + 675 T^{2} + 243)^{2}$$
$17$ $$(T^{3} - 24 T - 36)^{4}$$
$19$ $$(T^{2} + 4)^{6}$$
$23$ $$(T^{6} - 117 T^{4} + 4347 T^{2} + \cdots - 49923)^{2}$$
$29$ $$(T^{6} + 153 T^{4} + 7155 T^{2} + \cdots + 93987)^{2}$$
$31$ $$(T^{6} - 117 T^{4} + 2619 T^{2} + \cdots - 9747)^{2}$$
$37$ $$(T^{6} + 60 T^{4} + 1008 T^{2} + \cdots + 3888)^{2}$$
$41$ $$(T^{3} - 9 T^{2} + 3 T + 9)^{4}$$
$43$ $$(T^{6} + 123 T^{4} + 1935 T^{2} + \cdots + 6889)^{2}$$
$47$ $$(T^{6} - 297 T^{4} + 27135 T^{2} + \cdots - 717363)^{2}$$
$53$ $$(T^{6} + 180 T^{4} + 1728 T^{2} + \cdots + 432)^{2}$$
$59$ $$(T^{6} + 147 T^{4} + 711 T^{2} + 729)^{2}$$
$61$ $$(T^{6} + 153 T^{4} + 3267 T^{2} + \cdots + 4563)^{2}$$
$67$ $$(T^{6} + 231 T^{4} + 8091 T^{2} + \cdots + 18769)^{2}$$
$71$ $$(T^{6} - 288 T^{4} + 19872 T^{2} + \cdots - 124848)^{2}$$
$73$ $$(T^{3} - 84 T - 164)^{4}$$
$79$ $$(T^{6} - 93 T^{4} + 891 T^{2} - 2187)^{2}$$
$83$ $$(T^{6} + 171 T^{4} + 3159 T^{2} + \cdots + 6561)^{2}$$
$89$ $$(T^{3} - 12 T^{2} - 36 T + 108)^{4}$$
$97$ $$(T^{3} - 3 T^{2} - 213 T + 1187)^{4}$$