# Properties

 Label 114.4.e.e Level $114$ Weight $4$ Character orbit 114.e Analytic conductor $6.726$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$114 = 2 \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 114.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.72621774065$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.6967728.1 Defining polynomial: $$x^{6} - x^{5} + 8 x^{4} + 5 x^{3} + 50 x^{2} - 7 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{6}\cdot 3^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 + 2 \beta_{3} ) q^{2} + ( 3 + 3 \beta_{3} ) q^{3} + 4 \beta_{3} q^{4} + ( -1 + \beta_{1} - \beta_{3} - \beta_{5} ) q^{5} + 6 \beta_{3} q^{6} + ( -6 + \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{7} -8 q^{8} + 9 \beta_{3} q^{9} +O(q^{10})$$ $$q + ( 2 + 2 \beta_{3} ) q^{2} + ( 3 + 3 \beta_{3} ) q^{3} + 4 \beta_{3} q^{4} + ( -1 + \beta_{1} - \beta_{3} - \beta_{5} ) q^{5} + 6 \beta_{3} q^{6} + ( -6 + \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{7} -8 q^{8} + 9 \beta_{3} q^{9} + ( -2 \beta_{3} - 2 \beta_{5} ) q^{10} + ( -18 + 2 \beta_{1} - \beta_{2} + \beta_{4} ) q^{11} -12 q^{12} + ( 2 + 2 \beta_{2} + 23 \beta_{3} + 4 \beta_{4} ) q^{13} + ( -8 + 2 \beta_{1} + 8 \beta_{2} - 16 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{14} + ( -3 \beta_{3} - 3 \beta_{5} ) q^{15} + ( -16 - 16 \beta_{3} ) q^{16} + ( 16 - 3 \beta_{1} - 2 \beta_{2} + 18 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{17} -18 q^{18} + ( 48 - 2 \beta_{1} - 2 \beta_{2} + 58 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{19} + ( 4 - 4 \beta_{1} ) q^{20} + ( -12 + 3 \beta_{1} + 12 \beta_{2} - 24 \beta_{3} + 6 \beta_{4} - 3 \beta_{5} ) q^{21} + ( -38 + 4 \beta_{1} - 4 \beta_{2} - 34 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} ) q^{22} + ( -10 - 10 \beta_{2} - 71 \beta_{3} - 20 \beta_{4} - 5 \beta_{5} ) q^{23} + ( -24 - 24 \beta_{3} ) q^{24} + ( -7 - 7 \beta_{2} + 87 \beta_{3} - 14 \beta_{4} + 11 \beta_{5} ) q^{25} + ( -50 - 4 \beta_{2} + 4 \beta_{4} ) q^{26} -27 q^{27} + ( 8 + 8 \beta_{2} - 32 \beta_{3} + 16 \beta_{4} - 4 \beta_{5} ) q^{28} + ( 13 + 13 \beta_{2} - 16 \beta_{3} + 26 \beta_{4} - \beta_{5} ) q^{29} + ( 6 - 6 \beta_{1} ) q^{30} + ( 33 + 8 \beta_{1} + 3 \beta_{2} - 3 \beta_{4} ) q^{31} -32 \beta_{3} q^{32} + ( -57 + 6 \beta_{1} - 6 \beta_{2} - 51 \beta_{3} - 3 \beta_{4} - 6 \beta_{5} ) q^{33} + ( -2 - 2 \beta_{2} + 36 \beta_{3} - 4 \beta_{4} + 6 \beta_{5} ) q^{34} + ( 115 - 18 \beta_{1} + 22 \beta_{2} + 93 \beta_{3} + 11 \beta_{4} + 18 \beta_{5} ) q^{35} + ( -36 - 36 \beta_{3} ) q^{36} + ( 94 + 23 \beta_{1} + 13 \beta_{2} - 13 \beta_{4} ) q^{37} + ( -26 - 10 \beta_{1} - 10 \beta_{2} + 100 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{38} + ( -75 - 6 \beta_{2} + 6 \beta_{4} ) q^{39} + ( 8 - 8 \beta_{1} + 8 \beta_{3} + 8 \beta_{5} ) q^{40} + ( -30 - 4 \beta_{1} - 52 \beta_{2} + 22 \beta_{3} - 26 \beta_{4} + 4 \beta_{5} ) q^{41} + ( 12 + 12 \beta_{2} - 48 \beta_{3} + 24 \beta_{4} - 6 \beta_{5} ) q^{42} + ( 90 + 7 \beta_{1} - 36 \beta_{2} + 126 \beta_{3} - 18 \beta_{4} - 7 \beta_{5} ) q^{43} + ( -4 - 4 \beta_{2} - 68 \beta_{3} - 8 \beta_{4} - 8 \beta_{5} ) q^{44} + ( 9 - 9 \beta_{1} ) q^{45} + ( 162 - 10 \beta_{1} + 20 \beta_{2} - 20 \beta_{4} ) q^{46} + ( 18 + 18 \beta_{2} - 278 \beta_{3} + 36 \beta_{4} - 14 \beta_{5} ) q^{47} -48 \beta_{3} q^{48} + ( 377 + 11 \beta_{1} + 13 \beta_{2} - 13 \beta_{4} ) q^{49} + ( -160 + 22 \beta_{1} + 14 \beta_{2} - 14 \beta_{4} ) q^{50} + ( -3 - 3 \beta_{2} + 54 \beta_{3} - 6 \beta_{4} + 9 \beta_{5} ) q^{51} + ( -108 - 16 \beta_{2} - 92 \beta_{3} - 8 \beta_{4} ) q^{52} + ( -13 - 13 \beta_{2} - 17 \beta_{3} - 26 \beta_{4} + 28 \beta_{5} ) q^{53} + ( -54 - 54 \beta_{3} ) q^{54} + ( 456 - 37 \beta_{1} - 46 \beta_{2} + 502 \beta_{3} - 23 \beta_{4} + 37 \beta_{5} ) q^{55} + ( 48 - 8 \beta_{1} - 16 \beta_{2} + 16 \beta_{4} ) q^{56} + ( -39 - 15 \beta_{1} - 15 \beta_{2} + 150 \beta_{3} - 6 \beta_{4} + 6 \beta_{5} ) q^{57} + ( 6 - 2 \beta_{1} - 26 \beta_{2} + 26 \beta_{4} ) q^{58} + ( -291 + 9 \beta_{1} + 48 \beta_{2} - 339 \beta_{3} + 24 \beta_{4} - 9 \beta_{5} ) q^{59} + ( 12 - 12 \beta_{1} + 12 \beta_{3} + 12 \beta_{5} ) q^{60} + ( -2 - 2 \beta_{2} - 121 \beta_{3} - 4 \beta_{4} + 30 \beta_{5} ) q^{61} + ( 72 + 16 \beta_{1} + 12 \beta_{2} + 60 \beta_{3} + 6 \beta_{4} - 16 \beta_{5} ) q^{62} + ( 18 + 18 \beta_{2} - 72 \beta_{3} + 36 \beta_{4} - 9 \beta_{5} ) q^{63} + 64 q^{64} + ( 131 - 23 \beta_{1} - 18 \beta_{2} + 18 \beta_{4} ) q^{65} + ( -6 - 6 \beta_{2} - 102 \beta_{3} - 12 \beta_{4} - 12 \beta_{5} ) q^{66} + ( -5 - 5 \beta_{2} + 34 \beta_{3} - 10 \beta_{4} + 26 \beta_{5} ) q^{67} + ( -68 + 12 \beta_{1} + 4 \beta_{2} - 4 \beta_{4} ) q^{68} + ( 243 - 15 \beta_{1} + 30 \beta_{2} - 30 \beta_{4} ) q^{69} + ( 22 + 22 \beta_{2} + 186 \beta_{3} + 44 \beta_{4} + 36 \beta_{5} ) q^{70} + ( -336 + 31 \beta_{1} - 2 \beta_{2} - 334 \beta_{3} - \beta_{4} - 31 \beta_{5} ) q^{71} -72 \beta_{3} q^{72} + ( -75 + 14 \beta_{1} - 80 \beta_{2} + 5 \beta_{3} - 40 \beta_{4} - 14 \beta_{5} ) q^{73} + ( 214 + 46 \beta_{1} + 52 \beta_{2} + 162 \beta_{3} + 26 \beta_{4} - 46 \beta_{5} ) q^{74} + ( -240 + 33 \beta_{1} + 21 \beta_{2} - 21 \beta_{4} ) q^{75} + ( -244 - 12 \beta_{1} - 12 \beta_{2} - 32 \beta_{3} - 20 \beta_{4} + 20 \beta_{5} ) q^{76} + ( 11 - 69 \beta_{1} - 16 \beta_{2} + 16 \beta_{4} ) q^{77} + ( -162 - 24 \beta_{2} - 138 \beta_{3} - 12 \beta_{4} ) q^{78} + ( 144 + 18 \beta_{1} + 86 \beta_{2} + 58 \beta_{3} + 43 \beta_{4} - 18 \beta_{5} ) q^{79} + ( 16 \beta_{3} + 16 \beta_{5} ) q^{80} + ( -81 - 81 \beta_{3} ) q^{81} + ( -52 - 52 \beta_{2} + 44 \beta_{3} - 104 \beta_{4} + 8 \beta_{5} ) q^{82} + ( -709 - 21 \beta_{1} + 17 \beta_{2} - 17 \beta_{4} ) q^{83} + ( 72 - 12 \beta_{1} - 24 \beta_{2} + 24 \beta_{4} ) q^{84} + ( 12 + 12 \beta_{2} - 588 \beta_{3} + 24 \beta_{4} - 48 \beta_{5} ) q^{85} + ( -36 - 36 \beta_{2} + 252 \beta_{3} - 72 \beta_{4} - 14 \beta_{5} ) q^{86} + ( 9 - 3 \beta_{1} - 39 \beta_{2} + 39 \beta_{4} ) q^{87} + ( 144 - 16 \beta_{1} + 8 \beta_{2} - 8 \beta_{4} ) q^{88} + ( -1 - \beta_{2} + 381 \beta_{3} - 2 \beta_{4} + 24 \beta_{5} ) q^{89} + ( 18 - 18 \beta_{1} + 18 \beta_{3} + 18 \beta_{5} ) q^{90} + ( 62 + 62 \beta_{2} + 374 \beta_{3} + 124 \beta_{4} - 59 \beta_{5} ) q^{91} + ( 364 - 20 \beta_{1} + 80 \beta_{2} + 284 \beta_{3} + 40 \beta_{4} + 20 \beta_{5} ) q^{92} + ( 108 + 24 \beta_{1} + 18 \beta_{2} + 90 \beta_{3} + 9 \beta_{4} - 24 \beta_{5} ) q^{93} + ( 520 - 28 \beta_{1} - 36 \beta_{2} + 36 \beta_{4} ) q^{94} + ( -875 + 42 \beta_{1} + 4 \beta_{2} - 325 \beta_{3} - 25 \beta_{4} - 70 \beta_{5} ) q^{95} + 96 q^{96} + ( -470 + 7 \beta_{1} - 14 \beta_{2} - 456 \beta_{3} - 7 \beta_{4} - 7 \beta_{5} ) q^{97} + ( 780 + 22 \beta_{1} + 52 \beta_{2} + 728 \beta_{3} + 26 \beta_{4} - 22 \beta_{5} ) q^{98} + ( -9 - 9 \beta_{2} - 153 \beta_{3} - 18 \beta_{4} - 18 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 6q^{2} + 9q^{3} - 12q^{4} - 2q^{5} - 18q^{6} - 34q^{7} - 48q^{8} - 27q^{9} + O(q^{10})$$ $$6q + 6q^{2} + 9q^{3} - 12q^{4} - 2q^{5} - 18q^{6} - 34q^{7} - 48q^{8} - 27q^{9} + 4q^{10} - 104q^{11} - 72q^{12} - 75q^{13} - 34q^{14} + 6q^{15} - 48q^{16} + 48q^{17} - 108q^{18} + 104q^{19} + 16q^{20} - 51q^{21} - 104q^{22} + 238q^{23} - 72q^{24} - 229q^{25} - 300q^{26} - 162q^{27} + 68q^{28} + 8q^{29} + 24q^{30} + 214q^{31} + 96q^{32} - 156q^{33} - 96q^{34} + 294q^{35} - 108q^{36} + 610q^{37} - 430q^{38} - 450q^{39} + 16q^{40} - 16q^{41} + 102q^{42} + 331q^{43} + 208q^{44} + 36q^{45} + 952q^{46} + 766q^{47} + 144q^{48} + 2284q^{49} - 916q^{50} - 144q^{51} - 300q^{52} + 118q^{53} - 162q^{54} + 1400q^{55} + 272q^{56} - 645q^{57} + 32q^{58} - 936q^{59} + 24q^{60} + 399q^{61} + 214q^{62} + 153q^{63} + 384q^{64} + 740q^{65} + 312q^{66} - 61q^{67} - 384q^{68} + 1428q^{69} - 588q^{70} - 974q^{71} + 216q^{72} - 91q^{73} + 610q^{74} - 1374q^{75} - 1276q^{76} - 72q^{77} - 450q^{78} + 321q^{79} - 32q^{80} - 243q^{81} + 32q^{82} - 4296q^{83} + 408q^{84} + 1680q^{85} - 662q^{86} + 48q^{87} + 832q^{88} - 1116q^{89} + 36q^{90} - 1367q^{91} + 952q^{92} + 321q^{93} + 3064q^{94} - 4198q^{95} + 576q^{96} - 1382q^{97} + 2284q^{98} + 468q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 8 x^{4} + 5 x^{3} + 50 x^{2} - 7 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$10 \nu^{5} - 80 \nu^{4} + 116 \nu^{3} - 500 \nu^{2} + 70 \nu - 2525$$$$)/131$$ $$\beta_{2}$$ $$=$$ $$($$$$-52 \nu^{5} + 23 \nu^{4} - 184 \nu^{3} - 544 \nu^{2} - 1936 \nu + 161$$$$)/393$$ $$\beta_{3}$$ $$=$$ $$($$$$-56 \nu^{5} + 55 \nu^{4} - 440 \nu^{3} - 344 \nu^{2} - 2750 \nu - 8$$$$)/393$$ $$\beta_{4}$$ $$=$$ $$($$$$-58 \nu^{5} + 71 \nu^{4} - 568 \nu^{3} - 244 \nu^{2} - 1978 \nu - 289$$$$)/393$$ $$\beta_{5}$$ $$=$$ $$($$$$-1036 \nu^{5} + 821 \nu^{4} - 8140 \nu^{3} - 6364 \nu^{2} - 52840 \nu - 148$$$$)/393$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{4} - 3 \beta_{3} + \beta_{2} + 1$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{5} + \beta_{4} - 60 \beta_{3} + 2 \beta_{2} - 3 \beta_{1} - 58$$$$)/12$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{4} + 5 \beta_{2} - \beta_{1} - 25$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-6 \beta_{5} - 10 \beta_{4} + 126 \beta_{3} - 5 \beta_{2} - 5$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-21 \beta_{5} - 62 \beta_{4} + 537 \beta_{3} - 124 \beta_{2} + 21 \beta_{1} + 413$$$$)/6$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/114\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$97$$ $$\chi(n)$$ $$1$$ $$\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 0.0702177 − 0.121621i −1.13654 + 1.96854i 1.56632 − 2.71294i 0.0702177 + 0.121621i −1.13654 − 1.96854i 1.56632 + 2.71294i
1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i −10.1010 17.4954i −3.00000 + 5.19615i −22.8872 −8.00000 −4.50000 + 7.79423i 20.2020 34.9909i
7.2 1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i 2.60679 + 4.51510i −3.00000 + 5.19615i 31.4905 −8.00000 −4.50000 + 7.79423i −5.21359 + 9.03020i
7.3 1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i 6.49420 + 11.2483i −3.00000 + 5.19615i −25.6033 −8.00000 −4.50000 + 7.79423i −12.9884 + 22.4966i
49.1 1.00000 1.73205i 1.50000 2.59808i −2.00000 3.46410i −10.1010 + 17.4954i −3.00000 5.19615i −22.8872 −8.00000 −4.50000 7.79423i 20.2020 + 34.9909i
49.2 1.00000 1.73205i 1.50000 2.59808i −2.00000 3.46410i 2.60679 4.51510i −3.00000 5.19615i 31.4905 −8.00000 −4.50000 7.79423i −5.21359 9.03020i
49.3 1.00000 1.73205i 1.50000 2.59808i −2.00000 3.46410i 6.49420 11.2483i −3.00000 5.19615i −25.6033 −8.00000 −4.50000 7.79423i −12.9884 22.4966i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.4.e.e 6
3.b odd 2 1 342.4.g.g 6
19.c even 3 1 inner 114.4.e.e 6
19.c even 3 1 2166.4.a.s 3
19.d odd 6 1 2166.4.a.w 3
57.h odd 6 1 342.4.g.g 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.e.e 6 1.a even 1 1 trivial
114.4.e.e 6 19.c even 3 1 inner
342.4.g.g 6 3.b odd 2 1
342.4.g.g 6 57.h odd 6 1
2166.4.a.s 3 19.c even 3 1
2166.4.a.w 3 19.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + 2 T_{5}^{5} + 304 T_{5}^{4} - 3336 T_{5}^{3} + 87264 T_{5}^{2} - 410400 T_{5} + 1871424$$ acting on $$S_{4}^{\mathrm{new}}(114, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 - 2 T + T^{2} )^{3}$$
$3$ $$( 9 - 3 T + T^{2} )^{3}$$
$5$ $$1871424 - 410400 T + 87264 T^{2} - 3336 T^{3} + 304 T^{4} + 2 T^{5} + T^{6}$$
$7$ $$( -18453 - 941 T + 17 T^{2} + T^{3} )^{2}$$
$11$ $$( -32688 - 888 T + 52 T^{2} + T^{3} )^{2}$$
$13$ $$174424849 - 10816533 T + 1661286 T^{2} + 87839 T^{3} + 4806 T^{4} + 75 T^{5} + T^{6}$$
$17$ $$107495424 + 17915904 T + 2488320 T^{2} + 103680 T^{3} + 4032 T^{4} - 48 T^{5} + T^{6}$$
$19$ $$322687697779 - 4892771624 T + 57732203 T^{2} - 1103216 T^{3} + 8417 T^{4} - 104 T^{5} + T^{6}$$
$23$ $$37266922552896 - 140724714528 T + 1984304736 T^{2} - 6722952 T^{3} + 79696 T^{4} - 238 T^{5} + T^{6}$$
$29$ $$93900570624 - 13120192512 T + 1835661312 T^{2} - 270336 T^{3} + 42880 T^{4} - 8 T^{5} + T^{6}$$
$31$ $$( 1435247 - 14005 T - 107 T^{2} + T^{3} )^{2}$$
$37$ $$( 28900349 - 125173 T - 305 T^{2} + T^{3} )^{2}$$
$41$ $$49106682003456 - 1167188520960 T + 27630111744 T^{2} - 16680192 T^{3} + 166816 T^{4} + 16 T^{5} + T^{6}$$
$43$ $$9744392803201 + 262030309541 T + 6012841550 T^{2} + 34027673 T^{3} + 193502 T^{4} - 331 T^{5} + T^{6}$$
$47$ $$407898773822016 + 1844102387232 T + 23807672928 T^{2} - 110334936 T^{3} + 495448 T^{4} - 766 T^{5} + T^{6}$$
$53$ $$1664148477888576 - 8892270888480 T + 52328969568 T^{2} - 55866312 T^{3} + 231904 T^{4} - 118 T^{5} + T^{6}$$
$59$ $$1002956470182144 - 4754223538560 T + 52178655168 T^{2} + 203851296 T^{3} + 725976 T^{4} + 936 T^{5} + T^{6}$$
$61$ $$6110887068098569 - 16384025471007 T + 75118241958 T^{2} - 72718315 T^{3} + 368790 T^{4} - 399 T^{5} + T^{6}$$
$67$ $$762088088919249 - 5197122435123 T + 33758241598 T^{2} - 66695807 T^{3} + 191982 T^{4} + 61 T^{5} + T^{6}$$
$71$ $$288657109767744 - 363923915040 T + 17006990688 T^{2} + 54842904 T^{3} + 927256 T^{4} + 974 T^{5} + T^{6}$$
$73$ $$27959330910572721 + 95025859694139 T + 338182176550 T^{2} + 282705487 T^{3} + 576582 T^{4} + 91 T^{5} + T^{6}$$
$79$ $$4061834487945529 + 27250815916863 T + 162367371678 T^{2} + 264718547 T^{3} + 530622 T^{4} - 321 T^{5} + T^{6}$$
$83$ $$( 211415616 + 1271664 T + 2148 T^{2} + T^{3} )^{2}$$
$89$ $$121838150433024 + 2706083925120 T + 47784981888 T^{2} + 251522496 T^{3} + 1000296 T^{4} + 1116 T^{5} + T^{6}$$
$97$ $$6285015236935744 + 47654058696800 T + 251758892384 T^{2} + 672164024 T^{3} + 1308824 T^{4} + 1382 T^{5} + T^{6}$$