# Properties

 Label 8001.2.a.m Level $8001$ Weight $2$ Character orbit 8001.a Self dual yes Analytic conductor $63.888$ Analytic rank $1$ Dimension $11$ CM no Inner twists $1$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$8001 = 3^{2} \cdot 7 \cdot 127$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8001.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$63.8883066572$$ Analytic rank: $$1$$ Dimension: $$11$$ Coefficient field: $$\mathbb{Q}[x]/(x^{11} - \cdots)$$ Defining polynomial: $$x^{11} - 2 x^{10} - 15 x^{9} + 25 x^{8} + 88 x^{7} - 112 x^{6} - 247 x^{5} + 215 x^{4} + 313 x^{3} - 170 x^{2} - 121 x + 57$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2667) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{10}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + \beta_{7} q^{5} - q^{7} + ( 1 + \beta_{2} + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + \beta_{7} q^{5} - q^{7} + ( 1 + \beta_{2} + \beta_{3} ) q^{8} + ( -1 + \beta_{1} - \beta_{2} + \beta_{7} - \beta_{10} ) q^{10} + ( 1 - \beta_{7} - \beta_{9} ) q^{11} + ( -2 + \beta_{1} - \beta_{2} - \beta_{6} + \beta_{7} ) q^{13} -\beta_{1} q^{14} + ( -1 + 2 \beta_{1} + \beta_{3} - \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{16} + ( 2 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} + \beta_{10} ) q^{17} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} ) q^{19} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{7} + \beta_{9} - \beta_{10} ) q^{20} + ( -1 + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{22} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{23} + ( \beta_{1} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{25} + ( -3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{26} + ( -1 - \beta_{2} ) q^{28} + ( -\beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} ) q^{29} + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{31} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} + 4 \beta_{8} + 2 \beta_{9} ) q^{32} + ( -2 + \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{6} - 2 \beta_{7} ) q^{34} -\beta_{7} q^{35} + ( -2 - \beta_{2} + \beta_{4} + \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{37} + ( -3 \beta_{1} - \beta_{3} + \beta_{6} - 5 \beta_{7} - 4 \beta_{8} - 2 \beta_{9} ) q^{38} + ( -2 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{6} - \beta_{8} + \beta_{9} ) q^{40} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} - \beta_{9} ) q^{41} + ( -3 + 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} ) q^{43} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{7} - \beta_{9} + \beta_{10} ) q^{44} + ( -1 - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} ) q^{46} + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{8} - \beta_{10} ) q^{47} + q^{49} + ( 4 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} - 2 \beta_{7} - \beta_{8} - 3 \beta_{9} + \beta_{10} ) q^{50} + ( -6 - \beta_{1} - 3 \beta_{2} + \beta_{5} - 2 \beta_{8} - \beta_{10} ) q^{52} + ( 2 - 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{53} + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} ) q^{55} + ( -1 - \beta_{2} - \beta_{3} ) q^{56} + ( -2 - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{9} ) q^{58} + ( -3 + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{59} + ( -1 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{61} + ( -3 + \beta_{1} + 4 \beta_{3} + 2 \beta_{4} - \beta_{6} + 4 \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{62} + ( -1 + 3 \beta_{1} + \beta_{2} + 4 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{64} + ( 3 + 2 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - 4 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{65} + ( -2 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} ) q^{67} + ( 2 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + \beta_{6} - 3 \beta_{7} - \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{68} + ( 1 - \beta_{1} + \beta_{2} - \beta_{7} + \beta_{10} ) q^{70} + ( 1 + 2 \beta_{1} - \beta_{3} + \beta_{4} + 4 \beta_{7} + 3 \beta_{8} + 3 \beta_{9} ) q^{71} + ( -3 - 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{7} + 2 \beta_{8} + 3 \beta_{9} ) q^{73} + ( -5 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{7} - 3 \beta_{8} - 3 \beta_{9} + 4 \beta_{10} ) q^{74} + ( 3 - 4 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + \beta_{6} - 6 \beta_{7} - 5 \beta_{8} - 2 \beta_{9} + \beta_{10} ) q^{76} + ( -1 + \beta_{7} + \beta_{9} ) q^{77} + ( -1 - 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{79} + ( -4 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - 5 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} ) q^{80} + ( -6 \beta_{1} - \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - \beta_{5} + 2 \beta_{6} - 4 \beta_{7} - \beta_{9} + 4 \beta_{10} ) q^{82} + ( -2 - \beta_{1} + 3 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{83} + ( -3 - 6 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} ) q^{85} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} + 3 \beta_{7} + 4 \beta_{8} + 3 \beta_{9} + \beta_{10} ) q^{86} + ( \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{9} - \beta_{10} ) q^{88} + ( 2 \beta_{4} + \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - 3 \beta_{10} ) q^{89} + ( 2 - \beta_{1} + \beta_{2} + \beta_{6} - \beta_{7} ) q^{91} + ( 4 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - 3 \beta_{5} - 2 \beta_{8} - \beta_{10} ) q^{92} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{5} + 3 \beta_{7} + \beta_{8} + 3 \beta_{9} - 2 \beta_{10} ) q^{94} + ( -1 - 3 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} + 3 \beta_{7} + 4 \beta_{8} + 6 \beta_{9} - 2 \beta_{10} ) q^{95} + ( -4 - 2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{10} ) q^{97} + \beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$11q + 2q^{2} + 12q^{4} - q^{5} - 11q^{7} + 15q^{8} + O(q^{10})$$ $$11q + 2q^{2} + 12q^{4} - q^{5} - 11q^{7} + 15q^{8} - 12q^{10} + 7q^{11} - 24q^{13} - 2q^{14} - 6q^{16} + 15q^{17} - 19q^{19} - 3q^{20} - 3q^{22} + 11q^{23} + 10q^{25} - 10q^{26} - 12q^{28} + 10q^{29} - 20q^{31} + 27q^{32} - 9q^{34} + q^{35} - 22q^{37} - 8q^{38} - 29q^{40} - 9q^{41} - 17q^{43} + 9q^{44} - 18q^{46} + 7q^{47} + 11q^{49} + 47q^{50} - 66q^{52} + 28q^{53} - 24q^{55} - 15q^{56} - 39q^{58} - 35q^{59} - 6q^{61} - 18q^{62} + 11q^{64} + 43q^{65} - 22q^{67} + 12q^{68} + 12q^{70} + 22q^{71} - 29q^{73} - 14q^{74} + 10q^{76} - 7q^{77} - 20q^{79} - 66q^{80} - 24q^{82} - 17q^{83} - 50q^{85} + 12q^{86} + 2q^{88} + q^{89} + 24q^{91} + 22q^{92} + q^{94} - 10q^{95} - 45q^{97} + 2q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{11} - 2 x^{10} - 15 x^{9} + 25 x^{8} + 88 x^{7} - 112 x^{6} - 247 x^{5} + 215 x^{4} + 313 x^{3} - 170 x^{2} - 121 x + 57$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4 \nu + 2$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{10} + 6 \nu^{9} + 5 \nu^{8} - 66 \nu^{7} + 8 \nu^{6} + 255 \nu^{5} - 59 \nu^{4} - 399 \nu^{3} + 72 \nu^{2} + 190 \nu - 51$$$$)/7$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{10} - 5 \nu^{9} - 24 \nu^{8} + 55 \nu^{7} + 110 \nu^{6} - 209 \nu^{5} - 246 \nu^{4} + 308 \nu^{3} + 255 \nu^{2} - 114 \nu - 45$$$$)/7$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{10} - 11 \nu^{9} - 29 \nu^{8} + 128 \nu^{7} + 81 \nu^{6} - 520 \nu^{5} - 19 \nu^{4} + 840 \nu^{3} - 174 \nu^{2} - 402 \nu + 146$$$$)/7$$ $$\beta_{7}$$ $$=$$ $$($$$$2 \nu^{10} - 12 \nu^{9} - 10 \nu^{8} + 139 \nu^{7} - 37 \nu^{6} - 559 \nu^{5} + 272 \nu^{4} + 882 \nu^{3} - 431 \nu^{2} - 401 \nu + 193$$$$)/7$$ $$\beta_{8}$$ $$=$$ $$($$$$-4 \nu^{10} + 17 \nu^{9} + 34 \nu^{8} - 194 \nu^{7} - 66 \nu^{6} + 761 \nu^{5} - 89 \nu^{4} - 1162 \nu^{3} + 330 \nu^{2} + 515 \nu - 204$$$$)/7$$ $$\beta_{9}$$ $$=$$ $$($$$$8 \nu^{10} - 27 \nu^{9} - 82 \nu^{8} + 304 \nu^{7} + 279 \nu^{6} - 1172 \nu^{5} - 333 \nu^{4} + 1743 \nu^{3} - 9 \nu^{2} - 729 \nu + 191$$$$)/7$$ $$\beta_{10}$$ $$=$$ $$($$$$10 \nu^{10} - 32 \nu^{9} - 99 \nu^{8} + 352 \nu^{7} + 298 \nu^{6} - 1325 \nu^{5} - 180 \nu^{4} + 1939 \nu^{3} - 377 \nu^{2} - 829 \nu + 321$$$$)/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{9} + 2 \beta_{8} + \beta_{7} - \beta_{5} + \beta_{3} + 6 \beta_{2} + 2 \beta_{1} + 13$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{9} + 4 \beta_{8} + 3 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 9 \beta_{3} + 9 \beta_{2} + 21 \beta_{1} + 10$$ $$\nu^{6}$$ $$=$$ $$11 \beta_{9} + 23 \beta_{8} + 13 \beta_{7} - \beta_{6} - 9 \beta_{5} + \beta_{4} + 14 \beta_{3} + 37 \beta_{2} + 23 \beta_{1} + 65$$ $$\nu^{7}$$ $$=$$ $$25 \beta_{9} + 53 \beta_{8} + 39 \beta_{7} - 10 \beta_{6} - 12 \beta_{5} + 12 \beta_{4} + 71 \beta_{3} + 71 \beta_{2} + 127 \beta_{1} + 77$$ $$\nu^{8}$$ $$=$$ $$\beta_{10} + 94 \beta_{9} + 206 \beta_{8} + 127 \beta_{7} - 15 \beta_{6} - 65 \beta_{5} + 17 \beta_{4} + 140 \beta_{3} + 243 \beta_{2} + 206 \beta_{1} + 359$$ $$\nu^{9}$$ $$=$$ $$2 \beta_{10} + 231 \beta_{9} + 513 \beta_{8} + 372 \beta_{7} - 79 \beta_{6} - 108 \beta_{5} + 107 \beta_{4} + 544 \beta_{3} + 539 \beta_{2} + 838 \beta_{1} + 561$$ $$\nu^{10}$$ $$=$$ $$17 \beta_{10} + 745 \beta_{9} + 1696 \beta_{8} + 1103 \beta_{7} - 152 \beta_{6} - 449 \beta_{5} + 191 \beta_{4} + 1227 \beta_{3} + 1673 \beta_{2} + 1691 \beta_{1} + 2148$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.12999 −1.81808 −1.57007 −1.50746 −0.910036 0.455441 0.678644 1.68813 2.14674 2.22659 2.74009
−2.12999 0 2.53686 1.18187 0 −1.00000 −1.14352 0 −2.51737
1.2 −1.81808 0 1.30541 1.39765 0 −1.00000 1.26282 0 −2.54104
1.3 −1.57007 0 0.465107 2.53976 0 −1.00000 2.40988 0 −3.98759
1.4 −1.50746 0 0.272426 −0.476857 0 −1.00000 2.60424 0 0.718841
1.5 −0.910036 0 −1.17183 −2.84869 0 −1.00000 2.88648 0 2.59241
1.6 0.455441 0 −1.79257 0.749481 0 −1.00000 −1.72729 0 0.341344
1.7 0.678644 0 −1.53944 −2.84434 0 −1.00000 −2.40202 0 −1.93030
1.8 1.68813 0 0.849768 4.40419 0 −1.00000 −1.94174 0 7.43483
1.9 2.14674 0 2.60851 −3.76087 0 −1.00000 1.30632 0 −8.07362
1.10 2.22659 0 2.95768 0.700608 0 −1.00000 2.13237 0 1.55996
1.11 2.74009 0 5.50808 −2.04281 0 −1.00000 9.61245 0 −5.59747
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.11 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$1$$
$$127$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8001.2.a.m 11
3.b odd 2 1 2667.2.a.k 11

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.2.a.k 11 3.b odd 2 1
8001.2.a.m 11 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8001))$$:

 $$T_{2}^{11} - \cdots$$ $$T_{5}^{11} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$57 - 121 T - 170 T^{2} + 313 T^{3} + 215 T^{4} - 247 T^{5} - 112 T^{6} + 88 T^{7} + 25 T^{8} - 15 T^{9} - 2 T^{10} + T^{11}$$
$3$ $$T^{11}$$
$5$ $$288 - 400 T - 1192 T^{2} + 2212 T^{3} - 22 T^{4} - 1331 T^{5} + 197 T^{6} + 319 T^{7} - 32 T^{8} - 32 T^{9} + T^{10} + T^{11}$$
$7$ $$( 1 + T )^{11}$$
$11$ $$-36 - 29 T + 3986 T^{2} - 8290 T^{3} + 2721 T^{4} + 2976 T^{5} - 1720 T^{6} - 154 T^{7} + 220 T^{8} - 19 T^{9} - 7 T^{10} + T^{11}$$
$13$ $$-4206 - 28879 T - 40072 T^{2} + 26680 T^{3} + 34890 T^{4} - 3910 T^{5} - 8910 T^{6} - 1300 T^{7} + 560 T^{8} + 203 T^{9} + 24 T^{10} + T^{11}$$
$17$ $$-1027734 + 934417 T + 592196 T^{2} - 642702 T^{3} - 6636 T^{4} + 100401 T^{5} - 16043 T^{6} - 3647 T^{7} + 997 T^{8} - 2 T^{9} - 15 T^{10} + T^{11}$$
$19$ $$5452108 + 3313291 T - 2611332 T^{2} - 1580787 T^{3} + 224206 T^{4} + 219573 T^{5} + 12647 T^{6} - 9603 T^{7} - 1565 T^{8} + 25 T^{9} + 19 T^{10} + T^{11}$$
$23$ $$255744 - 945552 T + 1248080 T^{2} - 634240 T^{3} - 2422 T^{4} + 106785 T^{5} - 28022 T^{6} - 1049 T^{7} + 1115 T^{8} - 70 T^{9} - 11 T^{10} + T^{11}$$
$29$ $$-9221472 - 11240192 T - 973480 T^{2} + 2763640 T^{3} + 570370 T^{4} - 218507 T^{5} - 45140 T^{6} + 7677 T^{7} + 1190 T^{8} - 138 T^{9} - 10 T^{10} + T^{11}$$
$31$ $$260432712 + 140309189 T - 14131262 T^{2} - 18612657 T^{3} - 1384643 T^{4} + 800427 T^{5} + 119277 T^{6} - 10927 T^{7} - 2759 T^{8} - 25 T^{9} + 20 T^{10} + T^{11}$$
$37$ $$-1099454 + 3878325 T + 1207262 T^{2} - 1452847 T^{3} - 321373 T^{4} + 177525 T^{5} + 35617 T^{6} - 7651 T^{7} - 1653 T^{8} + 47 T^{9} + 22 T^{10} + T^{11}$$
$41$ $$-357543498 + 33775933 T + 109512291 T^{2} + 9262203 T^{3} - 7212393 T^{4} - 822612 T^{5} + 189872 T^{6} + 22508 T^{7} - 2179 T^{8} - 252 T^{9} + 9 T^{10} + T^{11}$$
$43$ $$926056896 + 642059344 T - 1415320 T^{2} - 81050000 T^{3} - 13637948 T^{4} + 1525617 T^{5} + 461260 T^{6} + 3583 T^{7} - 4939 T^{8} - 212 T^{9} + 17 T^{10} + T^{11}$$
$47$ $$-2774616 - 3856891 T + 3838993 T^{2} + 2419843 T^{3} - 327718 T^{4} - 296999 T^{5} - 6597 T^{6} + 11365 T^{7} + 626 T^{8} - 169 T^{9} - 7 T^{10} + T^{11}$$
$53$ $$-416 - 2544 T + 4256 T^{2} + 44412 T^{3} + 64028 T^{4} + 16793 T^{5} - 16727 T^{6} - 6827 T^{7} + 852 T^{8} + 177 T^{9} - 28 T^{10} + T^{11}$$
$59$ $$-1303104 - 9216464 T - 16687512 T^{2} - 4048552 T^{3} + 4136620 T^{4} + 1234729 T^{5} - 63912 T^{6} - 44579 T^{7} - 2719 T^{8} + 280 T^{9} + 35 T^{10} + T^{11}$$
$61$ $$-1347574 + 433429 T + 885546 T^{2} - 112235 T^{3} - 211359 T^{4} - 5325 T^{5} + 19263 T^{6} + 1917 T^{7} - 665 T^{8} - 93 T^{9} + 6 T^{10} + T^{11}$$
$67$ $$9891955904 + 13129762256 T + 2948406144 T^{2} - 444346976 T^{3} - 146278234 T^{4} + 2475449 T^{5} + 2199819 T^{6} + 48042 T^{7} - 12106 T^{8} - 454 T^{9} + 22 T^{10} + T^{11}$$
$71$ $$125943744 - 456179213 T + 424812369 T^{2} - 133824927 T^{3} + 7464734 T^{4} + 3586004 T^{5} - 535774 T^{6} - 16202 T^{7} + 6427 T^{8} - 162 T^{9} - 22 T^{10} + T^{11}$$
$73$ $$13520209618 + 9510367151 T + 745817410 T^{2} - 582096840 T^{3} - 75414841 T^{4} + 10127910 T^{5} + 1672678 T^{6} - 37994 T^{7} - 12812 T^{8} - 227 T^{9} + 29 T^{10} + T^{11}$$
$79$ $$-11637350848 + 3128912401 T + 1637287493 T^{2} - 120725523 T^{3} - 65389709 T^{4} + 481613 T^{5} + 1074427 T^{6} + 25149 T^{7} - 7734 T^{8} - 311 T^{9} + 20 T^{10} + T^{11}$$
$83$ $$205464384 - 271115952 T + 42003328 T^{2} + 30620768 T^{3} - 6914666 T^{4} - 975985 T^{5} + 244000 T^{6} + 16632 T^{7} - 3392 T^{8} - 186 T^{9} + 17 T^{10} + T^{11}$$
$89$ $$-518712672 - 1199824192 T - 146506128 T^{2} + 142899880 T^{3} + 12240104 T^{4} - 5766519 T^{5} - 206876 T^{6} + 88157 T^{7} + 914 T^{8} - 515 T^{9} - T^{10} + T^{11}$$
$97$ $$350624 - 1144320 T + 150240 T^{2} + 1572384 T^{3} + 6468 T^{4} - 571915 T^{5} - 152880 T^{6} + 1143 T^{7} + 5116 T^{8} + 753 T^{9} + 45 T^{10} + T^{11}$$
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