# Properties

 Label 3381.2.a.bl Level $3381$ Weight $2$ Character orbit 3381.a Self dual yes Analytic conductor $26.997$ Analytic rank $0$ Dimension $10$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$3381 = 3 \cdot 7^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3381.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.9974209234$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 4 x^{9} - 6 x^{8} + 36 x^{7} + x^{6} - 100 x^{5} + 26 x^{4} + 108 x^{3} - 33 x^{2} - 36 x + 14$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + q^{3} + ( \beta_{1} + \beta_{2} ) q^{4} -\beta_{7} q^{5} + \beta_{1} q^{6} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{8} - \beta_{9} ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + q^{3} + ( \beta_{1} + \beta_{2} ) q^{4} -\beta_{7} q^{5} + \beta_{1} q^{6} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{8} - \beta_{9} ) q^{8} + q^{9} + ( 1 + \beta_{1} - \beta_{8} - \beta_{9} ) q^{10} + ( \beta_{3} - \beta_{4} - \beta_{5} + \beta_{8} ) q^{11} + ( \beta_{1} + \beta_{2} ) q^{12} + ( -1 + \beta_{1} + \beta_{2} - \beta_{5} + \beta_{8} ) q^{13} -\beta_{7} q^{15} + ( \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{16} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{9} ) q^{17} + \beta_{1} q^{18} + ( 2 + \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{19} + ( 2 + \beta_{1} + \beta_{2} - \beta_{4} - \beta_{6} - \beta_{8} ) q^{20} + ( -\beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{22} - q^{23} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{8} - \beta_{9} ) q^{24} + ( -\beta_{4} - \beta_{9} ) q^{25} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{26} + q^{27} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{29} + ( 1 + \beta_{1} - \beta_{8} - \beta_{9} ) q^{30} + ( 2 - \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{31} + ( -2 \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{32} + ( \beta_{3} - \beta_{4} - \beta_{5} + \beta_{8} ) q^{33} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} ) q^{34} + ( \beta_{1} + \beta_{2} ) q^{36} + ( -\beta_{1} - 2 \beta_{2} - \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{37} + ( 1 + 5 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{38} + ( -1 + \beta_{1} + \beta_{2} - \beta_{5} + \beta_{8} ) q^{39} + ( -1 + 3 \beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{40} + ( 1 + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + 3 \beta_{7} + \beta_{8} + \beta_{9} ) q^{41} + ( -2 + \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{43} + ( -4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} + 3 \beta_{9} ) q^{44} -\beta_{7} q^{45} -\beta_{1} q^{46} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{8} + 2 \beta_{9} ) q^{47} + ( \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{48} + ( -\beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{6} - \beta_{7} - \beta_{8} ) q^{50} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{9} ) q^{51} + ( 2 - \beta_{1} + \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{8} + \beta_{9} ) q^{52} + ( 1 + \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{53} + \beta_{1} q^{54} + ( 2 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{55} + ( 2 + \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{57} + ( \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{58} + ( 4 - \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{7} + 2 \beta_{8} ) q^{59} + ( 2 + \beta_{1} + \beta_{2} - \beta_{4} - \beta_{6} - \beta_{8} ) q^{60} + ( 1 + \beta_{1} + \beta_{2} + \beta_{6} - 2 \beta_{9} ) q^{61} + ( 1 - \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 3 \beta_{4} + 2 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{62} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} - \beta_{9} ) q^{64} + ( 2 - 3 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{9} ) q^{65} + ( -\beta_{1} - \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{66} + ( 1 - 3 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{8} ) q^{67} + ( -2 + 5 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 5 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{68} - q^{69} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} ) q^{71} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{8} - \beta_{9} ) q^{72} + ( 2 - \beta_{1} - 3 \beta_{2} + \beta_{5} - 2 \beta_{6} + \beta_{9} ) q^{73} + ( 1 - 3 \beta_{1} - \beta_{2} + 4 \beta_{3} - 3 \beta_{4} + \beta_{7} + \beta_{9} ) q^{74} + ( -\beta_{4} - \beta_{9} ) q^{75} + ( 3 + 6 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{76} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{78} + ( -1 + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + 3 \beta_{6} + 3 \beta_{7} - \beta_{9} ) q^{79} + ( 2 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{9} ) q^{80} + q^{81} + ( -4 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{82} + ( 4 + \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{83} + ( -1 - \beta_{2} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{85} + ( 1 + 3 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + 5 \beta_{7} + 4 \beta_{8} ) q^{86} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{87} + ( -3 - 4 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{88} + ( 4 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{8} + \beta_{9} ) q^{89} + ( 1 + \beta_{1} - \beta_{8} - \beta_{9} ) q^{90} + ( -\beta_{1} - \beta_{2} ) q^{92} + ( 2 - \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{93} + ( 5 + \beta_{1} + \beta_{2} + 4 \beta_{3} - 6 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{9} ) q^{94} + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - 4 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{95} + ( -2 \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{96} + ( -3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 3 \beta_{6} - \beta_{7} + 3 \beta_{8} + 4 \beta_{9} ) q^{97} + ( \beta_{3} - \beta_{4} - \beta_{5} + \beta_{8} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + 4q^{2} + 10q^{3} + 8q^{4} + 4q^{5} + 4q^{6} + 12q^{8} + 10q^{9} + O(q^{10})$$ $$10q + 4q^{2} + 10q^{3} + 8q^{4} + 4q^{5} + 4q^{6} + 12q^{8} + 10q^{9} + 8q^{10} + 2q^{11} + 8q^{12} + 4q^{15} + 4q^{16} + 12q^{17} + 4q^{18} + 26q^{19} + 24q^{20} - 8q^{22} - 10q^{23} + 12q^{24} - 2q^{25} + 4q^{26} + 10q^{27} + 16q^{29} + 8q^{30} + 12q^{31} + 8q^{32} + 2q^{33} + 28q^{34} + 8q^{36} - 8q^{37} + 32q^{38} + 4q^{40} + 10q^{41} - 4q^{43} - 16q^{44} + 4q^{45} - 4q^{46} + 2q^{47} + 4q^{48} - 8q^{50} + 12q^{51} + 24q^{52} + 14q^{53} + 4q^{54} + 16q^{55} + 26q^{57} - 8q^{58} + 38q^{59} + 24q^{60} + 14q^{61} - 8q^{62} + 8q^{64} + 12q^{65} - 8q^{66} + 8q^{68} - 10q^{69} + 24q^{71} + 12q^{72} + 8q^{73} - 8q^{74} - 2q^{75} + 64q^{76} + 4q^{78} - 16q^{79} + 28q^{80} + 10q^{81} - 40q^{82} + 28q^{83} - 4q^{85} + 20q^{86} + 16q^{87} - 68q^{88} + 32q^{89} + 8q^{90} - 8q^{92} + 12q^{93} + 56q^{94} + 8q^{95} + 8q^{96} + 4q^{97} + 2q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 4 x^{9} - 6 x^{8} + 36 x^{7} + x^{6} - 100 x^{5} + 26 x^{4} + 108 x^{3} - 33 x^{2} - 36 x + 14$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{7} - 3 \nu^{6} - 7 \nu^{5} + 22 \nu^{4} + 12 \nu^{3} - 37 \nu^{2} - 9 \nu + 12$$ $$\beta_{4}$$ $$=$$ $$\nu^{9} - 3 \nu^{8} - 9 \nu^{7} + 28 \nu^{6} + 27 \nu^{5} - 82 \nu^{4} - 40 \nu^{3} + 91 \nu^{2} + 26 \nu - 28$$ $$\beta_{5}$$ $$=$$ $$-\nu^{9} + 3 \nu^{8} + 10 \nu^{7} - 31 \nu^{6} - 33 \nu^{5} + 103 \nu^{4} + 45 \nu^{3} - 123 \nu^{2} - 26 \nu + 37$$ $$\beta_{6}$$ $$=$$ $$-2 \nu^{9} + 7 \nu^{8} + 15 \nu^{7} - 63 \nu^{6} - 30 \nu^{5} + 173 \nu^{4} + 30 \nu^{3} - 176 \nu^{2} - 25 \nu + 46$$ $$\beta_{7}$$ $$=$$ $$3 \nu^{9} - 11 \nu^{8} - 21 \nu^{7} + 98 \nu^{6} + 34 \nu^{5} - 266 \nu^{4} - 26 \nu^{3} + 271 \nu^{2} + 30 \nu - 70$$ $$\beta_{8}$$ $$=$$ $$-4 \nu^{9} + 14 \nu^{8} + 31 \nu^{7} - 128 \nu^{6} - 69 \nu^{5} + 362 \nu^{4} + 83 \nu^{3} - 385 \nu^{2} - 67 \nu + 108$$ $$\beta_{9}$$ $$=$$ $$5 \nu^{9} - 17 \nu^{8} - 41 \nu^{7} + 159 \nu^{6} + 103 \nu^{5} - 466 \nu^{4} - 136 \nu^{3} + 514 \nu^{2} + 106 \nu - 149$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$-\beta_{9} - \beta_{8} + \beta_{4} - \beta_{3} + \beta_{2} + 5 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$-\beta_{9} - \beta_{8} - \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{4} - 2 \beta_{3} + 6 \beta_{2} + 7 \beta_{1} + 8$$ $$\nu^{5}$$ $$=$$ $$-8 \beta_{9} - 8 \beta_{8} - \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 9 \beta_{4} - 10 \beta_{3} + 8 \beta_{2} + 28 \beta_{1} + 8$$ $$\nu^{6}$$ $$=$$ $$-11 \beta_{9} - 10 \beta_{8} - 8 \beta_{7} - 18 \beta_{6} + 10 \beta_{5} + 13 \beta_{4} - 22 \beta_{3} + 37 \beta_{2} + 47 \beta_{1} + 41$$ $$\nu^{7}$$ $$=$$ $$-55 \beta_{9} - 52 \beta_{8} - 9 \beta_{7} - 24 \beta_{6} + 22 \beta_{5} + 68 \beta_{4} - 79 \beta_{3} + 60 \beta_{2} + 169 \beta_{1} + 53$$ $$\nu^{8}$$ $$=$$ $$-91 \beta_{9} - 75 \beta_{8} - 50 \beta_{7} - 131 \beta_{6} + 79 \beta_{5} + 122 \beta_{4} - 183 \beta_{3} + 237 \beta_{2} + 318 \beta_{1} + 230$$ $$\nu^{9}$$ $$=$$ $$-366 \beta_{9} - 319 \beta_{8} - 62 \beta_{7} - 215 \beta_{6} + 183 \beta_{5} + 494 \beta_{4} - 578 \beta_{3} + 440 \beta_{2} + 1060 \beta_{1} + 345$$

## 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
 −2.23864 −1.28424 −1.11439 −0.881528 0.507824 0.533756 1.46930 1.94871 2.42151 2.63770
−2.23864 1.00000 3.01150 2.46287 −2.23864 0 −2.26439 1.00000 −5.51348
1.2 −1.28424 1.00000 −0.350734 −1.91754 −1.28424 0 3.01890 1.00000 2.46257
1.3 −1.11439 1.00000 −0.758127 1.17161 −1.11439 0 3.07364 1.00000 −1.30563
1.4 −0.881528 1.00000 −1.22291 −2.92694 −0.881528 0 2.84108 1.00000 2.58018
1.5 0.507824 1.00000 −1.74212 3.36946 0.507824 0 −1.90033 1.00000 1.71109
1.6 0.533756 1.00000 −1.71510 −1.09368 0.533756 0 −1.98296 1.00000 −0.583758
1.7 1.46930 1.00000 0.158829 −2.14842 1.46930 0 −2.70522 1.00000 −3.15666
1.8 1.94871 1.00000 1.79746 1.40111 1.94871 0 −0.394699 1.00000 2.73035
1.9 2.42151 1.00000 3.86372 2.93909 2.42151 0 4.51302 1.00000 7.11705
1.10 2.63770 1.00000 4.95748 0.742420 2.63770 0 7.80096 1.00000 1.95828
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$1$$
$$23$$ $$1$$

## 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 3381.2.a.bl yes 10
7.b odd 2 1 3381.2.a.bk 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3381.2.a.bk 10 7.b odd 2 1
3381.2.a.bl yes 10 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3381))$$:

 $$T_{2}^{10} - \cdots$$ $$T_{5}^{10} - \cdots$$ $$T_{11}^{10} - \cdots$$ $$T_{13}^{10} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$14 - 36 T - 33 T^{2} + 108 T^{3} + 26 T^{4} - 100 T^{5} + T^{6} + 36 T^{7} - 6 T^{8} - 4 T^{9} + T^{10}$$
$3$ $$( -1 + T )^{10}$$
$5$ $$392 - 672 T - 377 T^{2} + 976 T^{3} - 24 T^{4} - 428 T^{5} + 71 T^{6} + 72 T^{7} - 16 T^{8} - 4 T^{9} + T^{10}$$
$7$ $$T^{10}$$
$11$ $$-12953 - 28550 T + 8629 T^{2} + 19900 T^{3} - 6314 T^{4} - 2936 T^{5} + 1038 T^{6} + 140 T^{7} - 57 T^{8} - 2 T^{9} + T^{10}$$
$13$ $$5598 - 1104 T - 12535 T^{2} + 10904 T^{3} - 350 T^{4} - 2584 T^{5} + 747 T^{6} + 96 T^{7} - 52 T^{8} + T^{10}$$
$17$ $$-226 + 1460 T - 2849 T^{2} + 852 T^{3} + 2296 T^{4} - 1232 T^{5} - 532 T^{6} + 260 T^{7} + 10 T^{8} - 12 T^{9} + T^{10}$$
$19$ $$-24113 + 558 T + 51489 T^{2} - 26920 T^{3} - 13486 T^{4} + 11628 T^{5} - 1202 T^{6} - 760 T^{7} + 239 T^{8} - 26 T^{9} + T^{10}$$
$23$ $$( 1 + T )^{10}$$
$29$ $$-486482 + 618948 T - 100417 T^{2} - 160168 T^{3} + 69248 T^{4} + 3452 T^{5} - 6620 T^{6} + 1112 T^{7} + 10 T^{8} - 16 T^{9} + T^{10}$$
$31$ $$8641252 - 1261844 T - 2812231 T^{2} + 764552 T^{3} + 168910 T^{4} - 69520 T^{5} + 446 T^{6} + 1672 T^{7} - 102 T^{8} - 12 T^{9} + T^{10}$$
$37$ $$679266 + 127620 T - 949907 T^{2} - 450124 T^{3} + 29746 T^{4} + 40004 T^{5} + 2602 T^{6} - 1060 T^{7} - 112 T^{8} + 8 T^{9} + T^{10}$$
$41$ $$1815793 - 2821190 T - 977659 T^{2} + 1622804 T^{3} + 6698 T^{4} - 116976 T^{5} + 7870 T^{6} + 2132 T^{7} - 191 T^{8} - 10 T^{9} + T^{10}$$
$43$ $$-14455202 + 3973512 T + 6036713 T^{2} - 659944 T^{3} - 646414 T^{4} + 34132 T^{5} + 20715 T^{6} - 676 T^{7} - 248 T^{8} + 4 T^{9} + T^{10}$$
$47$ $$-112744688 - 44845056 T + 21488656 T^{2} + 3553408 T^{3} - 1270868 T^{4} - 90344 T^{5} + 30892 T^{6} + 788 T^{7} - 305 T^{8} - 2 T^{9} + T^{10}$$
$53$ $$5454391 + 16885562 T - 14328281 T^{2} + 2381116 T^{3} + 565983 T^{4} - 176466 T^{5} + 957 T^{6} + 2996 T^{7} - 161 T^{8} - 14 T^{9} + T^{10}$$
$59$ $$42698601 - 67296570 T + 35004827 T^{2} - 6366492 T^{3} - 546479 T^{4} + 382190 T^{5} - 50159 T^{6} + 480 T^{7} + 441 T^{8} - 38 T^{9} + T^{10}$$
$61$ $$141967 - 121234 T - 131789 T^{2} + 69836 T^{3} + 29703 T^{4} - 14618 T^{5} - 1711 T^{6} + 1128 T^{7} - 51 T^{8} - 14 T^{9} + T^{10}$$
$67$ $$-376888 + 4785592 T + 1920121 T^{2} - 1010508 T^{3} - 359742 T^{4} + 57844 T^{5} + 19955 T^{6} - 508 T^{7} - 268 T^{8} + T^{10}$$
$71$ $$-472303438 + 515942472 T - 146773457 T^{2} - 5461680 T^{3} + 7275166 T^{4} - 755588 T^{5} - 36267 T^{6} + 8712 T^{7} - 186 T^{8} - 24 T^{9} + T^{10}$$
$73$ $$-2744 - 11480 T + 39409 T^{2} + 17912 T^{3} - 24066 T^{4} - 2720 T^{5} + 3302 T^{6} + 184 T^{7} - 138 T^{8} - 8 T^{9} + T^{10}$$
$79$ $$537938 - 2840156 T + 1640109 T^{2} + 2367236 T^{3} + 597966 T^{4} - 39044 T^{5} - 36862 T^{6} - 5236 T^{7} - 172 T^{8} + 16 T^{9} + T^{10}$$
$83$ $$11428 + 68124 T + 14767 T^{2} - 157108 T^{3} + 75080 T^{4} + 16352 T^{5} - 14916 T^{6} + 1988 T^{7} + 116 T^{8} - 28 T^{9} + T^{10}$$
$89$ $$1212644 - 3024116 T + 2563447 T^{2} - 715564 T^{3} - 136872 T^{4} + 117272 T^{5} - 21451 T^{6} + 392 T^{7} + 296 T^{8} - 32 T^{9} + T^{10}$$
$97$ $$-608495324 + 663569116 T - 251573807 T^{2} + 35049784 T^{3} + 1308600 T^{4} - 817304 T^{5} + 49774 T^{6} + 4296 T^{7} - 460 T^{8} - 4 T^{9} + T^{10}$$
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