# Properties

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

# Related objects

## 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: $$x^{10} - 3 x^{9} - 13 x^{8} + 41 x^{7} + 47 x^{6} - 165 x^{5} - 45 x^{4} + 207 x^{3} + 12 x^{2} - 76 x - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 483) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} - q^{3} + ( 2 + \beta_{2} ) q^{4} + ( -1 - \beta_{8} ) q^{5} -\beta_{1} q^{6} + ( 2 \beta_{1} + \beta_{3} + \beta_{7} - \beta_{8} ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} - q^{3} + ( 2 + \beta_{2} ) q^{4} + ( -1 - \beta_{8} ) q^{5} -\beta_{1} q^{6} + ( 2 \beta_{1} + \beta_{3} + \beta_{7} - \beta_{8} ) q^{8} + q^{9} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{10} + ( 1 - \beta_{5} - \beta_{6} ) q^{11} + ( -2 - \beta_{2} ) q^{12} + \beta_{6} q^{13} + ( 1 + \beta_{8} ) q^{15} + ( 4 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} - \beta_{9} ) q^{16} + ( -1 + \beta_{3} - \beta_{5} - \beta_{8} ) q^{17} + \beta_{1} q^{18} + ( \beta_{1} + \beta_{3} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{19} + ( -2 - \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{9} ) q^{20} + ( -\beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{22} + q^{23} + ( -2 \beta_{1} - \beta_{3} - \beta_{7} + \beta_{8} ) q^{24} + ( 2 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{25} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{26} - q^{27} + ( 2 + \beta_{2} - \beta_{4} - \beta_{8} ) q^{29} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{30} + ( -2 \beta_{1} + \beta_{2} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{31} + ( 1 + 5 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{9} ) q^{32} + ( -1 + \beta_{5} + \beta_{6} ) q^{33} + ( -1 - \beta_{2} - 3 \beta_{4} - 2 \beta_{5} ) q^{34} + ( 2 + \beta_{2} ) q^{36} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{7} + \beta_{8} ) q^{37} + ( 3 + 2 \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{9} ) q^{38} -\beta_{6} q^{39} + ( 5 - 6 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{40} + ( -3 + 3 \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{9} ) q^{41} + ( 3 - \beta_{1} - \beta_{3} + 2 \beta_{4} ) q^{43} + ( 1 - 2 \beta_{1} - 2 \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{9} ) q^{44} + ( -1 - \beta_{8} ) q^{45} + \beta_{1} q^{46} + ( \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{8} ) q^{47} + ( -4 - \beta_{1} - \beta_{2} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{9} ) q^{48} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{8} + \beta_{9} ) q^{50} + ( 1 - \beta_{3} + \beta_{5} + \beta_{8} ) q^{51} + ( 4 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{52} + ( -\beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{9} ) q^{53} -\beta_{1} q^{54} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{55} + ( -\beta_{1} - \beta_{3} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{57} + ( 1 + 4 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{58} + ( -2 + \beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{7} - \beta_{8} ) q^{59} + ( 2 + \beta_{3} - 3 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{9} ) q^{60} + ( 3 - 2 \beta_{1} - \beta_{5} - \beta_{6} ) q^{61} + ( -5 + \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + \beta_{9} ) q^{62} + ( 8 + 4 \beta_{1} + 2 \beta_{2} + \beta_{3} - 4 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} ) q^{64} + ( -1 - 2 \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{65} + ( \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{66} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{67} + ( -3 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{68} - q^{69} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + \beta_{6} - 3 \beta_{8} ) q^{71} + ( 2 \beta_{1} + \beta_{3} + \beta_{7} - \beta_{8} ) q^{72} + ( -1 + \beta_{1} + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{73} + ( 4 \beta_{1} - 2 \beta_{2} + \beta_{3} - 5 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} ) q^{74} + ( -2 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{75} + ( -3 + 6 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{9} ) q^{76} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{78} + ( 4 + \beta_{1} + 3 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{79} + ( -8 - 2 \beta_{2} - \beta_{3} + 5 \beta_{4} - 3 \beta_{6} - \beta_{7} - 3 \beta_{8} + 2 \beta_{9} ) q^{80} + q^{81} + ( 7 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} - 2 \beta_{9} ) q^{82} + ( -3 + 4 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{83} + ( 3 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{6} + 3 \beta_{8} - \beta_{9} ) q^{85} + ( -2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} ) q^{86} + ( -2 - \beta_{2} + \beta_{4} + \beta_{8} ) q^{87} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} + \beta_{8} + 3 \beta_{9} ) q^{88} + ( 2 \beta_{1} + \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{89} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{90} + ( 2 + \beta_{2} ) q^{92} + ( 2 \beta_{1} - \beta_{2} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{93} + ( 3 + \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{94} + ( 4 - 3 \beta_{1} - 3 \beta_{3} + \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{95} + ( -1 - 5 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{9} ) q^{96} + ( 2 \beta_{1} - 4 \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{97} + ( 1 - \beta_{5} - \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q + 3q^{2} - 10q^{3} + 15q^{4} - 5q^{5} - 3q^{6} + 9q^{8} + 10q^{9} + O(q^{10})$$ $$10q + 3q^{2} - 10q^{3} + 15q^{4} - 5q^{5} - 3q^{6} + 9q^{8} + 10q^{9} + 11q^{10} + 8q^{11} - 15q^{12} + 5q^{15} + 37q^{16} - 11q^{17} + 3q^{18} + q^{19} - 15q^{20} + 6q^{22} + 10q^{23} - 9q^{24} + 21q^{25} + q^{26} - 10q^{27} + 22q^{29} - 11q^{30} - 3q^{31} + 11q^{32} - 8q^{33} - 3q^{34} + 15q^{36} - 3q^{37} + 16q^{38} + 39q^{40} - 26q^{41} + 27q^{43} + 16q^{44} - 5q^{45} + 3q^{46} + 11q^{47} - 37q^{48} + 2q^{50} + 11q^{51} + 29q^{52} + 5q^{53} - 3q^{54} - 18q^{55} - q^{57} + 16q^{58} - 10q^{59} + 15q^{60} + 22q^{61} - 32q^{62} + 69q^{64} - 11q^{65} - 6q^{66} - 2q^{67} - 21q^{68} - 10q^{69} + 27q^{71} + 9q^{72} - 8q^{73} + 14q^{74} - 21q^{75} - 22q^{76} - q^{78} + 21q^{79} - 53q^{80} + 10q^{81} + 36q^{82} - 12q^{83} + 23q^{85} + 18q^{86} - 22q^{87} - 10q^{88} + 6q^{89} + 11q^{90} + 15q^{92} + 3q^{93} + 35q^{94} + 44q^{95} - 11q^{96} + 6q^{97} + 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 3 x^{9} - 13 x^{8} + 41 x^{7} + 47 x^{6} - 165 x^{5} - 45 x^{4} + 207 x^{3} + 12 x^{2} - 76 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$($$$$3 \nu^{9} - \nu^{8} - 24 \nu^{7} + 6 \nu^{6} - 55 \nu^{5} - 41 \nu^{4} + 586 \nu^{3} + 364 \nu^{2} - 654 \nu - 276$$$$)/106$$ $$\beta_{4}$$ $$=$$ $$($$$$4 \nu^{9} - 19 \nu^{8} - 32 \nu^{7} + 220 \nu^{6} - 38 \nu^{5} - 567 \nu^{4} + 428 \nu^{3} - 186 \nu^{2} - 130 \nu + 374$$$$)/106$$ $$\beta_{5}$$ $$=$$ $$($$$$-13 \nu^{9} + 22 \nu^{8} + 157 \nu^{7} - 238 \nu^{6} - 433 \nu^{5} + 478 \nu^{4} - 119 \nu^{3} + 578 \nu^{2} + 290 \nu - 394$$$$)/106$$ $$\beta_{6}$$ $$=$$ $$($$$$6 \nu^{9} - 2 \nu^{8} - 101 \nu^{7} + 12 \nu^{6} + 579 \nu^{5} + 77 \nu^{4} - 1319 \nu^{3} - 438 \nu^{2} + 918 \nu + 243$$$$)/53$$ $$\beta_{7}$$ $$=$$ $$($$$$21 \nu^{9} - 60 \nu^{8} - 274 \nu^{7} + 784 \nu^{6} + 993 \nu^{5} - 2778 \nu^{4} - 880 \nu^{3} + 2124 \nu^{2} - 126 \nu - 130$$$$)/106$$ $$\beta_{8}$$ $$=$$ $$($$$$24 \nu^{9} - 61 \nu^{8} - 298 \nu^{7} + 790 \nu^{6} + 938 \nu^{5} - 2819 \nu^{4} - 400 \nu^{3} + 2488 \nu^{2} - 144 \nu - 406$$$$)/106$$ $$\beta_{9}$$ $$=$$ $$($$$$29 \nu^{9} - 45 \nu^{8} - 444 \nu^{7} + 588 \nu^{6} + 2189 \nu^{5} - 2163 \nu^{4} - 3946 \nu^{3} + 2176 \nu^{2} + 1946 \nu - 442$$$$)/106$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$-\beta_{8} + \beta_{7} + \beta_{3} + 6 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-\beta_{9} + \beta_{7} + \beta_{6} - \beta_{4} + 7 \beta_{2} + \beta_{1} + 24$$ $$\nu^{5}$$ $$=$$ $$-\beta_{9} - 8 \beta_{8} + 9 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \beta_{4} + 10 \beta_{3} + \beta_{2} + 41 \beta_{1} + 1$$ $$\nu^{6}$$ $$=$$ $$-13 \beta_{9} + 2 \beta_{8} + 12 \beta_{7} + 13 \beta_{6} + 2 \beta_{5} - 14 \beta_{4} + \beta_{3} + 48 \beta_{2} + 14 \beta_{1} + 160$$ $$\nu^{7}$$ $$=$$ $$-16 \beta_{9} - 57 \beta_{8} + 73 \beta_{7} + 28 \beta_{6} + 26 \beta_{5} + 10 \beta_{4} + 87 \beta_{3} + 12 \beta_{2} + 296 \beta_{1} + 12$$ $$\nu^{8}$$ $$=$$ $$-129 \beta_{9} + 28 \beta_{8} + 113 \beta_{7} + 131 \beta_{6} + 28 \beta_{5} - 143 \beta_{4} + 20 \beta_{3} + 337 \beta_{2} + 143 \beta_{1} + 1116$$ $$\nu^{9}$$ $$=$$ $$-177 \beta_{9} - 402 \beta_{8} + 581 \beta_{7} + 292 \beta_{6} + 250 \beta_{5} + 65 \beta_{4} + 724 \beta_{3} + 105 \beta_{2} + 2199 \beta_{1} + 101$$

## 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
 −2.73999 −2.06506 −1.05351 −0.769091 −0.0262565 0.864859 1.19920 2.27502 2.51520 2.79962
−2.73999 −1.00000 5.50755 −3.38181 2.73999 0 −9.61066 1.00000 9.26614
1.2 −2.06506 −1.00000 2.26446 −0.608853 2.06506 0 −0.546135 1.00000 1.25732
1.3 −1.05351 −1.00000 −0.890116 2.17612 1.05351 0 3.04477 1.00000 −2.29256
1.4 −0.769091 −1.00000 −1.40850 −4.02709 0.769091 0 2.62145 1.00000 3.09720
1.5 −0.0262565 −1.00000 −1.99931 2.77828 0.0262565 0 0.105008 1.00000 −0.0729480
1.6 0.864859 −1.00000 −1.25202 −2.49557 −0.864859 0 −2.81254 1.00000 −2.15832
1.7 1.19920 −1.00000 −0.561916 −0.572653 −1.19920 0 −3.07225 1.00000 −0.686727
1.8 2.27502 −1.00000 3.17573 −1.23869 −2.27502 0 2.67481 1.00000 −2.81806
1.9 2.51520 −1.00000 4.32625 4.31735 −2.51520 0 5.85100 1.00000 10.8590
1.10 2.79962 −1.00000 5.83786 −1.94708 −2.79962 0 10.7446 1.00000 −5.45107
 $$n$$: 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.bi 10
7.b odd 2 1 3381.2.a.bj 10
7.c even 3 2 483.2.i.h 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.i.h 20 7.c even 3 2
3381.2.a.bi 10 1.a even 1 1 trivial
3381.2.a.bj 10 7.b odd 2 1

## 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

$p$ $F_p(T)$
$2$ $$-2 - 76 T + 12 T^{2} + 207 T^{3} - 45 T^{4} - 165 T^{5} + 47 T^{6} + 41 T^{7} - 13 T^{8} - 3 T^{9} + T^{10}$$
$3$ $$( 1 + T )^{10}$$
$5$ $$-746 - 3434 T - 5156 T^{2} - 2057 T^{3} + 1431 T^{4} + 1191 T^{5} + 21 T^{6} - 151 T^{7} - 23 T^{8} + 5 T^{9} + T^{10}$$
$7$ $$T^{10}$$
$11$ $$5632 - 8192 T - 6144 T^{2} + 9040 T^{3} + 768 T^{4} - 2520 T^{5} + 120 T^{6} + 256 T^{7} - 24 T^{8} - 8 T^{9} + T^{10}$$
$13$ $$30667 - 60372 T + 23181 T^{2} + 15500 T^{3} - 9422 T^{4} - 1256 T^{5} + 1150 T^{6} + 32 T^{7} - 57 T^{8} + T^{10}$$
$17$ $$-230282 - 623142 T - 471414 T^{2} - 73699 T^{3} + 48833 T^{4} + 16709 T^{5} - 791 T^{6} - 787 T^{7} - 41 T^{8} + 11 T^{9} + T^{10}$$
$19$ $$-114788 + 388680 T - 459072 T^{2} + 217704 T^{3} - 20574 T^{4} - 14466 T^{5} + 3350 T^{6} + 263 T^{7} - 103 T^{8} - T^{9} + T^{10}$$
$23$ $$( -1 + T )^{10}$$
$29$ $$198784 + 29696 T - 366848 T^{2} - 47904 T^{3} + 91552 T^{4} - 2824 T^{5} - 6224 T^{6} + 714 T^{7} + 106 T^{8} - 22 T^{9} + T^{10}$$
$31$ $$-1835888 + 1430160 T + 977104 T^{2} - 268336 T^{3} - 143796 T^{4} + 14784 T^{5} + 7654 T^{6} - 339 T^{7} - 159 T^{8} + 3 T^{9} + T^{10}$$
$37$ $$18584768 + 18444544 T + 2113120 T^{2} - 1903760 T^{3} - 384104 T^{4} + 59456 T^{5} + 14744 T^{6} - 721 T^{7} - 209 T^{8} + 3 T^{9} + T^{10}$$
$41$ $$-30634016 - 18552576 T + 2829792 T^{2} + 3978080 T^{3} + 736096 T^{4} - 63296 T^{5} - 33088 T^{6} - 2874 T^{7} + 102 T^{8} + 26 T^{9} + T^{10}$$
$43$ $$59884 + 185336 T - 362252 T^{2} - 163764 T^{3} + 130702 T^{4} + 17650 T^{5} - 14896 T^{6} + 1397 T^{7} + 151 T^{8} - 27 T^{9} + T^{10}$$
$47$ $$-2272 - 19104 T - 54804 T^{2} - 58141 T^{3} - 6661 T^{4} + 17839 T^{5} + 7867 T^{6} + 645 T^{7} - 131 T^{8} - 11 T^{9} + T^{10}$$
$53$ $$2709352 + 42216132 T - 37631902 T^{2} + 10197799 T^{3} - 275005 T^{4} - 270159 T^{5} + 26829 T^{6} + 2139 T^{7} - 303 T^{8} - 5 T^{9} + T^{10}$$
$59$ $$345888 + 113376 T - 693856 T^{2} - 180096 T^{3} + 241192 T^{4} + 88528 T^{5} + 1760 T^{6} - 2126 T^{7} - 166 T^{8} + 10 T^{9} + T^{10}$$
$61$ $$-394624 + 399296 T + 121696 T^{2} - 257488 T^{3} + 76496 T^{4} + 10056 T^{5} - 7664 T^{6} + 776 T^{7} + 104 T^{8} - 22 T^{9} + T^{10}$$
$67$ $$-1855397 - 307438 T + 1068155 T^{2} + 20974 T^{3} - 169322 T^{4} + 5842 T^{5} + 9358 T^{6} - 344 T^{7} - 187 T^{8} + 2 T^{9} + T^{10}$$
$71$ $$5367896 + 4154628 T - 44520154 T^{2} + 10898271 T^{3} + 1657557 T^{4} - 467797 T^{5} - 14987 T^{6} + 6221 T^{7} - 73 T^{8} - 27 T^{9} + T^{10}$$
$73$ $$7759903 + 716996 T - 11850479 T^{2} - 6590004 T^{3} - 415662 T^{4} + 306740 T^{5} + 39550 T^{6} - 2956 T^{7} - 385 T^{8} + 8 T^{9} + T^{10}$$
$79$ $$-46702692 - 214839072 T - 15099168 T^{2} + 22960992 T^{3} + 105850 T^{4} - 662222 T^{5} + 15278 T^{6} + 6805 T^{7} - 265 T^{8} - 21 T^{9} + T^{10}$$
$83$ $$-947876608 + 972671872 T - 3040064 T^{2} - 63451744 T^{3} - 2806928 T^{4} + 1054760 T^{5} + 67052 T^{6} - 6282 T^{7} - 468 T^{8} + 12 T^{9} + T^{10}$$
$89$ $$-419224 - 395208 T + 2475888 T^{2} - 487316 T^{3} - 527512 T^{4} + 30828 T^{5} + 27748 T^{6} + 652 T^{7} - 300 T^{8} - 6 T^{9} + T^{10}$$
$97$ $$-21133952 + 161556160 T + 64563616 T^{2} - 23002288 T^{3} - 10632736 T^{4} - 273816 T^{5} + 150064 T^{6} + 2948 T^{7} - 684 T^{8} - 6 T^{9} + T^{10}$$