# Properties

 Label 403.2.a.e Level 403 Weight 2 Character orbit 403.a Self dual yes Analytic conductor 3.218 Analytic rank 0 Dimension 8 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$403 = 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 403.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.21797120146$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{4} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + ( 1 + \beta_{3} + \beta_{7} ) q^{5} + ( -\beta_{1} - \beta_{4} - \beta_{6} ) q^{6} + ( -\beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{7} + ( -\beta_{2} - \beta_{3} ) q^{8} + ( 1 + \beta_{3} + \beta_{4} + \beta_{5} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 1 + \beta_{4} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + ( 1 + \beta_{3} + \beta_{7} ) q^{5} + ( -\beta_{1} - \beta_{4} - \beta_{6} ) q^{6} + ( -\beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{7} + ( -\beta_{2} - \beta_{3} ) q^{8} + ( 1 + \beta_{3} + \beta_{4} + \beta_{5} ) q^{9} + ( -\beta_{1} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{10} + ( -\beta_{3} + \beta_{4} + \beta_{6} ) q^{11} + ( 2 + 2 \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{7} ) q^{12} + q^{13} + ( 1 + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{14} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{5} - \beta_{6} ) q^{15} + ( -1 + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{16} + ( 1 + \beta_{3} - \beta_{7} ) q^{17} + ( -2 - \beta_{2} - \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{18} + ( -1 - \beta_{1} - \beta_{2} + \beta_{5} + \beta_{7} ) q^{19} + ( 1 + \beta_{1} + \beta_{2} + 2 \beta_{4} ) q^{20} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{21} + ( -\beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{22} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} ) q^{23} + ( -2 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{24} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{7} ) q^{25} -\beta_{1} q^{26} + ( 1 + \beta_{1} + \beta_{2} + \beta_{5} - \beta_{7} ) q^{27} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{28} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{29} + ( -1 - 3 \beta_{1} - 2 \beta_{2} - \beta_{6} ) q^{30} - q^{31} + ( -3 + 2 \beta_{1} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{32} + ( 1 + \beta_{1} + \beta_{5} + \beta_{6} ) q^{33} + ( -2 + \beta_{1} - 2 \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{34} + ( 3 \beta_{1} - 2 \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{35} + ( 1 + 2 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{36} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{37} + ( 3 + \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{38} + ( 1 + \beta_{4} ) q^{39} + ( -3 - 2 \beta_{2} - \beta_{3} + 2 \beta_{7} ) q^{40} + ( 2 + \beta_{2} - \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{41} + ( -3 + \beta_{1} - 2 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{42} + ( -1 - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{43} + ( 3 - \beta_{1} + \beta_{2} + 2 \beta_{6} ) q^{44} + ( 2 - \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{45} + ( 2 - \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{46} + ( 2 + \beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{47} + ( 1 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - 2 \beta_{7} ) q^{48} + ( 3 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{49} + ( -3 + \beta_{3} - \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{50} + ( 3 + \beta_{1} + 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{51} + ( 1 + \beta_{2} ) q^{52} + ( 3 + 2 \beta_{1} + 3 \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{53} + ( -5 - \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{7} ) q^{54} + ( -2 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{55} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{56} + ( -4 - \beta_{1} - \beta_{2} - 3 \beta_{3} - 2 \beta_{6} ) q^{57} + ( -4 - \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{58} + ( -2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{59} + ( 8 + \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{7} ) q^{60} + ( -1 + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{61} + \beta_{1} q^{62} + ( -4 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{5} - 3 \beta_{7} ) q^{63} + ( -2 + 3 \beta_{1} - \beta_{2} - 2 \beta_{5} - \beta_{6} ) q^{64} + ( 1 + \beta_{3} + \beta_{7} ) q^{65} + ( -5 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{6} ) q^{66} + ( 3 - 4 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{67} + ( -1 + 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{68} + ( -2 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} + 5 \beta_{4} + \beta_{5} - \beta_{7} ) q^{69} + ( -3 - 2 \beta_{1} + 2 \beta_{3} + \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{70} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{6} ) q^{71} + ( -4 - 3 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} - \beta_{6} - \beta_{7} ) q^{72} + ( -4 + 4 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{73} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{74} + ( -3 + 3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} - 4 \beta_{5} - \beta_{6} - \beta_{7} ) q^{75} + ( -4 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{76} + ( 4 - \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{77} + ( -\beta_{1} - \beta_{4} - \beta_{6} ) q^{78} + ( -3 - \beta_{1} - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{79} + ( 1 + 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{80} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{81} + ( 3 - 3 \beta_{1} + \beta_{3} + 3 \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{82} + ( -2 + 3 \beta_{1} - 4 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{83} + ( -7 + 2 \beta_{1} - \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - 4 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{84} + ( 2 - 3 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} - 3 \beta_{4} - 2 \beta_{7} ) q^{85} + ( 1 + 3 \beta_{1} + \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} ) q^{86} + ( 3 - 2 \beta_{2} + 3 \beta_{4} + \beta_{5} + \beta_{7} ) q^{87} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{5} ) q^{88} + ( 2 \beta_{1} + 4 \beta_{2} - 4 \beta_{4} + \beta_{6} ) q^{89} + ( -2 - 2 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{90} + ( -\beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{91} + ( -2 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{92} + ( -1 - \beta_{4} ) q^{93} + ( -4 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{94} + ( -1 - \beta_{1} - 3 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{7} ) q^{95} + ( -7 - \beta_{2} - 3 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{96} + ( -2 - 4 \beta_{1} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{97} + ( 5 - 5 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{98} + ( -1 + 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - q^{2} + 7q^{3} + 7q^{4} + 11q^{5} - 2q^{7} + 9q^{9} + O(q^{10})$$ $$8q - q^{2} + 7q^{3} + 7q^{4} + 11q^{5} - 2q^{7} + 9q^{9} - 2q^{10} - 2q^{11} + 19q^{12} + 8q^{13} + 3q^{14} + 6q^{15} - 7q^{16} + 7q^{17} - 12q^{18} - 5q^{19} + 6q^{20} + 8q^{21} - 4q^{22} + 14q^{23} - 17q^{24} + 17q^{25} - q^{26} + 7q^{27} - 9q^{28} + 12q^{29} - 9q^{30} - 8q^{31} - 21q^{32} + 10q^{33} - 12q^{34} - q^{35} + 11q^{36} + 2q^{37} + 24q^{38} + 7q^{39} - 19q^{40} + 13q^{41} - 27q^{42} - 5q^{43} + 22q^{44} + 19q^{45} + 17q^{46} + 23q^{47} + 3q^{48} + 26q^{49} - 26q^{50} + 18q^{51} + 7q^{52} + 25q^{53} - 36q^{54} - 17q^{55} + 8q^{56} - 35q^{57} - 29q^{58} - 5q^{59} + 71q^{60} - 9q^{61} + q^{62} - 37q^{63} - 14q^{64} + 11q^{65} - 41q^{66} + 22q^{67} - 6q^{68} - 7q^{69} - 29q^{70} - 17q^{71} - 34q^{72} - 27q^{73} + 14q^{74} - 33q^{75} - 36q^{76} + 31q^{77} - 23q^{79} + 9q^{80} - 12q^{81} + 18q^{82} - 25q^{83} - 62q^{84} + 13q^{85} + 11q^{86} + 26q^{87} + 5q^{88} + 2q^{89} - 14q^{90} - 2q^{91} - 20q^{92} - 7q^{93} - 38q^{94} + 3q^{95} - 52q^{96} - 15q^{97} + 39q^{98} - 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 11 x^{6} + 10 x^{5} + 37 x^{4} - 33 x^{3} - 36 x^{2} + 33 x - 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4 \nu + 3$$ $$\beta_{4}$$ $$=$$ $$\nu^{5} - \nu^{4} - 7 \nu^{3} + 5 \nu^{2} + 10 \nu - 5$$ $$\beta_{5}$$ $$=$$ $$-\nu^{6} + \nu^{5} + 8 \nu^{4} - 6 \nu^{3} - 15 \nu^{2} + 9 \nu + 2$$ $$\beta_{6}$$ $$=$$ $$\nu^{6} - 2 \nu^{5} - 6 \nu^{4} + 12 \nu^{3} + 5 \nu^{2} - 15 \nu + 5$$ $$\beta_{7}$$ $$=$$ $$\nu^{7} - 10 \nu^{5} - \nu^{4} + 28 \nu^{3} + \nu^{2} - 21 \nu + 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 6 \beta_{2} + 13$$ $$\nu^{5}$$ $$=$$ $$\beta_{6} + \beta_{5} + 2 \beta_{4} + 8 \beta_{3} + 8 \beta_{2} + 18 \beta_{1} + 3$$ $$\nu^{6}$$ $$=$$ $$9 \beta_{6} + 8 \beta_{5} + 10 \beta_{4} + 10 \beta_{3} + 35 \beta_{2} + 3 \beta_{1} + 64$$ $$\nu^{7}$$ $$=$$ $$\beta_{7} + 11 \beta_{6} + 11 \beta_{5} + 21 \beta_{4} + 53 \beta_{3} + 57 \beta_{2} + 89 \beta_{1} + 36$$

## 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
 2.53815 1.71590 1.53950 0.656875 0.147670 −1.38904 −2.06827 −2.14079
−2.53815 2.96968 4.44218 2.45818 −7.53747 −1.29216 −6.19862 5.81898 −6.23922
1.2 −1.71590 −1.27835 0.944315 −2.00831 2.19352 −3.76625 1.81145 −1.36583 3.44606
1.3 −1.53950 0.734778 0.370072 3.72666 −1.13119 3.41650 2.50928 −2.46010 −5.73720
1.4 −0.656875 2.67827 −1.56852 −1.55873 −1.75929 2.17607 2.34407 4.17313 1.02389
1.5 −0.147670 −2.43721 −1.97819 4.40045 0.359904 −2.90659 0.587461 2.94000 −0.649816
1.6 1.38904 1.62343 −0.0705747 3.01447 2.25501 2.19250 −2.87611 −0.364464 4.18722
1.7 2.06827 2.49221 2.27774 1.40222 5.15456 −5.08294 0.574435 3.21112 2.90017
1.8 2.14079 0.217190 2.58298 −0.434938 0.464957 3.26287 1.24803 −2.95283 −0.931109
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 403.2.a.e 8
3.b odd 2 1 3627.2.a.p 8
4.b odd 2 1 6448.2.a.bd 8
13.b even 2 1 5239.2.a.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.a.e 8 1.a even 1 1 trivial
3627.2.a.p 8 3.b odd 2 1
5239.2.a.i 8 13.b even 2 1
6448.2.a.bd 8 4.b odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$13$$ $$-1$$
$$31$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(403))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + 5 T^{2} + 4 T^{3} + 17 T^{4} + 17 T^{5} + 48 T^{6} + 45 T^{7} + 100 T^{8} + 90 T^{9} + 192 T^{10} + 136 T^{11} + 272 T^{12} + 128 T^{13} + 320 T^{14} + 128 T^{15} + 256 T^{16}$$
$3$ $$1 - 7 T + 32 T^{2} - 105 T^{3} + 289 T^{4} - 678 T^{5} + 1449 T^{6} - 2804 T^{7} + 5074 T^{8} - 8412 T^{9} + 13041 T^{10} - 18306 T^{11} + 23409 T^{12} - 25515 T^{13} + 23328 T^{14} - 15309 T^{15} + 6561 T^{16}$$
$5$ $$1 - 11 T + 72 T^{2} - 350 T^{3} + 1397 T^{4} - 4774 T^{5} + 14277 T^{6} - 37831 T^{7} + 89438 T^{8} - 189155 T^{9} + 356925 T^{10} - 596750 T^{11} + 873125 T^{12} - 1093750 T^{13} + 1125000 T^{14} - 859375 T^{15} + 390625 T^{16}$$
$7$ $$1 + 2 T + 17 T^{2} + 56 T^{3} + 261 T^{4} + 773 T^{5} + 2582 T^{6} + 7565 T^{7} + 21254 T^{8} + 52955 T^{9} + 126518 T^{10} + 265139 T^{11} + 626661 T^{12} + 941192 T^{13} + 2000033 T^{14} + 1647086 T^{15} + 5764801 T^{16}$$
$11$ $$1 + 2 T + 57 T^{2} + 103 T^{3} + 1580 T^{4} + 2544 T^{5} + 28112 T^{6} + 39952 T^{7} + 359040 T^{8} + 439472 T^{9} + 3401552 T^{10} + 3386064 T^{11} + 23132780 T^{12} + 16588253 T^{13} + 100978977 T^{14} + 38974342 T^{15} + 214358881 T^{16}$$
$13$ $$( 1 - T )^{8}$$
$17$ $$1 - 7 T + 88 T^{2} - 484 T^{3} + 3785 T^{4} - 17682 T^{5} + 105707 T^{6} - 423093 T^{7} + 2094890 T^{8} - 7192581 T^{9} + 30549323 T^{10} - 86871666 T^{11} + 316126985 T^{12} - 687210788 T^{13} + 2124106072 T^{14} - 2872370711 T^{15} + 6975757441 T^{16}$$
$19$ $$1 + 5 T + 79 T^{2} + 433 T^{3} + 3319 T^{4} + 17629 T^{5} + 100366 T^{6} + 462785 T^{7} + 2242286 T^{8} + 8792915 T^{9} + 36232126 T^{10} + 120917311 T^{11} + 432535399 T^{12} + 1072150867 T^{13} + 3716624599 T^{14} + 4469358695 T^{15} + 16983563041 T^{16}$$
$23$ $$1 - 14 T + 186 T^{2} - 1576 T^{3} + 13203 T^{4} - 86804 T^{5} + 558681 T^{6} - 2976046 T^{7} + 15525890 T^{8} - 68449058 T^{9} + 295542249 T^{10} - 1056144268 T^{11} + 3694740723 T^{12} - 10143676568 T^{13} + 27534675354 T^{14} - 47667556258 T^{15} + 78310985281 T^{16}$$
$29$ $$1 - 12 T + 218 T^{2} - 1903 T^{3} + 19562 T^{4} - 134879 T^{5} + 1012631 T^{6} - 5780334 T^{7} + 35047928 T^{8} - 167629686 T^{9} + 851622671 T^{10} - 3289563931 T^{11} + 13835830922 T^{12} - 39032716547 T^{13} + 129671483978 T^{14} - 206998515708 T^{15} + 500246412961 T^{16}$$
$31$ $$( 1 + T )^{8}$$
$37$ $$1 - 2 T + 230 T^{2} - 357 T^{3} + 24787 T^{4} - 30372 T^{5} + 1640477 T^{6} - 1626076 T^{7} + 73077860 T^{8} - 60164812 T^{9} + 2245813013 T^{10} - 1538432916 T^{11} + 46454828707 T^{12} - 24755792649 T^{13} + 590117074070 T^{14} - 189863754266 T^{15} + 3512479453921 T^{16}$$
$41$ $$1 - 13 T + 257 T^{2} - 2755 T^{3} + 32456 T^{4} - 279296 T^{5} + 2480988 T^{6} - 17390736 T^{7} + 124312468 T^{8} - 713020176 T^{9} + 4170540828 T^{10} - 19249359616 T^{11} + 91712899016 T^{12} - 319183833755 T^{13} + 1220776789937 T^{14} - 2531805560453 T^{15} + 7984925229121 T^{16}$$
$43$ $$1 + 5 T + 217 T^{2} + 765 T^{3} + 21840 T^{4} + 54132 T^{5} + 1417198 T^{6} + 2632544 T^{7} + 68767656 T^{8} + 113199392 T^{9} + 2620399102 T^{10} + 4303872924 T^{11} + 74666613840 T^{12} + 112461458895 T^{13} + 1371735781633 T^{14} + 1359093055535 T^{15} + 11688200277601 T^{16}$$
$47$ $$1 - 23 T + 461 T^{2} - 6449 T^{3} + 79244 T^{4} - 808340 T^{5} + 7419988 T^{6} - 59462726 T^{7} + 433337284 T^{8} - 2794748122 T^{9} + 16390753492 T^{10} - 83924283820 T^{11} + 386685441164 T^{12} - 1479045950143 T^{13} + 4969218266669 T^{14} - 11652331770649 T^{15} + 23811286661761 T^{16}$$
$53$ $$1 - 25 T + 494 T^{2} - 7208 T^{3} + 90047 T^{4} - 958140 T^{5} + 9122429 T^{6} - 77029789 T^{7} + 592048214 T^{8} - 4082578817 T^{9} + 25624903061 T^{10} - 142645008780 T^{11} + 710514142607 T^{12} - 3014353113544 T^{13} + 10949194397726 T^{14} - 29367778495925 T^{15} + 62259690411361 T^{16}$$
$59$ $$1 + 5 T + 220 T^{2} + 1532 T^{3} + 28399 T^{4} + 212720 T^{5} + 2537683 T^{6} + 18481679 T^{7} + 171856562 T^{8} + 1090419061 T^{9} + 8833674523 T^{10} + 43688220880 T^{11} + 344120935039 T^{12} + 1095264026068 T^{13} + 9279717401020 T^{14} + 12443257424095 T^{15} + 146830437604321 T^{16}$$
$61$ $$1 + 9 T + 248 T^{2} + 2059 T^{3} + 32311 T^{4} + 241390 T^{5} + 2989495 T^{6} + 19910716 T^{7} + 209511118 T^{8} + 1214553676 T^{9} + 11123910895 T^{10} + 54790943590 T^{11} + 447372968551 T^{12} + 1739023783759 T^{13} + 12777052841528 T^{14} + 28284685524189 T^{15} + 191707312997281 T^{16}$$
$67$ $$1 - 22 T + 438 T^{2} - 5894 T^{3} + 78940 T^{4} - 831393 T^{5} + 8698949 T^{6} - 76822007 T^{7} + 682981360 T^{8} - 5147074469 T^{9} + 39049582061 T^{10} - 250052252859 T^{11} + 1590729491740 T^{12} - 7957637380658 T^{13} + 39620771390022 T^{14} - 133335655317106 T^{15} + 406067677556641 T^{16}$$
$71$ $$1 + 17 T + 445 T^{2} + 6341 T^{3} + 96767 T^{4} + 1106115 T^{5} + 12767580 T^{6} + 118530765 T^{7} + 1105292078 T^{8} + 8415684315 T^{9} + 64361370780 T^{10} + 395890725765 T^{11} + 2459012135327 T^{12} + 11440618314691 T^{13} + 57004626344845 T^{14} + 154617042692647 T^{15} + 645753531245761 T^{16}$$
$73$ $$1 + 27 T + 644 T^{2} + 9278 T^{3} + 121477 T^{4} + 1137746 T^{5} + 10431667 T^{6} + 75319832 T^{7} + 672560488 T^{8} + 5498347736 T^{9} + 55590353443 T^{10} + 442602535682 T^{11} + 3449733121957 T^{12} + 19233958239854 T^{13} + 97459241730116 T^{14} + 298279760015619 T^{15} + 806460091894081 T^{16}$$
$79$ $$1 + 23 T + 631 T^{2} + 9223 T^{3} + 151697 T^{4} + 1714229 T^{5} + 21624126 T^{6} + 201926223 T^{7} + 2071566706 T^{8} + 15952171617 T^{9} + 134956170366 T^{10} + 845181751931 T^{11} + 5908610437457 T^{12} + 28379691167977 T^{13} + 153388184433751 T^{14} + 441689906681657 T^{15} + 1517108809906561 T^{16}$$
$83$ $$1 + 25 T + 601 T^{2} + 10469 T^{3} + 161546 T^{4} + 2131884 T^{5} + 25331676 T^{6} + 268856216 T^{7} + 2572721864 T^{8} + 22315065928 T^{9} + 174509915964 T^{10} + 1218983556708 T^{11} + 7666701924266 T^{12} + 41237816491567 T^{13} + 196491164394769 T^{14} + 678401274740675 T^{15} + 2252292232139041 T^{16}$$
$89$ $$1 - 2 T + 277 T^{2} + 284 T^{3} + 45671 T^{4} + 80366 T^{5} + 5706761 T^{6} + 13758075 T^{7} + 542670086 T^{8} + 1224468675 T^{9} + 45203253881 T^{10} + 56655538654 T^{11} + 2865500888711 T^{12} + 1585872883516 T^{13} + 137663817596197 T^{14} - 88462669791058 T^{15} + 3936588805702081 T^{16}$$
$97$ $$1 + 15 T + 426 T^{2} + 4739 T^{3} + 84178 T^{4} + 734525 T^{5} + 10845149 T^{6} + 82622183 T^{7} + 1125533024 T^{8} + 8014351751 T^{9} + 102042006941 T^{10} + 670381135325 T^{11} + 7452217816018 T^{12} + 40695405477923 T^{13} + 354846074099754 T^{14} + 1211974267171695 T^{15} + 7837433594376961 T^{16}$$